1\begin{patch}{ugIntroCalcDerivPagePatch1} 2\begin{paste}{ugIntroCalcDerivPageFull1}{ugIntroCalcDerivPageEmpty1} 3\pastebutton{ugIntroCalcDerivPageFull1}{\hidepaste} 4\tab{5}\spadcommand{f := exp exp x\bound{f }} 5\indentrel{3}\begin{verbatim} 6 x 7 %e 8 (1) %e 9 Type: Expression(Integer) 10\end{verbatim} 11\indentrel{-3}\end{paste}\end{patch} 12 13\begin{patch}{ugIntroCalcDerivPageEmpty1} 14\begin{paste}{ugIntroCalcDerivPageEmpty1}{ugIntroCalcDerivPagePatch1} 15\pastebutton{ugIntroCalcDerivPageEmpty1}{\showpaste} 16\tab{5}\spadcommand{f := exp exp x\bound{f }} 17\end{paste}\end{patch} 18 19\begin{patch}{ugIntroCalcDerivPagePatch2} 20\begin{paste}{ugIntroCalcDerivPageFull2}{ugIntroCalcDerivPageEmpty2} 21\pastebutton{ugIntroCalcDerivPageFull2}{\hidepaste} 22\tab{5}\spadcommand{D(f, x)\free{f }} 23\indentrel{3}\begin{verbatim} 24 x 25 x %e 26 (2) %e %e 27 Type: Expression(Integer) 28\end{verbatim} 29\indentrel{-3}\end{paste}\end{patch} 30 31\begin{patch}{ugIntroCalcDerivPageEmpty2} 32\begin{paste}{ugIntroCalcDerivPageEmpty2}{ugIntroCalcDerivPagePatch2} 33\pastebutton{ugIntroCalcDerivPageEmpty2}{\showpaste} 34\tab{5}\spadcommand{D(f, x)\free{f }} 35\end{paste}\end{patch} 36 37\begin{patch}{ugIntroCalcDerivPagePatch3} 38\begin{paste}{ugIntroCalcDerivPageFull3}{ugIntroCalcDerivPageEmpty3} 39\pastebutton{ugIntroCalcDerivPageFull3}{\hidepaste} 40\tab{5}\spadcommand{D(f, x, 4)\free{f }} 41\indentrel{3}\begin{verbatim} 42 x 43 x 4 x 3 x 2 x %e 44 (3) ((%e ) + 6 (%e ) + 7 (%e ) + %e )%e 45 Type: Expression(Integer) 46\end{verbatim} 47\indentrel{-3}\end{paste}\end{patch} 48 49\begin{patch}{ugIntroCalcDerivPageEmpty3} 50\begin{paste}{ugIntroCalcDerivPageEmpty3}{ugIntroCalcDerivPagePatch3} 51\pastebutton{ugIntroCalcDerivPageEmpty3}{\showpaste} 52\tab{5}\spadcommand{D(f, x, 4)\free{f }} 53\end{paste}\end{patch} 54 55\begin{patch}{ugIntroCalcDerivPagePatch4} 56\begin{paste}{ugIntroCalcDerivPageFull4}{ugIntroCalcDerivPageEmpty4} 57\pastebutton{ugIntroCalcDerivPageFull4}{\hidepaste} 58\tab{5}\spadcommand{g := sin(x^2 + y)\bound{g }} 59\indentrel{3}\begin{verbatim} 60 2 61 (4) sin(y + x ) 62 Type: Expression(Integer) 63\end{verbatim} 64\indentrel{-3}\end{paste}\end{patch} 65 66\begin{patch}{ugIntroCalcDerivPageEmpty4} 67\begin{paste}{ugIntroCalcDerivPageEmpty4}{ugIntroCalcDerivPagePatch4} 68\pastebutton{ugIntroCalcDerivPageEmpty4}{\showpaste} 69\tab{5}\spadcommand{g := sin(x^2 + y)\bound{g }} 70\end{paste}\end{patch} 71 72\begin{patch}{ugIntroCalcDerivPagePatch5} 73\begin{paste}{ugIntroCalcDerivPageFull5}{ugIntroCalcDerivPageEmpty5} 74\pastebutton{ugIntroCalcDerivPageFull5}{\hidepaste} 75\tab{5}\spadcommand{D(g, y)\free{g }} 76\indentrel{3}\begin{verbatim} 77 2 78 (5) cos(y + x ) 79 Type: Expression(Integer) 80\end{verbatim} 81\indentrel{-3}\end{paste}\end{patch} 82 83\begin{patch}{ugIntroCalcDerivPageEmpty5} 84\begin{paste}{ugIntroCalcDerivPageEmpty5}{ugIntroCalcDerivPagePatch5} 85\pastebutton{ugIntroCalcDerivPageEmpty5}{\showpaste} 86\tab{5}\spadcommand{D(g, y)\free{g }} 87\end{paste}\end{patch} 88 89\begin{patch}{ugIntroCalcDerivPagePatch6} 90\begin{paste}{ugIntroCalcDerivPageFull6}{ugIntroCalcDerivPageEmpty6} 91\pastebutton{ugIntroCalcDerivPageFull6}{\hidepaste} 92\tab{5}\spadcommand{D(g, [y, y, x, x])\free{g }} 93\indentrel{3}\begin{verbatim} 94 2 2 2 95 (6) 4 x sin(y + x ) - 2 cos(y + x ) 96 Type: Expression(Integer) 97\end{verbatim} 98\indentrel{-3}\end{paste}\end{patch} 99 100\begin{patch}{ugIntroCalcDerivPageEmpty6} 101\begin{paste}{ugIntroCalcDerivPageEmpty6}{ugIntroCalcDerivPagePatch6} 102\pastebutton{ugIntroCalcDerivPageEmpty6}{\showpaste} 103\tab{5}\spadcommand{D(g, [y, y, x, x])\free{g }} 104\end{paste}\end{patch} 105 106\begin{patch}{ugIntroCalcDerivPagePatch7} 107\begin{paste}{ugIntroCalcDerivPageFull7}{ugIntroCalcDerivPageEmpty7} 108\pastebutton{ugIntroCalcDerivPageFull7}{\hidepaste} 109\tab{5}\spadcommand{D(F,x)} 110\indentrel{3}\begin{verbatim} 111 (7) 0 112 Type: Polynomial(Integer) 113\end{verbatim} 114\indentrel{-3}\end{paste}\end{patch} 115 116\begin{patch}{ugIntroCalcDerivPageEmpty7} 117\begin{paste}{ugIntroCalcDerivPageEmpty7}{ugIntroCalcDerivPagePatch7} 118\pastebutton{ugIntroCalcDerivPageEmpty7}{\showpaste} 119\tab{5}\spadcommand{D(F,x)} 120\end{paste}\end{patch} 121 122\begin{patch}{ugIntroCalcDerivPagePatch8} 123\begin{paste}{ugIntroCalcDerivPageFull8}{ugIntroCalcDerivPageEmpty8} 124\pastebutton{ugIntroCalcDerivPageFull8}{\hidepaste} 125\tab{5}\spadcommand{F := operator 'F; x := operator 'x; y := operator 'y\bound{F x y }} 126\indentrel{3}\begin{verbatim} 127 (8) y 128 Type: BasicOperator 129\end{verbatim} 130\indentrel{-3}\end{paste}\end{patch} 131 132\begin{patch}{ugIntroCalcDerivPageEmpty8} 133\begin{paste}{ugIntroCalcDerivPageEmpty8}{ugIntroCalcDerivPagePatch8} 134\pastebutton{ugIntroCalcDerivPageEmpty8}{\showpaste} 135\tab{5}\spadcommand{F := operator 'F; x := operator 'x; y := operator 'y\bound{F x y }} 136\end{paste}\end{patch} 137 138\begin{patch}{ugIntroCalcDerivPagePatch9} 139\begin{paste}{ugIntroCalcDerivPageFull9}{ugIntroCalcDerivPageEmpty9} 140\pastebutton{ugIntroCalcDerivPageFull9}{\hidepaste} 141\tab{5}\spadcommand{a := F(x z, y z, z^2) + x y(z+1)\bound{a }\free{F }\free{x }\free{y }} 142\indentrel{3}\begin{verbatim} 143 2 144 (9) x(y(z + 1)) + F(x(z),y(z),z ) 145 Type: Expression(Integer) 146\end{verbatim} 147\indentrel{-3}\end{paste}\end{patch} 148 149\begin{patch}{ugIntroCalcDerivPageEmpty9} 150\begin{paste}{ugIntroCalcDerivPageEmpty9}{ugIntroCalcDerivPagePatch9} 151\pastebutton{ugIntroCalcDerivPageEmpty9}{\showpaste} 152\tab{5}\spadcommand{a := F(x z, y z, z^2) + x y(z+1)\bound{a }\free{F }\free{x }\free{y }} 153\end{paste}\end{patch} 154 155\begin{patch}{ugIntroCalcDerivPagePatch10} 156\begin{paste}{ugIntroCalcDerivPageFull10}{ugIntroCalcDerivPageEmpty10} 157\pastebutton{ugIntroCalcDerivPageFull10}{\hidepaste} 158\tab{5}\spadcommand{dadz := D(a, z)\bound{da }\free{a }} 159\indentrel{3}\begin{verbatim} 160 (10) 161 2 , 2 162 2 z F (x(z),y(z),z ) + y (z)F (x(z),y(z),z ) 163 ,3 ,2 164 + 165 , 2 , , 166 x (z)F (x(z),y(z),z ) + x (y(z + 1))y (z + 1) 167 ,1 168 Type: Expression(Integer) 169\end{verbatim} 170\indentrel{-3}\end{paste}\end{patch} 171 172\begin{patch}{ugIntroCalcDerivPageEmpty10} 173\begin{paste}{ugIntroCalcDerivPageEmpty10}{ugIntroCalcDerivPagePatch10} 174\pastebutton{ugIntroCalcDerivPageEmpty10}{\showpaste} 175\tab{5}\spadcommand{dadz := D(a, z)\bound{da }\free{a }} 176\end{paste}\end{patch} 177 178\begin{patch}{ugIntroCalcDerivPagePatch11} 179\begin{paste}{ugIntroCalcDerivPageFull11}{ugIntroCalcDerivPageEmpty11} 180\pastebutton{ugIntroCalcDerivPageFull11}{\hidepaste} 181\tab{5}\spadcommand{eval(eval(dadz, 'x, z +-> exp z), 'y, z +-> log(z+1))\free{da }} 182\indentrel{3}\begin{verbatim} 183 (11) 184 2 z 2 185 (2 z + 2 z)F (%e ,log(z + 1),z ) 186 ,3 187 + 188 z 2 189 F (%e ,log(z + 1),z ) 190 ,2 191 + 192 z z 2 193 (z + 1)%e F (%e ,log(z + 1),z ) + z + 1 194 ,1 195 / 196 z + 1 197 Type: Expression(Integer) 198\end{verbatim} 199\indentrel{-3}\end{paste}\end{patch} 200 201\begin{patch}{ugIntroCalcDerivPageEmpty11} 202\begin{paste}{ugIntroCalcDerivPageEmpty11}{ugIntroCalcDerivPagePatch11} 203\pastebutton{ugIntroCalcDerivPageEmpty11}{\showpaste} 204\tab{5}\spadcommand{eval(eval(dadz, 'x, z +-> exp z), 'y, z +-> log(z+1))\free{da }} 205\end{paste}\end{patch} 206 207\begin{patch}{ugIntroCalcDerivPagePatch12} 208\begin{paste}{ugIntroCalcDerivPageFull12}{ugIntroCalcDerivPageEmpty12} 209\pastebutton{ugIntroCalcDerivPageFull12}{\hidepaste} 210\tab{5}\spadcommand{eval(eval(a, 'x, z +-> exp z), 'y, z +-> log(z+1))\free{a }\bound{eva }} 211\indentrel{3}\begin{verbatim} 212 z 2 213 (12) F(%e ,log(z + 1),z ) + z + 2 214 Type: Expression(Integer) 215\end{verbatim} 216\indentrel{-3}\end{paste}\end{patch} 217 218\begin{patch}{ugIntroCalcDerivPageEmpty12} 219\begin{paste}{ugIntroCalcDerivPageEmpty12}{ugIntroCalcDerivPagePatch12} 220\pastebutton{ugIntroCalcDerivPageEmpty12}{\showpaste} 221\tab{5}\spadcommand{eval(eval(a, 'x, z +-> exp z), 'y, z +-> log(z+1))\free{a }\bound{eva }} 222\end{paste}\end{patch} 223 224\begin{patch}{ugIntroCalcDerivPagePatch13} 225\begin{paste}{ugIntroCalcDerivPageFull13}{ugIntroCalcDerivPageEmpty13} 226\pastebutton{ugIntroCalcDerivPageFull13}{\hidepaste} 227\tab{5}\spadcommand{D(\%, z)\free{eva }} 228\indentrel{3}\begin{verbatim} 229 (13) 230 2 z 2 231 (2 z + 2 z)F (%e ,log(z + 1),z ) 232 ,3 233 + 234 z 2 235 F (%e ,log(z + 1),z ) 236 ,2 237 + 238 z z 2 239 (z + 1)%e F (%e ,log(z + 1),z ) + z + 1 240 ,1 241 / 242 z + 1 243 Type: Expression(Integer) 244\end{verbatim} 245\indentrel{-3}\end{paste}\end{patch} 246 247\begin{patch}{ugIntroCalcDerivPageEmpty13} 248\begin{paste}{ugIntroCalcDerivPageEmpty13}{ugIntroCalcDerivPagePatch13} 249\pastebutton{ugIntroCalcDerivPageEmpty13}{\showpaste} 250\tab{5}\spadcommand{D(\%, z)\free{eva }} 251\end{paste}\end{patch} 252 253\begin{patch}{ugIntroVariablesPagePatch1} 254\begin{paste}{ugIntroVariablesPageFull1}{ugIntroVariablesPageEmpty1} 255\pastebutton{ugIntroVariablesPageFull1}{\hidepaste} 256\tab{5}\spadcommand{(x^2 - x*y^3 +3*y)^2} 257\indentrel{3}\begin{verbatim} 258 2 6 4 3 3 2 2 4 259 (1) x y - 6 x y - 2 x y + 9 y + 6 x y + x 260 Type: Polynomial(Integer) 261\end{verbatim} 262\indentrel{-3}\end{paste}\end{patch} 263 264\begin{patch}{ugIntroVariablesPageEmpty1} 265\begin{paste}{ugIntroVariablesPageEmpty1}{ugIntroVariablesPagePatch1} 266\pastebutton{ugIntroVariablesPageEmpty1}{\showpaste} 267\tab{5}\spadcommand{(x^2 - x*y^3 +3*y)^2} 268\end{paste}\end{patch} 269 270\begin{patch}{ugIntroVariablesPagePatch2} 271\begin{paste}{ugIntroVariablesPageFull2}{ugIntroVariablesPageEmpty2} 272\pastebutton{ugIntroVariablesPageFull2}{\hidepaste} 273\tab{5}\spadcommand{p: UP(x,INT) := (3*x-1)^2 * (2*x + 8)} 274\indentrel{3}\begin{verbatim} 275 3 2 276 (2) 18 x + 60 x - 46 x + 8 277 Type: UnivariatePolynomial(x,Integer) 278\end{verbatim} 279\indentrel{-3}\end{paste}\end{patch} 280 281\begin{patch}{ugIntroVariablesPageEmpty2} 282\begin{paste}{ugIntroVariablesPageEmpty2}{ugIntroVariablesPagePatch2} 283\pastebutton{ugIntroVariablesPageEmpty2}{\showpaste} 284\tab{5}\spadcommand{p: UP(x,INT) := (3*x-1)^2 * (2*x + 8)} 285\end{paste}\end{patch} 286 287\begin{patch}{ugIntroVariablesPagePatch3} 288\begin{paste}{ugIntroVariablesPageFull3}{ugIntroVariablesPageEmpty3} 289\pastebutton{ugIntroVariablesPageFull3}{\hidepaste} 290\tab{5}\spadcommand{m: MPOLY([x,y],INT) := (x^2-x*y^3+3*y)^2\bound{m }} 291\indentrel{3}\begin{verbatim} 292 4 3 3 6 2 4 2 293 (3) x - 2 y x + (y + 6 y)x - 6 y x + 9 y 294 Type: MultivariatePolynomial([x,y],Integer) 295\end{verbatim} 296\indentrel{-3}\end{paste}\end{patch} 297 298\begin{patch}{ugIntroVariablesPageEmpty3} 299\begin{paste}{ugIntroVariablesPageEmpty3}{ugIntroVariablesPagePatch3} 300\pastebutton{ugIntroVariablesPageEmpty3}{\showpaste} 301\tab{5}\spadcommand{m: MPOLY([x,y],INT) := (x^2-x*y^3+3*y)^2\bound{m }} 302\end{paste}\end{patch} 303 304\begin{patch}{ugIntroVariablesPagePatch4} 305\begin{paste}{ugIntroVariablesPageFull4}{ugIntroVariablesPageEmpty4} 306\pastebutton{ugIntroVariablesPageFull4}{\hidepaste} 307\tab{5}\spadcommand{m :: MPOLY([y,x],INT)\free{m }} 308\indentrel{3}\begin{verbatim} 309 2 6 4 3 3 2 2 4 310 (4) x y - 6 x y - 2 x y + 9 y + 6 x y + x 311 Type: MultivariatePolynomial([y,x],Integer) 312\end{verbatim} 313\indentrel{-3}\end{paste}\end{patch} 314 315\begin{patch}{ugIntroVariablesPageEmpty4} 316\begin{paste}{ugIntroVariablesPageEmpty4}{ugIntroVariablesPagePatch4} 317\pastebutton{ugIntroVariablesPageEmpty4}{\showpaste} 318\tab{5}\spadcommand{m :: MPOLY([y,x],INT)\free{m }} 319\end{paste}\end{patch} 320 321\begin{patch}{ugIntroVariablesPagePatch5} 322\begin{paste}{ugIntroVariablesPageFull5}{ugIntroVariablesPageEmpty5} 323\pastebutton{ugIntroVariablesPageFull5}{\hidepaste} 324\tab{5}\spadcommand{m :: DMP([y,x],INT)\free{m }} 325\indentrel{3}\begin{verbatim} 326 6 2 4 3 3 2 2 4 327 (5) y x - 6 y x - 2 y x + 9 y + 6 y x + x 328 Type: DistributedMultivariatePolynomial([y,x],Integer) 329\end{verbatim} 330\indentrel{-3}\end{paste}\end{patch} 331 332\begin{patch}{ugIntroVariablesPageEmpty5} 333\begin{paste}{ugIntroVariablesPageEmpty5}{ugIntroVariablesPagePatch5} 334\pastebutton{ugIntroVariablesPageEmpty5}{\showpaste} 335\tab{5}\spadcommand{m :: DMP([y,x],INT)\free{m }} 336\end{paste}\end{patch} 337 338\begin{patch}{ugIntroVariablesPagePatch6} 339\begin{paste}{ugIntroVariablesPageFull6}{ugIntroVariablesPageEmpty6} 340\pastebutton{ugIntroVariablesPageFull6}{\hidepaste} 341\tab{5}\spadcommand{m :: HDMP([y,x],INT)\free{m }} 342\indentrel{3}\begin{verbatim} 343 6 2 3 3 4 4 2 2 344 (6) y x - 2 y x - 6 y x + x + 6 y x + 9 y 345Type: HomogeneousDistributedMultivariatePolynomial([y,x],Integer) 346\end{verbatim} 347\indentrel{-3}\end{paste}\end{patch} 348 349\begin{patch}{ugIntroVariablesPageEmpty6} 350\begin{paste}{ugIntroVariablesPageEmpty6}{ugIntroVariablesPagePatch6} 351\pastebutton{ugIntroVariablesPageEmpty6}{\showpaste} 352\tab{5}\spadcommand{m :: HDMP([y,x],INT)\free{m }} 353\end{paste}\end{patch} 354 355\begin{patch}{ugIntroIntegratePagePatch1} 356\begin{paste}{ugIntroIntegratePageFull1}{ugIntroIntegratePageEmpty1} 357\pastebutton{ugIntroIntegratePageFull1}{\hidepaste} 358\tab{5}\spadcommand{integrate((x^2+2*x+1)/((x+1)^6+1),x)} 359\indentrel{3}\begin{verbatim} 360 3 2 361 atan(x + 3 x + 3 x + 1) 362 (1) ------------------------- 363 3 364 Type: Union(Expression(Integer),...) 365\end{verbatim} 366\indentrel{-3}\end{paste}\end{patch} 367 368\begin{patch}{ugIntroIntegratePageEmpty1} 369\begin{paste}{ugIntroIntegratePageEmpty1}{ugIntroIntegratePagePatch1} 370\pastebutton{ugIntroIntegratePageEmpty1}{\showpaste} 371\tab{5}\spadcommand{integrate((x^2+2*x+1)/((x+1)^6+1),x)} 372\end{paste}\end{patch} 373 374\begin{patch}{ugIntroIntegratePagePatch2} 375\begin{paste}{ugIntroIntegratePageFull2}{ugIntroIntegratePageEmpty2} 376\pastebutton{ugIntroIntegratePageFull2}{\hidepaste} 377\tab{5}\spadcommand{integrate(1/(x^2 + a),x)} 378\indentrel{3}\begin{verbatim} 379 2 +---+ 380 (x - a)\|- a + 2 a x +-+ 381 log(----------------------) x\|a 382 2 atan(-----) 383 x + a a 384 (2) [---------------------------, -----------] 385 +---+ +-+ 386 2 \|- a \|a 387 Type: Union(List(Expression(Integer)),...) 388\end{verbatim} 389\indentrel{-3}\end{paste}\end{patch} 390 391\begin{patch}{ugIntroIntegratePageEmpty2} 392\begin{paste}{ugIntroIntegratePageEmpty2}{ugIntroIntegratePagePatch2} 393\pastebutton{ugIntroIntegratePageEmpty2}{\showpaste} 394\tab{5}\spadcommand{integrate(1/(x^2 + a),x)} 395\end{paste}\end{patch} 396 397\begin{patch}{ugIntroIntegratePagePatch3} 398\begin{paste}{ugIntroIntegratePageFull3}{ugIntroIntegratePageEmpty3} 399\pastebutton{ugIntroIntegratePageFull3}{\hidepaste} 400\tab{5}\spadcommand{complexIntegrate(1/(x^2 + a),x)} 401\indentrel{3}\begin{verbatim} 402 (3) 403 +---+ +---+ +---+ +---+ 404 | 1 | 1 | 1 | 1 405 |- - log(a |- - + x) - |- - log(- a |- - + x) 406 \| a \| a \| a \| a 407 ------------------------------------------------- 408 2 409 Type: Expression(Integer) 410\end{verbatim} 411\indentrel{-3}\end{paste}\end{patch} 412 413\begin{patch}{ugIntroIntegratePageEmpty3} 414\begin{paste}{ugIntroIntegratePageEmpty3}{ugIntroIntegratePagePatch3} 415\pastebutton{ugIntroIntegratePageEmpty3}{\showpaste} 416\tab{5}\spadcommand{complexIntegrate(1/(x^2 + a),x)} 417\end{paste}\end{patch} 418 419\begin{patch}{ugIntroIntegratePagePatch4} 420\begin{paste}{ugIntroIntegratePageFull4}{ugIntroIntegratePageEmpty4} 421\pastebutton{ugIntroIntegratePageFull4}{\hidepaste} 422\tab{5}\spadcommand{integrate(x^3 / (a+b*x)^(1/3),x)} 423\indentrel{3}\begin{verbatim} 424 (4) 425 3 3 2 2 2 3 3+-------+2 426 (120 b x - 135 a b x + 162 a b x - 243 a )\|b x + a 427 ------------------------------------------------------- 428 4 429 440 b 430 Type: Union(Expression(Integer),...) 431\end{verbatim} 432\indentrel{-3}\end{paste}\end{patch} 433 434\begin{patch}{ugIntroIntegratePageEmpty4} 435\begin{paste}{ugIntroIntegratePageEmpty4}{ugIntroIntegratePagePatch4} 436\pastebutton{ugIntroIntegratePageEmpty4}{\showpaste} 437\tab{5}\spadcommand{integrate(x^3 / (a+b*x)^(1/3),x)} 438\end{paste}\end{patch} 439 440\begin{patch}{ugIntroIntegratePagePatch5} 441\begin{paste}{ugIntroIntegratePageFull5}{ugIntroIntegratePageEmpty5} 442\pastebutton{ugIntroIntegratePageFull5}{\hidepaste} 443\tab{5}\spadcommand{integrate(1 / (x^3 * (a+b*x)^(1/3)),x)} 444\indentrel{3}\begin{verbatim} 445 (5) 446 - 447 2 2 +-+ 448 2 b x \|3 449 * 450 3+-+3+-------+2 3+-+2 3+-------+ 451 log(\|a \|b x + a + \|a \|b x + a + a) 452 + 453 2 2 +-+ 3+-+2 3+-------+ 454 4 b x \|3 log(\|a \|b x + a - a) 455 + 456 +-+3+-+2 3+-------+ +-+ 457 2 2 2 \|3 \|a \|b x + a + a\|3 458 12 b x atan(------------------------------) 459 3 a 460 + 461 +-+3+-+3+-------+2 462 (12 b x - 9 a)\|3 \|a \|b x + a 463 / 464 2 2 +-+3+-+ 465 18 a x \|3 \|a 466 Type: Union(Expression(Integer),...) 467\end{verbatim} 468\indentrel{-3}\end{paste}\end{patch} 469 470\begin{patch}{ugIntroIntegratePageEmpty5} 471\begin{paste}{ugIntroIntegratePageEmpty5}{ugIntroIntegratePagePatch5} 472\pastebutton{ugIntroIntegratePageEmpty5}{\showpaste} 473\tab{5}\spadcommand{integrate(1 / (x^3 * (a+b*x)^(1/3)),x)} 474\end{paste}\end{patch} 475 476\begin{patch}{ugIntroIntegratePagePatch6} 477\begin{paste}{ugIntroIntegratePageFull6}{ugIntroIntegratePageEmpty6} 478\pastebutton{ugIntroIntegratePageFull6}{\hidepaste} 479\tab{5}\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)} 480\indentrel{3}\begin{verbatim} 481 x +--------+ 482 ++ log(\|b + %Y a + 1) 483 (6) | -------------------- d%Y 484 ++ %Y 485 Type: Union(Expression(Integer),...) 486\end{verbatim} 487\indentrel{-3}\end{paste}\end{patch} 488 489\begin{patch}{ugIntroIntegratePageEmpty6} 490\begin{paste}{ugIntroIntegratePageEmpty6}{ugIntroIntegratePagePatch6} 491\pastebutton{ugIntroIntegratePageEmpty6}{\showpaste} 492\tab{5}\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)} 493\end{paste}\end{patch} 494 495\begin{patch}{ugIntroIntegratePagePatch7} 496\begin{paste}{ugIntroIntegratePageFull7}{ugIntroIntegratePageEmpty7} 497\pastebutton{ugIntroIntegratePageFull7}{\hidepaste} 498\tab{5}\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b)) / (sqrt(x+b) * (x + cosh(1+sqrt(x + b)))), x)} 499\indentrel{3}\begin{verbatim} 500 (7) 501 +-----+ 502 - 2 cosh(\|x + b + 1) - 2 x 503 2 log(---------------------------------------) 504 +-----+ +-----+ 505 sinh(\|x + b + 1) - cosh(\|x + b + 1) 506 + 507 +-----+ 508 - 2 \|x + b 509 Type: Union(Expression(Integer),...) 510\end{verbatim} 511\indentrel{-3}\end{paste}\end{patch} 512 513\begin{patch}{ugIntroIntegratePageEmpty7} 514\begin{paste}{ugIntroIntegratePageEmpty7}{ugIntroIntegratePagePatch7} 515\pastebutton{ugIntroIntegratePageEmpty7}{\showpaste} 516\tab{5}\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b)) / (sqrt(x+b) * (x + cosh(1+sqrt(x + b)))), x)} 517\end{paste}\end{patch} 518 519\begin{patch}{ugIntroIntegratePagePatch8} 520\begin{paste}{ugIntroIntegratePageFull8}{ugIntroIntegratePageEmpty8} 521\pastebutton{ugIntroIntegratePageFull8}{\hidepaste} 522\tab{5}\spadcommand{integrate(tan(atan(x)/3),x)} 523\indentrel{3}\begin{verbatim} 524 (8) 525 atan(x) 2 atan(x) 2 526 8 log(3 tan(-------) - 1) - 3 tan(-------) 527 3 3 528 + 529 atan(x) 530 18 x tan(-------) 531 3 532 / 533 18 534 Type: Union(Expression(Integer),...) 535\end{verbatim} 536\indentrel{-3}\end{paste}\end{patch} 537 538\begin{patch}{ugIntroIntegratePageEmpty8} 539\begin{paste}{ugIntroIntegratePageEmpty8}{ugIntroIntegratePagePatch8} 540\pastebutton{ugIntroIntegratePageEmpty8}{\showpaste} 541\tab{5}\spadcommand{integrate(tan(atan(x)/3),x)} 542\end{paste}\end{patch} 543 544\begin{patch}{ugIntroIntegratePagePatch9} 545\begin{paste}{ugIntroIntegratePageFull9}{ugIntroIntegratePageEmpty9} 546\pastebutton{ugIntroIntegratePageFull9}{\hidepaste} 547\tab{5}\spadcommand{integrate((x + 1) / (x*(x + log x) ^ (3/2)), x)} 548\indentrel{3}\begin{verbatim} 549 +----------+ 550 2 \|log(x) + x 551 (9) - --------------- 552 log(x) + x 553 Type: Union(Expression(Integer),...) 554\end{verbatim} 555\indentrel{-3}\end{paste}\end{patch} 556 557\begin{patch}{ugIntroIntegratePageEmpty9} 558\begin{paste}{ugIntroIntegratePageEmpty9}{ugIntroIntegratePagePatch9} 559\pastebutton{ugIntroIntegratePageEmpty9}{\showpaste} 560\tab{5}\spadcommand{integrate((x + 1) / (x*(x + log x) ^ (3/2)), x)} 561\end{paste}\end{patch} 562 563\begin{patch}{ugIntroIntegratePagePatch10} 564\begin{paste}{ugIntroIntegratePageFull10}{ugIntroIntegratePageEmpty10} 565\pastebutton{ugIntroIntegratePageFull10}{\hidepaste} 566\tab{5}\spadcommand{integrate(exp(-x^2) * erf(x) / (erf(x)^3 - erf(x)^2 - erf(x) + 1),x)} 567\indentrel{3}\begin{verbatim} 568 +---+ erf(x) - 1 +---+ 569 (erf(x) - 1)\|%pi log(----------) - 2 \|%pi 570 erf(x) + 1 571 (10) -------------------------------------------- 572 8 erf(x) - 8 573 Type: Union(Expression(Integer),...) 574\end{verbatim} 575\indentrel{-3}\end{paste}\end{patch} 576 577\begin{patch}{ugIntroIntegratePageEmpty10} 578\begin{paste}{ugIntroIntegratePageEmpty10}{ugIntroIntegratePagePatch10} 579\pastebutton{ugIntroIntegratePageEmpty10}{\showpaste} 580\tab{5}\spadcommand{integrate(exp(-x^2) * erf(x) / (erf(x)^3 - erf(x)^2 - erf(x) + 1),x)} 581\end{paste}\end{patch} 582 583\begin{patch}{ugIntroCalcLimitsPagePatch1} 584\begin{paste}{ugIntroCalcLimitsPageFull1}{ugIntroCalcLimitsPageEmpty1} 585\pastebutton{ugIntroCalcLimitsPageFull1}{\hidepaste} 586\tab{5}\spadcommand{g := csc(a*x) / csch(b*x)\bound{g }} 587\indentrel{3}\begin{verbatim} 588 csc(a x) 589 (1) --------- 590 csch(b x) 591 Type: Expression(Integer) 592\end{verbatim} 593\indentrel{-3}\end{paste}\end{patch} 594 595\begin{patch}{ugIntroCalcLimitsPageEmpty1} 596\begin{paste}{ugIntroCalcLimitsPageEmpty1}{ugIntroCalcLimitsPagePatch1} 597\pastebutton{ugIntroCalcLimitsPageEmpty1}{\showpaste} 598\tab{5}\spadcommand{g := csc(a*x) / csch(b*x)\bound{g }} 599\end{paste}\end{patch} 600 601\begin{patch}{ugIntroCalcLimitsPagePatch2} 602\begin{paste}{ugIntroCalcLimitsPageFull2}{ugIntroCalcLimitsPageEmpty2} 603\pastebutton{ugIntroCalcLimitsPageFull2}{\hidepaste} 604\tab{5}\spadcommand{limit(g,x=0)\free{g }} 605\indentrel{3}\begin{verbatim} 606 b 607 (2) - 608 a 609Type: Union(OrderedCompletion(Expression(Integer)),...) 610\end{verbatim} 611\indentrel{-3}\end{paste}\end{patch} 612 613\begin{patch}{ugIntroCalcLimitsPageEmpty2} 614\begin{paste}{ugIntroCalcLimitsPageEmpty2}{ugIntroCalcLimitsPagePatch2} 615\pastebutton{ugIntroCalcLimitsPageEmpty2}{\showpaste} 616\tab{5}\spadcommand{limit(g,x=0)\free{g }} 617\end{paste}\end{patch} 618 619\begin{patch}{ugIntroCalcLimitsPagePatch3} 620\begin{paste}{ugIntroCalcLimitsPageFull3}{ugIntroCalcLimitsPageEmpty3} 621\pastebutton{ugIntroCalcLimitsPageFull3}{\hidepaste} 622\tab{5}\spadcommand{h := (1 + k/x)^x\bound{h }} 623\indentrel{3}\begin{verbatim} 624 x + k x 625 (3) (-----) 626 x 627 Type: Expression(Integer) 628\end{verbatim} 629\indentrel{-3}\end{paste}\end{patch} 630 631\begin{patch}{ugIntroCalcLimitsPageEmpty3} 632\begin{paste}{ugIntroCalcLimitsPageEmpty3}{ugIntroCalcLimitsPagePatch3} 633\pastebutton{ugIntroCalcLimitsPageEmpty3}{\showpaste} 634\tab{5}\spadcommand{h := (1 + k/x)^x\bound{h }} 635\end{paste}\end{patch} 636 637\begin{patch}{ugIntroCalcLimitsPagePatch4} 638\begin{paste}{ugIntroCalcLimitsPageFull4}{ugIntroCalcLimitsPageEmpty4} 639\pastebutton{ugIntroCalcLimitsPageFull4}{\hidepaste} 640\tab{5}\spadcommand{limit(h,x=\%plusInfinity)\free{h }} 641\indentrel{3}\begin{verbatim} 642 k 643 (4) %e 644Type: Union(OrderedCompletion(Expression(Integer)),...) 645\end{verbatim} 646\indentrel{-3}\end{paste}\end{patch} 647 648\begin{patch}{ugIntroCalcLimitsPageEmpty4} 649\begin{paste}{ugIntroCalcLimitsPageEmpty4}{ugIntroCalcLimitsPagePatch4} 650\pastebutton{ugIntroCalcLimitsPageEmpty4}{\showpaste} 651\tab{5}\spadcommand{limit(h,x=\%plusInfinity)\free{h }} 652\end{paste}\end{patch} 653 654\begin{patch}{ugIntroCalcLimitsPagePatch5} 655\begin{paste}{ugIntroCalcLimitsPageFull5}{ugIntroCalcLimitsPageEmpty5} 656\pastebutton{ugIntroCalcLimitsPageFull5}{\hidepaste} 657\tab{5}\spadcommand{limit(sqrt(y^2)/y,y = 0)} 658\indentrel{3}\begin{verbatim} 659 (5) [leftHandLimit = - 1, rightHandLimit = 1] 660Type: Union(Record(leftHandLimit: Union(OrderedCompletion(Expression(Integer)),"failed"),rightHandLimit: Union(OrderedCompletion(Expression(Integer)),"failed")),...) 661\end{verbatim} 662\indentrel{-3}\end{paste}\end{patch} 663 664\begin{patch}{ugIntroCalcLimitsPageEmpty5} 665\begin{paste}{ugIntroCalcLimitsPageEmpty5}{ugIntroCalcLimitsPagePatch5} 666\pastebutton{ugIntroCalcLimitsPageEmpty5}{\showpaste} 667\tab{5}\spadcommand{limit(sqrt(y^2)/y,y = 0)} 668\end{paste}\end{patch} 669 670\begin{patch}{ugIntroCalcLimitsPagePatch6} 671\begin{paste}{ugIntroCalcLimitsPageFull6}{ugIntroCalcLimitsPageEmpty6} 672\pastebutton{ugIntroCalcLimitsPageFull6}{\hidepaste} 673\tab{5}\spadcommand{limit(exp(-1/x^2),x = 0)} 674\indentrel{3}\begin{verbatim} 675 (6) 0 676Type: Union(OrderedCompletion(Expression(Integer)),...) 677\end{verbatim} 678\indentrel{-3}\end{paste}\end{patch} 679 680\begin{patch}{ugIntroCalcLimitsPageEmpty6} 681\begin{paste}{ugIntroCalcLimitsPageEmpty6}{ugIntroCalcLimitsPagePatch6} 682\pastebutton{ugIntroCalcLimitsPageEmpty6}{\showpaste} 683\tab{5}\spadcommand{limit(exp(-1/x^2),x = 0)} 684\end{paste}\end{patch} 685 686\begin{patch}{ugIntroCalcLimitsPagePatch7} 687\begin{paste}{ugIntroCalcLimitsPageFull7}{ugIntroCalcLimitsPageEmpty7} 688\pastebutton{ugIntroCalcLimitsPageFull7}{\hidepaste} 689\tab{5}\spadcommand{complexLimit(exp(-1/x^2),x = 0)} 690\indentrel{3}\begin{verbatim} 691 (7) "failed" 692 Type: Union("failed",...) 693\end{verbatim} 694\indentrel{-3}\end{paste}\end{patch} 695 696\begin{patch}{ugIntroCalcLimitsPageEmpty7} 697\begin{paste}{ugIntroCalcLimitsPageEmpty7}{ugIntroCalcLimitsPagePatch7} 698\pastebutton{ugIntroCalcLimitsPageEmpty7}{\showpaste} 699\tab{5}\spadcommand{complexLimit(exp(-1/x^2),x = 0)} 700\end{paste}\end{patch} 701 702\begin{patch}{ugIntroSysCmmandsPagePatch1} 703\begin{paste}{ugIntroSysCmmandsPageFull1}{ugIntroSysCmmandsPageEmpty1} 704\pastebutton{ugIntroSysCmmandsPageFull1}{\hidepaste} 705\tab{5}\spadcommand{)what operations integrate} 706\indentrel{3}\begin{verbatim} 707\end{verbatim} 708\indentrel{-3}\end{paste}\end{patch} 709 710\begin{patch}{ugIntroSysCmmandsPageEmpty1} 711\begin{paste}{ugIntroSysCmmandsPageEmpty1}{ugIntroSysCmmandsPagePatch1} 712\pastebutton{ugIntroSysCmmandsPageEmpty1}{\showpaste} 713\tab{5}\spadcommand{)what operations integrate} 714\end{paste}\end{patch} 715 716\begin{patch}{ugIntroSysCmmandsPagePatch2} 717\begin{paste}{ugIntroSysCmmandsPageFull2}{ugIntroSysCmmandsPageEmpty2} 718\pastebutton{ugIntroSysCmmandsPageFull2}{\hidepaste} 719\tab{5}\spadcommand{f(0) == 1; g(0) == 1\bound{u1 }} 720\indentrel{3}\begin{verbatim} 721 Type: Void 722\end{verbatim} 723\indentrel{-3}\end{paste}\end{patch} 724 725\begin{patch}{ugIntroSysCmmandsPageEmpty2} 726\begin{paste}{ugIntroSysCmmandsPageEmpty2}{ugIntroSysCmmandsPagePatch2} 727\pastebutton{ugIntroSysCmmandsPageEmpty2}{\showpaste} 728\tab{5}\spadcommand{f(0) == 1; g(0) == 1\bound{u1 }} 729\end{paste}\end{patch} 730 731\begin{patch}{ugIntroSysCmmandsPagePatch3} 732\begin{paste}{ugIntroSysCmmandsPageFull3}{ugIntroSysCmmandsPageEmpty3} 733\pastebutton{ugIntroSysCmmandsPageFull3}{\hidepaste} 734\tab{5}\spadcommand{f(n) == e/2*f(n-1) - x*g(n-1)\bound{u2 }\free{u1 }} 735\indentrel{3}\begin{verbatim} 736 Type: Void 737\end{verbatim} 738\indentrel{-3}\end{paste}\end{patch} 739 740\begin{patch}{ugIntroSysCmmandsPageEmpty3} 741\begin{paste}{ugIntroSysCmmandsPageEmpty3}{ugIntroSysCmmandsPagePatch3} 742\pastebutton{ugIntroSysCmmandsPageEmpty3}{\showpaste} 743\tab{5}\spadcommand{f(n) == e/2*f(n-1) - x*g(n-1)\bound{u2 }\free{u1 }} 744\end{paste}\end{patch} 745 746\begin{patch}{ugIntroSysCmmandsPagePatch4} 747\begin{paste}{ugIntroSysCmmandsPageFull4}{ugIntroSysCmmandsPageEmpty4} 748\pastebutton{ugIntroSysCmmandsPageFull4}{\hidepaste} 749\tab{5}\spadcommand{g(n) == -x*f(n-1) + d/3*g(n-1)\bound{u3 }\free{u2 }} 750\indentrel{3}\begin{verbatim} 751 Type: Void 752\end{verbatim} 753\indentrel{-3}\end{paste}\end{patch} 754 755\begin{patch}{ugIntroSysCmmandsPageEmpty4} 756\begin{paste}{ugIntroSysCmmandsPageEmpty4}{ugIntroSysCmmandsPagePatch4} 757\pastebutton{ugIntroSysCmmandsPageEmpty4}{\showpaste} 758\tab{5}\spadcommand{g(n) == -x*f(n-1) + d/3*g(n-1)\bound{u3 }\free{u2 }} 759\end{paste}\end{patch} 760 761\begin{patch}{ugIntroSysCmmandsPagePatch5} 762\begin{paste}{ugIntroSysCmmandsPageFull5}{ugIntroSysCmmandsPageEmpty5} 763\pastebutton{ugIntroSysCmmandsPageFull5}{\hidepaste} 764\tab{5}\spadcommand{f(3)\bound{u4 }\free{u3 }} 765\indentrel{3}\begin{verbatim} 766 (4) 767 3 1 2 1 2 1 1 2 1 3 768 - x + (e + - d)x + (- - e - - d e - - d )x + - e 769 3 4 6 9 8 770 Type: Polynomial(Fraction(Integer)) 771\end{verbatim} 772\indentrel{-3}\end{paste}\end{patch} 773 774\begin{patch}{ugIntroSysCmmandsPageEmpty5} 775\begin{paste}{ugIntroSysCmmandsPageEmpty5}{ugIntroSysCmmandsPagePatch5} 776\pastebutton{ugIntroSysCmmandsPageEmpty5}{\showpaste} 777\tab{5}\spadcommand{f(3)\bound{u4 }\free{u3 }} 778\end{paste}\end{patch} 779 780\begin{patch}{ugIntroSysCmmandsPagePatch6} 781\begin{paste}{ugIntroSysCmmandsPageFull6}{ugIntroSysCmmandsPageEmpty6} 782\pastebutton{ugIntroSysCmmandsPageFull6}{\hidepaste} 783\tab{5}\spadcommand{)undo 2\bound{u5 }\free{u4 }} 784\indentrel{3}\begin{verbatim} 785\end{verbatim} 786\indentrel{-3}\end{paste}\end{patch} 787 788\begin{patch}{ugIntroSysCmmandsPageEmpty6} 789\begin{paste}{ugIntroSysCmmandsPageEmpty6}{ugIntroSysCmmandsPagePatch6} 790\pastebutton{ugIntroSysCmmandsPageEmpty6}{\showpaste} 791\tab{5}\spadcommand{)undo 2\bound{u5 }\free{u4 }} 792\end{paste}\end{patch} 793 794\begin{patch}{ugIntroSysCmmandsPagePatch7} 795\begin{paste}{ugIntroSysCmmandsPageFull7}{ugIntroSysCmmandsPageEmpty7} 796\pastebutton{ugIntroSysCmmandsPageFull7}{\hidepaste} 797\tab{5}\spadcommand{f(n) == d/3*f(n-1) - x*g(n-1)\bound{u6 }\free{u5 }} 798\indentrel{3}\begin{verbatim} 799 Type: Void 800\end{verbatim} 801\indentrel{-3}\end{paste}\end{patch} 802 803\begin{patch}{ugIntroSysCmmandsPageEmpty7} 804\begin{paste}{ugIntroSysCmmandsPageEmpty7}{ugIntroSysCmmandsPagePatch7} 805\pastebutton{ugIntroSysCmmandsPageEmpty7}{\showpaste} 806\tab{5}\spadcommand{f(n) == d/3*f(n-1) - x*g(n-1)\bound{u6 }\free{u5 }} 807\end{paste}\end{patch} 808 809\begin{patch}{ugIntroSysCmmandsPagePatch8} 810\begin{paste}{ugIntroSysCmmandsPageFull8}{ugIntroSysCmmandsPageEmpty8} 811\pastebutton{ugIntroSysCmmandsPageFull8}{\hidepaste} 812\tab{5}\spadcommand{)undo )redo\bound{u7 }\free{u6 }} 813\indentrel{3}\begin{verbatim} 814\end{verbatim} 815\indentrel{-3}\end{paste}\end{patch} 816 817\begin{patch}{ugIntroSysCmmandsPageEmpty8} 818\begin{paste}{ugIntroSysCmmandsPageEmpty8}{ugIntroSysCmmandsPagePatch8} 819\pastebutton{ugIntroSysCmmandsPageEmpty8}{\showpaste} 820\tab{5}\spadcommand{)undo )redo\bound{u7 }\free{u6 }} 821\end{paste}\end{patch} 822 823\begin{patch}{ugIntroSysCmmandsPagePatch9} 824\begin{paste}{ugIntroSysCmmandsPageFull9}{ugIntroSysCmmandsPageEmpty9} 825\pastebutton{ugIntroSysCmmandsPageFull9}{\hidepaste} 826\tab{5}\spadcommand{)undo 4\bound{u8 }\free{u7 }} 827\indentrel{3}\begin{verbatim} 828\end{verbatim} 829\indentrel{-3}\end{paste}\end{patch} 830 831\begin{patch}{ugIntroSysCmmandsPageEmpty9} 832\begin{paste}{ugIntroSysCmmandsPageEmpty9}{ugIntroSysCmmandsPagePatch9} 833\pastebutton{ugIntroSysCmmandsPageEmpty9}{\showpaste} 834\tab{5}\spadcommand{)undo 4\bound{u8 }\free{u7 }} 835\end{paste}\end{patch} 836 837\begin{patch}{ugIntroSysCmmandsPagePatch10} 838\begin{paste}{ugIntroSysCmmandsPageFull10}{ugIntroSysCmmandsPageEmpty10} 839\pastebutton{ugIntroSysCmmandsPageFull10}{\hidepaste} 840\tab{5}\spadcommand{f(3)\bound{u9 }\free{u8 }} 841\indentrel{3}\begin{verbatim} 842 (6) 843 3 1 2 1 2 1 1 2 1 3 844 - x + (e + - d)x + (- - e - - d e - - d )x + - e 845 3 4 6 9 8 846 Type: Polynomial(Fraction(Integer)) 847\end{verbatim} 848\indentrel{-3}\end{paste}\end{patch} 849 850\begin{patch}{ugIntroSysCmmandsPageEmpty10} 851\begin{paste}{ugIntroSysCmmandsPageEmpty10}{ugIntroSysCmmandsPagePatch10} 852\pastebutton{ugIntroSysCmmandsPageEmpty10}{\showpaste} 853\tab{5}\spadcommand{f(3)\bound{u9 }\free{u8 }} 854\end{paste}\end{patch} 855 856\begin{patch}{ugIntroArithmeticPagePatch1} 857\begin{paste}{ugIntroArithmeticPageFull1}{ugIntroArithmeticPageEmpty1} 858\pastebutton{ugIntroArithmeticPageFull1}{\hidepaste} 859\tab{5}\spadcommand{1 + 2 - 3 / 4 * 3 ^ 2 - 1} 860\indentrel{3}\begin{verbatim} 861 19 862 (1) - -- 863 4 864 Type: Fraction(Integer) 865\end{verbatim} 866\indentrel{-3}\end{paste}\end{patch} 867 868\begin{patch}{ugIntroArithmeticPageEmpty1} 869\begin{paste}{ugIntroArithmeticPageEmpty1}{ugIntroArithmeticPagePatch1} 870\pastebutton{ugIntroArithmeticPageEmpty1}{\showpaste} 871\tab{5}\spadcommand{1 + 2 - 3 / 4 * 3 ^ 2 - 1} 872\end{paste}\end{patch} 873 874\begin{patch}{ugIntroArithmeticPagePatch2} 875\begin{paste}{ugIntroArithmeticPageFull2}{ugIntroArithmeticPageEmpty2} 876\pastebutton{ugIntroArithmeticPageFull2}{\hidepaste} 877\tab{5}\spadcommand{((1 + 2) - ((3 / 4) * (3 ^ 2))) - 1} 878\indentrel{3}\begin{verbatim} 879 19 880 (2) - -- 881 4 882 Type: Fraction(Integer) 883\end{verbatim} 884\indentrel{-3}\end{paste}\end{patch} 885 886\begin{patch}{ugIntroArithmeticPageEmpty2} 887\begin{paste}{ugIntroArithmeticPageEmpty2}{ugIntroArithmeticPagePatch2} 888\pastebutton{ugIntroArithmeticPageEmpty2}{\showpaste} 889\tab{5}\spadcommand{((1 + 2) - ((3 / 4) * (3 ^ 2))) - 1} 890\end{paste}\end{patch} 891 892\begin{patch}{ugIntroArithmeticPagePatch3} 893\begin{paste}{ugIntroArithmeticPageFull3}{ugIntroArithmeticPageEmpty3} 894\pastebutton{ugIntroArithmeticPageFull3}{\hidepaste} 895\tab{5}\spadcommand{1 + 2 - 3/ (4 * 3 ^ (2 - 1))} 896\indentrel{3}\begin{verbatim} 897 11 898 (3) -- 899 4 900 Type: Fraction(Integer) 901\end{verbatim} 902\indentrel{-3}\end{paste}\end{patch} 903 904\begin{patch}{ugIntroArithmeticPageEmpty3} 905\begin{paste}{ugIntroArithmeticPageEmpty3}{ugIntroArithmeticPagePatch3} 906\pastebutton{ugIntroArithmeticPageEmpty3}{\showpaste} 907\tab{5}\spadcommand{1 + 2 - 3/ (4 * 3 ^ (2 - 1))} 908\end{paste}\end{patch} 909 910\begin{patch}{ugIntroConversionPagePatch1} 911\begin{paste}{ugIntroConversionPageFull1}{ugIntroConversionPageEmpty1} 912\pastebutton{ugIntroConversionPageFull1}{\hidepaste} 913\tab{5}\spadcommand{p := r^2 + 2/3\bound{p }} 914\indentrel{3}\begin{verbatim} 915 2 2 916 (1) r + - 917 3 918 Type: Polynomial(Fraction(Integer)) 919\end{verbatim} 920\indentrel{-3}\end{paste}\end{patch} 921 922\begin{patch}{ugIntroConversionPageEmpty1} 923\begin{paste}{ugIntroConversionPageEmpty1}{ugIntroConversionPagePatch1} 924\pastebutton{ugIntroConversionPageEmpty1}{\showpaste} 925\tab{5}\spadcommand{p := r^2 + 2/3\bound{p }} 926\end{paste}\end{patch} 927 928\begin{patch}{ugIntroConversionPagePatch2} 929\begin{paste}{ugIntroConversionPageFull2}{ugIntroConversionPageEmpty2} 930\pastebutton{ugIntroConversionPageFull2}{\hidepaste} 931\tab{5}\spadcommand{p :: Fraction Polynomial Integer\free{p }} 932\indentrel{3}\begin{verbatim} 933 2 934 3 r + 2 935 (2) -------- 936 3 937 Type: Fraction(Polynomial(Integer)) 938\end{verbatim} 939\indentrel{-3}\end{paste}\end{patch} 940 941\begin{patch}{ugIntroConversionPageEmpty2} 942\begin{paste}{ugIntroConversionPageEmpty2}{ugIntroConversionPagePatch2} 943\pastebutton{ugIntroConversionPageEmpty2}{\showpaste} 944\tab{5}\spadcommand{p :: Fraction Polynomial Integer\free{p }} 945\end{paste}\end{patch} 946 947\begin{patch}{ugIntroYouPagePatch1} 948\begin{paste}{ugIntroYouPageFull1}{ugIntroYouPageEmpty1} 949\pastebutton{ugIntroYouPageFull1}{\hidepaste} 950\tab{5}\spadcommand{fact(0) == 1\bound{fact }} 951\indentrel{3}\begin{verbatim} 952 Type: Void 953\end{verbatim} 954\indentrel{-3}\end{paste}\end{patch} 955 956\begin{patch}{ugIntroYouPageEmpty1} 957\begin{paste}{ugIntroYouPageEmpty1}{ugIntroYouPagePatch1} 958\pastebutton{ugIntroYouPageEmpty1}{\showpaste} 959\tab{5}\spadcommand{fact(0) == 1\bound{fact }} 960\end{paste}\end{patch} 961 962\begin{patch}{ugIntroYouPagePatch2} 963\begin{paste}{ugIntroYouPageFull2}{ugIntroYouPageEmpty2} 964\pastebutton{ugIntroYouPageFull2}{\hidepaste} 965\tab{5}\spadcommand{fact(n) == n*fact(n-1)\bound{facta }\free{fact }} 966\indentrel{3}\begin{verbatim} 967 Type: Void 968\end{verbatim} 969\indentrel{-3}\end{paste}\end{patch} 970 971\begin{patch}{ugIntroYouPageEmpty2} 972\begin{paste}{ugIntroYouPageEmpty2}{ugIntroYouPagePatch2} 973\pastebutton{ugIntroYouPageEmpty2}{\showpaste} 974\tab{5}\spadcommand{fact(n) == n*fact(n-1)\bound{facta }\free{fact }} 975\end{paste}\end{patch} 976 977\begin{patch}{ugIntroYouPagePatch3} 978\begin{paste}{ugIntroYouPageFull3}{ugIntroYouPageEmpty3} 979\pastebutton{ugIntroYouPageFull3}{\hidepaste} 980\tab{5}\spadcommand{fact(50)\free{facta }} 981\indentrel{3}\begin{verbatim} 982 (3) 983 304140932017133780436126081660647688443776415689605120_ 984 00000000000 985 Type: PositiveInteger 986\end{verbatim} 987\indentrel{-3}\end{paste}\end{patch} 988 989\begin{patch}{ugIntroYouPageEmpty3} 990\begin{paste}{ugIntroYouPageEmpty3}{ugIntroYouPagePatch3} 991\pastebutton{ugIntroYouPageEmpty3}{\showpaste} 992\tab{5}\spadcommand{fact(50)\free{facta }} 993\end{paste}\end{patch} 994 995\begin{patch}{ugIntroYouPagePatch4} 996\begin{paste}{ugIntroYouPageFull4}{ugIntroYouPageEmpty4} 997\pastebutton{ugIntroYouPageFull4}{\hidepaste} 998\tab{5}\spadcommand{fac(n) == if n < 3 then n else n * fac(n - 1)\bound{fac }} 999\indentrel{3}\begin{verbatim} 1000 Type: Void 1001\end{verbatim} 1002\indentrel{-3}\end{paste}\end{patch} 1003 1004\begin{patch}{ugIntroYouPageEmpty4} 1005\begin{paste}{ugIntroYouPageEmpty4}{ugIntroYouPagePatch4} 1006\pastebutton{ugIntroYouPageEmpty4}{\showpaste} 1007\tab{5}\spadcommand{fac(n) == if n < 3 then n else n * fac(n - 1)\bound{fac }} 1008\end{paste}\end{patch} 1009 1010\begin{patch}{ugIntroYouPagePatch5} 1011\begin{paste}{ugIntroYouPageFull5}{ugIntroYouPageEmpty5} 1012\pastebutton{ugIntroYouPageFull5}{\hidepaste} 1013\tab{5}\spadcommand{fac(50)\free{fac }} 1014\indentrel{3}\begin{verbatim} 1015 (5) 1016 304140932017133780436126081660647688443776415689605120_ 1017 00000000000 1018 Type: PositiveInteger 1019\end{verbatim} 1020\indentrel{-3}\end{paste}\end{patch} 1021 1022\begin{patch}{ugIntroYouPageEmpty5} 1023\begin{paste}{ugIntroYouPageEmpty5}{ugIntroYouPagePatch5} 1024\pastebutton{ugIntroYouPageEmpty5}{\showpaste} 1025\tab{5}\spadcommand{fac(50)\free{fac }} 1026\end{paste}\end{patch} 1027 1028\begin{patch}{ugIntroYouPagePatch6} 1029\begin{paste}{ugIntroYouPageFull6}{ugIntroYouPageEmpty6} 1030\pastebutton{ugIntroYouPageFull6}{\hidepaste} 1031\tab{5}\spadcommand{fa(n) == (a := 1; for i in 2..n repeat a := a*i; a)\bound{fa }} 1032\indentrel{3}\begin{verbatim} 1033 Type: Void 1034\end{verbatim} 1035\indentrel{-3}\end{paste}\end{patch} 1036 1037\begin{patch}{ugIntroYouPageEmpty6} 1038\begin{paste}{ugIntroYouPageEmpty6}{ugIntroYouPagePatch6} 1039\pastebutton{ugIntroYouPageEmpty6}{\showpaste} 1040\tab{5}\spadcommand{fa(n) == (a := 1; for i in 2..n repeat a := a*i; a)\bound{fa }} 1041\end{paste}\end{patch} 1042 1043\begin{patch}{ugIntroYouPagePatch7} 1044\begin{paste}{ugIntroYouPageFull7}{ugIntroYouPageEmpty7} 1045\pastebutton{ugIntroYouPageFull7}{\hidepaste} 1046\tab{5}\spadcommand{fa(50)\free{fa }} 1047\indentrel{3}\begin{verbatim} 1048 (7) 1049 304140932017133780436126081660647688443776415689605120_ 1050 00000000000 1051 Type: PositiveInteger 1052\end{verbatim} 1053\indentrel{-3}\end{paste}\end{patch} 1054 1055\begin{patch}{ugIntroYouPageEmpty7} 1056\begin{paste}{ugIntroYouPageEmpty7}{ugIntroYouPagePatch7} 1057\pastebutton{ugIntroYouPageEmpty7}{\showpaste} 1058\tab{5}\spadcommand{fa(50)\free{fa }} 1059\end{paste}\end{patch} 1060 1061\begin{patch}{ugIntroYouPagePatch8} 1062\begin{paste}{ugIntroYouPageFull8}{ugIntroYouPageEmpty8} 1063\pastebutton{ugIntroYouPageFull8}{\hidepaste} 1064\tab{5}\spadcommand{f(n) == reduce(*,[i for i in 2..n])\bound{f }} 1065\indentrel{3}\begin{verbatim} 1066 Type: Void 1067\end{verbatim} 1068\indentrel{-3}\end{paste}\end{patch} 1069 1070\begin{patch}{ugIntroYouPageEmpty8} 1071\begin{paste}{ugIntroYouPageEmpty8}{ugIntroYouPagePatch8} 1072\pastebutton{ugIntroYouPageEmpty8}{\showpaste} 1073\tab{5}\spadcommand{f(n) == reduce(*,[i for i in 2..n])\bound{f }} 1074\end{paste}\end{patch} 1075 1076\begin{patch}{ugIntroYouPagePatch9} 1077\begin{paste}{ugIntroYouPageFull9}{ugIntroYouPageEmpty9} 1078\pastebutton{ugIntroYouPageFull9}{\hidepaste} 1079\tab{5}\spadcommand{f(50)\free{f }} 1080\indentrel{3}\begin{verbatim} 1081 (9) 1082 304140932017133780436126081660647688443776415689605120_ 1083 00000000000 1084 Type: PositiveInteger 1085\end{verbatim} 1086\indentrel{-3}\end{paste}\end{patch} 1087 1088\begin{patch}{ugIntroYouPageEmpty9} 1089\begin{paste}{ugIntroYouPageEmpty9}{ugIntroYouPagePatch9} 1090\pastebutton{ugIntroYouPageEmpty9}{\showpaste} 1091\tab{5}\spadcommand{f(50)\free{f }} 1092\end{paste}\end{patch} 1093 1094\begin{patch}{ugIntroYouPagePatch10} 1095\begin{paste}{ugIntroYouPageFull10}{ugIntroYouPageEmpty10} 1096\pastebutton{ugIntroYouPageFull10}{\hidepaste} 1097\tab{5}\spadcommand{factorial(50)} 1098\indentrel{3}\begin{verbatim} 1099 (10) 1100 304140932017133780436126081660647688443776415689605120_ 1101 00000000000 1102 Type: PositiveInteger 1103\end{verbatim} 1104\indentrel{-3}\end{paste}\end{patch} 1105 1106\begin{patch}{ugIntroYouPageEmpty10} 1107\begin{paste}{ugIntroYouPageEmpty10}{ugIntroYouPagePatch10} 1108\pastebutton{ugIntroYouPageEmpty10}{\showpaste} 1109\tab{5}\spadcommand{factorial(50)} 1110\end{paste}\end{patch} 1111 1112\begin{patch}{ugIntroYouPagePatch11} 1113\begin{paste}{ugIntroYouPageFull11}{ugIntroYouPageEmpty11} 1114\pastebutton{ugIntroYouPageFull11}{\hidepaste} 1115\tab{5}\spadcommand{permMat(n, i, j) == 1116 m := diagonalMatrix 1117 [(if i = k or j = k then 0 else 1) 1118 for k in 1..n] 1119 m(i,j) := 1 1120 m(j,i) := 1 1121 m 1122\bound{permMat }} 1123\indentrel{3}\begin{verbatim} 1124 Type: Void 1125\end{verbatim} 1126\indentrel{-3}\end{paste}\end{patch} 1127 1128\begin{patch}{ugIntroYouPageEmpty11} 1129\begin{paste}{ugIntroYouPageEmpty11}{ugIntroYouPagePatch11} 1130\pastebutton{ugIntroYouPageEmpty11}{\showpaste} 1131\tab{5}\spadcommand{permMat(n, i, j) == 1132 m := diagonalMatrix 1133 [(if i = k or j = k then 0 else 1) 1134 for k in 1..n] 1135 m(i,j) := 1 1136 m(j,i) := 1 1137 m 1138\bound{permMat }} 1139\end{paste}\end{patch} 1140 1141\begin{patch}{ugIntroYouPagePatch12} 1142\begin{paste}{ugIntroYouPageFull12}{ugIntroYouPageEmpty12} 1143\pastebutton{ugIntroYouPageFull12}{\hidepaste} 1144\tab{5}\spadcommand{p := permMat(4,2,3)\free{permMat }\bound{p }} 1145\indentrel{3}\begin{verbatim} 1146 +1 0 0 0+ 1147 | | 1148 |0 0 1 0| 1149 (12) | | 1150 |0 1 0 0| 1151 | | 1152 +0 0 0 1+ 1153 Type: Matrix(NonNegativeInteger) 1154\end{verbatim} 1155\indentrel{-3}\end{paste}\end{patch} 1156 1157\begin{patch}{ugIntroYouPageEmpty12} 1158\begin{paste}{ugIntroYouPageEmpty12}{ugIntroYouPagePatch12} 1159\pastebutton{ugIntroYouPageEmpty12}{\showpaste} 1160\tab{5}\spadcommand{p := permMat(4,2,3)\free{permMat }\bound{p }} 1161\end{paste}\end{patch} 1162 1163\begin{patch}{ugIntroYouPagePatch13} 1164\begin{paste}{ugIntroYouPageFull13}{ugIntroYouPageEmpty13} 1165\pastebutton{ugIntroYouPageFull13}{\hidepaste} 1166\tab{5}\spadcommand{m := matrix [[4*i + j for j in 1..4] for i in 0..3]\bound{m }} 1167\indentrel{3}\begin{verbatim} 1168 +1 2 3 4 + 1169 | | 1170 |5 6 7 8 | 1171 (13) | | 1172 |9 10 11 12| 1173 | | 1174 +13 14 15 16+ 1175 Type: Matrix(NonNegativeInteger) 1176\end{verbatim} 1177\indentrel{-3}\end{paste}\end{patch} 1178 1179\begin{patch}{ugIntroYouPageEmpty13} 1180\begin{paste}{ugIntroYouPageEmpty13}{ugIntroYouPagePatch13} 1181\pastebutton{ugIntroYouPageEmpty13}{\showpaste} 1182\tab{5}\spadcommand{m := matrix [[4*i + j for j in 1..4] for i in 0..3]\bound{m }} 1183\end{paste}\end{patch} 1184 1185\begin{patch}{ugIntroYouPagePatch14} 1186\begin{paste}{ugIntroYouPageFull14}{ugIntroYouPageEmpty14} 1187\pastebutton{ugIntroYouPageFull14}{\hidepaste} 1188\tab{5}\spadcommand{permMat(4,2,3) * m\free{p m }} 1189\indentrel{3}\begin{verbatim} 1190 +1 2 3 4 + 1191 | | 1192 |9 10 11 12| 1193 (14) | | 1194 |5 6 7 8 | 1195 | | 1196 +13 14 15 16+ 1197 Type: Matrix(NonNegativeInteger) 1198\end{verbatim} 1199\indentrel{-3}\end{paste}\end{patch} 1200 1201\begin{patch}{ugIntroYouPageEmpty14} 1202\begin{paste}{ugIntroYouPageEmpty14}{ugIntroYouPagePatch14} 1203\pastebutton{ugIntroYouPageEmpty14}{\showpaste} 1204\tab{5}\spadcommand{permMat(4,2,3) * m\free{p m }} 1205\end{paste}\end{patch} 1206 1207\begin{patch}{ugIntroYouPagePatch15} 1208\begin{paste}{ugIntroYouPageFull15}{ugIntroYouPageEmpty15} 1209\pastebutton{ugIntroYouPageFull15}{\hidepaste} 1210\tab{5}\spadcommand{t : (Float -> Float, Float) -> Float\bound{tdecl }} 1211\indentrel{3}\begin{verbatim} 1212 Type: Void 1213\end{verbatim} 1214\indentrel{-3}\end{paste}\end{patch} 1215 1216\begin{patch}{ugIntroYouPageEmpty15} 1217\begin{paste}{ugIntroYouPageEmpty15}{ugIntroYouPagePatch15} 1218\pastebutton{ugIntroYouPageEmpty15}{\showpaste} 1219\tab{5}\spadcommand{t : (Float -> Float, Float) -> Float\bound{tdecl }} 1220\end{paste}\end{patch} 1221 1222\begin{patch}{ugIntroYouPagePatch16} 1223\begin{paste}{ugIntroYouPageFull16}{ugIntroYouPageEmpty16} 1224\pastebutton{ugIntroYouPageFull16}{\hidepaste} 1225\tab{5}\spadcommand{t(fun, x) == fun(x)^2 + sin(x)^2\free{tdecl }\bound{t }} 1226\indentrel{3}\begin{verbatim} 1227 Type: Void 1228\end{verbatim} 1229\indentrel{-3}\end{paste}\end{patch} 1230 1231\begin{patch}{ugIntroYouPageEmpty16} 1232\begin{paste}{ugIntroYouPageEmpty16}{ugIntroYouPagePatch16} 1233\pastebutton{ugIntroYouPageEmpty16}{\showpaste} 1234\tab{5}\spadcommand{t(fun, x) == fun(x)^2 + sin(x)^2\free{tdecl }\bound{t }} 1235\end{paste}\end{patch} 1236 1237\begin{patch}{ugIntroYouPagePatch17} 1238\begin{paste}{ugIntroYouPageFull17}{ugIntroYouPageEmpty17} 1239\pastebutton{ugIntroYouPageFull17}{\hidepaste} 1240\tab{5}\spadcommand{t(cos, 5.2058)\free{t }} 1241\indentrel{3}\begin{verbatim} 1242 (17) 1.0 1243 Type: Float 1244\end{verbatim} 1245\indentrel{-3}\end{paste}\end{patch} 1246 1247\begin{patch}{ugIntroYouPageEmpty17} 1248\begin{paste}{ugIntroYouPageEmpty17}{ugIntroYouPagePatch17} 1249\pastebutton{ugIntroYouPageEmpty17}{\showpaste} 1250\tab{5}\spadcommand{t(cos, 5.2058)\free{t }} 1251\end{paste}\end{patch} 1252 1253\begin{patch}{ugIntroYouPagePatch18} 1254\begin{paste}{ugIntroYouPageFull18}{ugIntroYouPageEmpty18} 1255\pastebutton{ugIntroYouPageFull18}{\hidepaste} 1256\tab{5}\spadcommand{cosinv(y) == cos(1/y)\bound{cosinv }} 1257\indentrel{3}\begin{verbatim} 1258 Type: Void 1259\end{verbatim} 1260\indentrel{-3}\end{paste}\end{patch} 1261 1262\begin{patch}{ugIntroYouPageEmpty18} 1263\begin{paste}{ugIntroYouPageEmpty18}{ugIntroYouPagePatch18} 1264\pastebutton{ugIntroYouPageEmpty18}{\showpaste} 1265\tab{5}\spadcommand{cosinv(y) == cos(1/y)\bound{cosinv }} 1266\end{paste}\end{patch} 1267 1268\begin{patch}{ugIntroYouPagePatch19} 1269\begin{paste}{ugIntroYouPageFull19}{ugIntroYouPageEmpty19} 1270\pastebutton{ugIntroYouPageFull19}{\hidepaste} 1271\tab{5}\spadcommand{t(cosinv, 5.2058)\free{t }\free{cosinv }} 1272\indentrel{3}\begin{verbatim} 1273 (19) 1.7392237241_800516493 1274 Type: Float 1275\end{verbatim} 1276\indentrel{-3}\end{paste}\end{patch} 1277 1278\begin{patch}{ugIntroYouPageEmpty19} 1279\begin{paste}{ugIntroYouPageEmpty19}{ugIntroYouPagePatch19} 1280\pastebutton{ugIntroYouPageEmpty19}{\showpaste} 1281\tab{5}\spadcommand{t(cosinv, 5.2058)\free{t }\free{cosinv }} 1282\end{paste}\end{patch} 1283 1284\begin{patch}{ugIntroYouPagePatch20} 1285\begin{paste}{ugIntroYouPageFull20}{ugIntroYouPageEmpty20} 1286\pastebutton{ugIntroYouPageFull20}{\hidepaste} 1287\tab{5}\spadcommand{groupSqrt := rule(sqrt(a) * sqrt(b) == sqrt(a*b))\bound{g }} 1288\indentrel{3}\begin{verbatim} 1289 +-+ +-+ +---+ 1290 (20) %DD\|a \|b == %DD\|a b 1291 Type: RewriteRule(Integer,Integer,Expression(Integer)) 1292\end{verbatim} 1293\indentrel{-3}\end{paste}\end{patch} 1294 1295\begin{patch}{ugIntroYouPageEmpty20} 1296\begin{paste}{ugIntroYouPageEmpty20}{ugIntroYouPagePatch20} 1297\pastebutton{ugIntroYouPageEmpty20}{\showpaste} 1298\tab{5}\spadcommand{groupSqrt := rule(sqrt(a) * sqrt(b) == sqrt(a*b))\bound{g }} 1299\end{paste}\end{patch} 1300 1301\begin{patch}{ugIntroYouPagePatch21} 1302\begin{paste}{ugIntroYouPageFull21}{ugIntroYouPageEmpty21} 1303\pastebutton{ugIntroYouPageFull21}{\hidepaste} 1304\tab{5}\spadcommand{a := (sqrt(x) + sqrt(y) + sqrt(z))^4\bound{sxy }} 1305\indentrel{3}\begin{verbatim} 1306 (21) 1307 +-+ +-+ +-+ 1308 ((4 z + 4 y + 12 x)\|y + (4 z + 12 y + 4 x)\|x )\|z 1309 + 1310 +-+ +-+ 2 2 1311 (12 z + 4 y + 4 x)\|x \|y + z + (6 y + 6 x)z + y 1312 + 1313 2 1314 6 x y + x 1315 Type: Expression(Integer) 1316\end{verbatim} 1317\indentrel{-3}\end{paste}\end{patch} 1318 1319\begin{patch}{ugIntroYouPageEmpty21} 1320\begin{paste}{ugIntroYouPageEmpty21}{ugIntroYouPagePatch21} 1321\pastebutton{ugIntroYouPageEmpty21}{\showpaste} 1322\tab{5}\spadcommand{a := (sqrt(x) + sqrt(y) + sqrt(z))^4\bound{sxy }} 1323\end{paste}\end{patch} 1324 1325\begin{patch}{ugIntroYouPagePatch22} 1326\begin{paste}{ugIntroYouPageFull22}{ugIntroYouPageEmpty22} 1327\pastebutton{ugIntroYouPageFull22}{\hidepaste} 1328\tab{5}\spadcommand{groupSqrt a\free{sxy }\free{g }} 1329\indentrel{3}\begin{verbatim} 1330 (22) 1331 +---+ +---+ 1332 (4 z + 4 y + 12 x)\|y z + (4 z + 12 y + 4 x)\|x z 1333 + 1334 +---+ 2 2 1335 (12 z + 4 y + 4 x)\|x y + z + (6 y + 6 x)z + y 1336 + 1337 2 1338 6 x y + x 1339 Type: Expression(Integer) 1340\end{verbatim} 1341\indentrel{-3}\end{paste}\end{patch} 1342 1343\begin{patch}{ugIntroYouPageEmpty22} 1344\begin{paste}{ugIntroYouPageEmpty22}{ugIntroYouPagePatch22} 1345\pastebutton{ugIntroYouPageEmpty22}{\showpaste} 1346\tab{5}\spadcommand{groupSqrt a\free{sxy }\free{g }} 1347\end{paste}\end{patch} 1348 1349\begin{patch}{ugIntroTypesPagePatch1} 1350\begin{paste}{ugIntroTypesPageFull1}{ugIntroTypesPageEmpty1} 1351\pastebutton{ugIntroTypesPageFull1}{\hidepaste} 1352\tab{5}\spadcommand{8} 1353\indentrel{3}\begin{verbatim} 1354 (1) 8 1355 Type: PositiveInteger 1356\end{verbatim} 1357\indentrel{-3}\end{paste}\end{patch} 1358 1359\begin{patch}{ugIntroTypesPageEmpty1} 1360\begin{paste}{ugIntroTypesPageEmpty1}{ugIntroTypesPagePatch1} 1361\pastebutton{ugIntroTypesPageEmpty1}{\showpaste} 1362\tab{5}\spadcommand{8} 1363\end{paste}\end{patch} 1364 1365\begin{patch}{ugIntroTypesPagePatch2} 1366\begin{paste}{ugIntroTypesPageFull2}{ugIntroTypesPageEmpty2} 1367\pastebutton{ugIntroTypesPageFull2}{\hidepaste} 1368\tab{5}\spadcommand{-8} 1369\indentrel{3}\begin{verbatim} 1370 (2) - 8 1371 Type: Integer 1372\end{verbatim} 1373\indentrel{-3}\end{paste}\end{patch} 1374 1375\begin{patch}{ugIntroTypesPageEmpty2} 1376\begin{paste}{ugIntroTypesPageEmpty2}{ugIntroTypesPagePatch2} 1377\pastebutton{ugIntroTypesPageEmpty2}{\showpaste} 1378\tab{5}\spadcommand{-8} 1379\end{paste}\end{patch} 1380 1381\begin{patch}{ugIntroTypesPagePatch3} 1382\begin{paste}{ugIntroTypesPageFull3}{ugIntroTypesPageEmpty3} 1383\pastebutton{ugIntroTypesPageFull3}{\hidepaste} 1384\tab{5}\spadcommand{x^8} 1385\indentrel{3}\begin{verbatim} 1386 8 1387 (3) x 1388 Type: Polynomial(Integer) 1389\end{verbatim} 1390\indentrel{-3}\end{paste}\end{patch} 1391 1392\begin{patch}{ugIntroTypesPageEmpty3} 1393\begin{paste}{ugIntroTypesPageEmpty3}{ugIntroTypesPagePatch3} 1394\pastebutton{ugIntroTypesPageEmpty3}{\showpaste} 1395\tab{5}\spadcommand{x^8} 1396\end{paste}\end{patch} 1397 1398\begin{patch}{ugIntroTypesPagePatch4} 1399\begin{paste}{ugIntroTypesPageFull4}{ugIntroTypesPageEmpty4} 1400\pastebutton{ugIntroTypesPageFull4}{\hidepaste} 1401\tab{5}\spadcommand{x^(-8)} 1402\indentrel{3}\begin{verbatim} 1403 1 1404 (4) -- 1405 8 1406 x 1407 Type: Fraction(Polynomial(Integer)) 1408\end{verbatim} 1409\indentrel{-3}\end{paste}\end{patch} 1410 1411\begin{patch}{ugIntroTypesPageEmpty4} 1412\begin{paste}{ugIntroTypesPageEmpty4}{ugIntroTypesPagePatch4} 1413\pastebutton{ugIntroTypesPageEmpty4}{\showpaste} 1414\tab{5}\spadcommand{x^(-8)} 1415\end{paste}\end{patch} 1416 1417\begin{patch}{ugIntroTwoDimPagePatch1} 1418\begin{paste}{ugIntroTwoDimPageFull1}{ugIntroTwoDimPageEmpty1} 1419\pastebutton{ugIntroTwoDimPageFull1}{\hidepaste} 1420\tab{5}\spadcommand{m := matrix([[1,2], [3,4]])\bound{m }} 1421\indentrel{3}\begin{verbatim} 1422 +1 2+ 1423 (1) | | 1424 +3 4+ 1425 Type: Matrix(Integer) 1426\end{verbatim} 1427\indentrel{-3}\end{paste}\end{patch} 1428 1429\begin{patch}{ugIntroTwoDimPageEmpty1} 1430\begin{paste}{ugIntroTwoDimPageEmpty1}{ugIntroTwoDimPagePatch1} 1431\pastebutton{ugIntroTwoDimPageEmpty1}{\showpaste} 1432\tab{5}\spadcommand{m := matrix([[1,2], [3,4]])\bound{m }} 1433\end{paste}\end{patch} 1434 1435\begin{patch}{ugIntroTwoDimPagePatch2} 1436\begin{paste}{ugIntroTwoDimPageFull2}{ugIntroTwoDimPageEmpty2} 1437\pastebutton{ugIntroTwoDimPageFull2}{\hidepaste} 1438\tab{5}\spadcommand{matrix([[1/(i + j - x) for i in 1..4] for j in 1..4])\bound{hilb }} 1439\indentrel{3}\begin{verbatim} 1440 + 1 1 1 1 + 1441 |- ----- - ----- - ----- - -----| 1442 | x - 2 x - 3 x - 4 x - 5| 1443 | | 1444 | 1 1 1 1 | 1445 |- ----- - ----- - ----- - -----| 1446 | x - 3 x - 4 x - 5 x - 6| 1447 (2) | | 1448 | 1 1 1 1 | 1449 |- ----- - ----- - ----- - -----| 1450 | x - 4 x - 5 x - 6 x - 7| 1451 | | 1452 | 1 1 1 1 | 1453 |- ----- - ----- - ----- - -----| 1454 + x - 5 x - 6 x - 7 x - 8+ 1455 Type: Matrix(Fraction(Polynomial(Integer))) 1456\end{verbatim} 1457\indentrel{-3}\end{paste}\end{patch} 1458 1459\begin{patch}{ugIntroTwoDimPageEmpty2} 1460\begin{paste}{ugIntroTwoDimPageEmpty2}{ugIntroTwoDimPagePatch2} 1461\pastebutton{ugIntroTwoDimPageEmpty2}{\showpaste} 1462\tab{5}\spadcommand{matrix([[1/(i + j - x) for i in 1..4] for j in 1..4])\bound{hilb }} 1463\end{paste}\end{patch} 1464 1465\begin{patch}{ugIntroTwoDimPagePatch3} 1466\begin{paste}{ugIntroTwoDimPageFull3}{ugIntroTwoDimPageEmpty3} 1467\pastebutton{ugIntroTwoDimPageFull3}{\hidepaste} 1468\tab{5}\spadcommand{vm := matrix [[1,1,1], [x,y,z], [x*x,y*y,z*z]]\bound{vm }} 1469\indentrel{3}\begin{verbatim} 1470 +1 1 1 + 1471 | | 1472 (3) |x y z | 1473 | | 1474 | 2 2 2| 1475 +x y z + 1476 Type: Matrix(Polynomial(Integer)) 1477\end{verbatim} 1478\indentrel{-3}\end{paste}\end{patch} 1479 1480\begin{patch}{ugIntroTwoDimPageEmpty3} 1481\begin{paste}{ugIntroTwoDimPageEmpty3}{ugIntroTwoDimPagePatch3} 1482\pastebutton{ugIntroTwoDimPageEmpty3}{\showpaste} 1483\tab{5}\spadcommand{vm := matrix [[1,1,1], [x,y,z], [x*x,y*y,z*z]]\bound{vm }} 1484\end{paste}\end{patch} 1485 1486\begin{patch}{ugIntroTwoDimPagePatch4} 1487\begin{paste}{ugIntroTwoDimPageFull4}{ugIntroTwoDimPageEmpty4} 1488\pastebutton{ugIntroTwoDimPageFull4}{\hidepaste} 1489\tab{5}\spadcommand{vm(3,3)\free{vm }} 1490\indentrel{3}\begin{verbatim} 1491 2 1492 (4) z 1493 Type: Polynomial(Integer) 1494\end{verbatim} 1495\indentrel{-3}\end{paste}\end{patch} 1496 1497\begin{patch}{ugIntroTwoDimPageEmpty4} 1498\begin{paste}{ugIntroTwoDimPageEmpty4}{ugIntroTwoDimPagePatch4} 1499\pastebutton{ugIntroTwoDimPageEmpty4}{\showpaste} 1500\tab{5}\spadcommand{vm(3,3)\free{vm }} 1501\end{paste}\end{patch} 1502 1503\begin{patch}{ugIntroTwoDimPagePatch5} 1504\begin{paste}{ugIntroTwoDimPageFull5}{ugIntroTwoDimPageEmpty5} 1505\pastebutton{ugIntroTwoDimPageFull5}{\hidepaste} 1506\tab{5}\spadcommand{column(vm,2)\free{vm }} 1507\indentrel{3}\begin{verbatim} 1508 2 1509 (5) [1, y, y ] 1510 Type: Vector(Polynomial(Integer)) 1511\end{verbatim} 1512\indentrel{-3}\end{paste}\end{patch} 1513 1514\begin{patch}{ugIntroTwoDimPageEmpty5} 1515\begin{paste}{ugIntroTwoDimPageEmpty5}{ugIntroTwoDimPagePatch5} 1516\pastebutton{ugIntroTwoDimPageEmpty5}{\showpaste} 1517\tab{5}\spadcommand{column(vm,2)\free{vm }} 1518\end{paste}\end{patch} 1519 1520\begin{patch}{ugIntroTwoDimPagePatch6} 1521\begin{paste}{ugIntroTwoDimPageFull6}{ugIntroTwoDimPageEmpty6} 1522\pastebutton{ugIntroTwoDimPageFull6}{\hidepaste} 1523\tab{5}\spadcommand{vm * vm\free{vm }} 1524\indentrel{3}\begin{verbatim} 1525 (6) 1526 + 2 2 2 + 1527 | x + x + 1 y + y + 1 z + z + 1 | 1528 | | 1529 | 2 2 2 3 | 1530 | x z + x y + x y z + y + x z + y z + x | 1531 | | 1532 | 2 2 2 2 2 2 3 2 4 2 2| 1533 +x z + x y + x y z + y + x z + y z + x + 1534 Type: Matrix(Polynomial(Integer)) 1535\end{verbatim} 1536\indentrel{-3}\end{paste}\end{patch} 1537 1538\begin{patch}{ugIntroTwoDimPageEmpty6} 1539\begin{paste}{ugIntroTwoDimPageEmpty6}{ugIntroTwoDimPagePatch6} 1540\pastebutton{ugIntroTwoDimPageEmpty6}{\showpaste} 1541\tab{5}\spadcommand{vm * vm\free{vm }} 1542\end{paste}\end{patch} 1543 1544\begin{patch}{ugIntroTwoDimPagePatch7} 1545\begin{paste}{ugIntroTwoDimPageFull7}{ugIntroTwoDimPageEmpty7} 1546\pastebutton{ugIntroTwoDimPageFull7}{\hidepaste} 1547\tab{5}\spadcommand{factor determinant vm\free{vm }\bound{d }} 1548\indentrel{3}\begin{verbatim} 1549 (7) (y - x)(z - y)(z - x) 1550 Type: Factored(Polynomial(Integer)) 1551\end{verbatim} 1552\indentrel{-3}\end{paste}\end{patch} 1553 1554\begin{patch}{ugIntroTwoDimPageEmpty7} 1555\begin{paste}{ugIntroTwoDimPageEmpty7}{ugIntroTwoDimPagePatch7} 1556\pastebutton{ugIntroTwoDimPageEmpty7}{\showpaste} 1557\tab{5}\spadcommand{factor determinant vm\free{vm }\bound{d }} 1558\end{paste}\end{patch} 1559 1560\begin{patch}{ugIntroAssignPagePatch1} 1561\begin{paste}{ugIntroAssignPageFull1}{ugIntroAssignPageEmpty1} 1562\pastebutton{ugIntroAssignPageFull1}{\hidepaste} 1563\tab{5}\spadcommand{(x - y*z)^2} 1564\indentrel{3}\begin{verbatim} 1565 2 2 2 1566 (1) y z - 2 x y z + x 1567 Type: Polynomial(Integer) 1568\end{verbatim} 1569\indentrel{-3}\end{paste}\end{patch} 1570 1571\begin{patch}{ugIntroAssignPageEmpty1} 1572\begin{paste}{ugIntroAssignPageEmpty1}{ugIntroAssignPagePatch1} 1573\pastebutton{ugIntroAssignPageEmpty1}{\showpaste} 1574\tab{5}\spadcommand{(x - y*z)^2} 1575\end{paste}\end{patch} 1576 1577\begin{patch}{ugIntroAssignPagePatch2} 1578\begin{paste}{ugIntroAssignPageFull2}{ugIntroAssignPageEmpty2} 1579\pastebutton{ugIntroAssignPageFull2}{\hidepaste} 1580\tab{5}\spadcommand{x := 4} 1581\indentrel{3}\begin{verbatim} 1582 (2) 4 1583 Type: PositiveInteger 1584\end{verbatim} 1585\indentrel{-3}\end{paste}\end{patch} 1586 1587\begin{patch}{ugIntroAssignPageEmpty2} 1588\begin{paste}{ugIntroAssignPageEmpty2}{ugIntroAssignPagePatch2} 1589\pastebutton{ugIntroAssignPageEmpty2}{\showpaste} 1590\tab{5}\spadcommand{x := 4} 1591\end{paste}\end{patch} 1592 1593\begin{patch}{ugIntroAssignPagePatch3} 1594\begin{paste}{ugIntroAssignPageFull3}{ugIntroAssignPageEmpty3} 1595\pastebutton{ugIntroAssignPageFull3}{\hidepaste} 1596\tab{5}\spadcommand{x := z + 3/5} 1597\indentrel{3}\begin{verbatim} 1598 3 1599 (3) z + - 1600 5 1601 Type: Polynomial(Fraction(Integer)) 1602\end{verbatim} 1603\indentrel{-3}\end{paste}\end{patch} 1604 1605\begin{patch}{ugIntroAssignPageEmpty3} 1606\begin{paste}{ugIntroAssignPageEmpty3}{ugIntroAssignPagePatch3} 1607\pastebutton{ugIntroAssignPageEmpty3}{\showpaste} 1608\tab{5}\spadcommand{x := z + 3/5} 1609\end{paste}\end{patch} 1610 1611\begin{patch}{ugIntroAssignPagePatch4} 1612\begin{paste}{ugIntroAssignPageFull4}{ugIntroAssignPageEmpty4} 1613\pastebutton{ugIntroAssignPageFull4}{\hidepaste} 1614\tab{5}\spadcommand{y : Integer\bound{y }} 1615\indentrel{3}\begin{verbatim} 1616 Type: Void 1617\end{verbatim} 1618\indentrel{-3}\end{paste}\end{patch} 1619 1620\begin{patch}{ugIntroAssignPageEmpty4} 1621\begin{paste}{ugIntroAssignPageEmpty4}{ugIntroAssignPagePatch4} 1622\pastebutton{ugIntroAssignPageEmpty4}{\showpaste} 1623\tab{5}\spadcommand{y : Integer\bound{y }} 1624\end{paste}\end{patch} 1625 1626\begin{patch}{ugIntroAssignPagePatch5} 1627\begin{paste}{ugIntroAssignPageFull5}{ugIntroAssignPageEmpty5} 1628\pastebutton{ugIntroAssignPageFull5}{\hidepaste} 1629\tab{5}\spadcommand{y := 89\bound{y1 }\free{y }} 1630\indentrel{3}\begin{verbatim} 1631 (5) 89 1632 Type: Integer 1633\end{verbatim} 1634\indentrel{-3}\end{paste}\end{patch} 1635 1636\begin{patch}{ugIntroAssignPageEmpty5} 1637\begin{paste}{ugIntroAssignPageEmpty5}{ugIntroAssignPagePatch5} 1638\pastebutton{ugIntroAssignPageEmpty5}{\showpaste} 1639\tab{5}\spadcommand{y := 89\bound{y1 }\free{y }} 1640\end{paste}\end{patch} 1641 1642\begin{patch}{ugIntroAssignPagePatch6} 1643\begin{paste}{ugIntroAssignPageFull6}{ugIntroAssignPageEmpty6} 1644\pastebutton{ugIntroAssignPageFull6}{\hidepaste} 1645\tab{5}\spadcommand{y := sin \%pi} 1646\indentrel{3}\begin{verbatim} 1647 (6) 0 1648 Type: Integer 1649\end{verbatim} 1650\indentrel{-3}\end{paste}\end{patch} 1651 1652\begin{patch}{ugIntroAssignPageEmpty6} 1653\begin{paste}{ugIntroAssignPageEmpty6}{ugIntroAssignPagePatch6} 1654\pastebutton{ugIntroAssignPageEmpty6}{\showpaste} 1655\tab{5}\spadcommand{y := sin \%pi} 1656\end{paste}\end{patch} 1657 1658\begin{patch}{ugIntroAssignPagePatch7} 1659\begin{paste}{ugIntroAssignPageFull7}{ugIntroAssignPageEmpty7} 1660\pastebutton{ugIntroAssignPageFull7}{\hidepaste} 1661\tab{5}\spadcommand{y := 2/3} 1662\indentrel{3}\begin{verbatim} 1663error 1664\end{verbatim} 1665\indentrel{-3}\end{paste}\end{patch} 1666 1667\begin{patch}{ugIntroAssignPageEmpty7} 1668\begin{paste}{ugIntroAssignPageEmpty7}{ugIntroAssignPagePatch7} 1669\pastebutton{ugIntroAssignPageEmpty7}{\showpaste} 1670\tab{5}\spadcommand{y := 2/3} 1671\end{paste}\end{patch} 1672 1673\begin{patch}{ugIntroAssignPagePatch8} 1674\begin{paste}{ugIntroAssignPageFull8}{ugIntroAssignPageEmpty8} 1675\pastebutton{ugIntroAssignPageFull8}{\hidepaste} 1676\tab{5}\spadcommand{f : Float := 2/3} 1677\indentrel{3}\begin{verbatim} 1678 (7) 0.6666666666_6666666667 1679 Type: Float 1680\end{verbatim} 1681\indentrel{-3}\end{paste}\end{patch} 1682 1683\begin{patch}{ugIntroAssignPageEmpty8} 1684\begin{paste}{ugIntroAssignPageEmpty8}{ugIntroAssignPagePatch8} 1685\pastebutton{ugIntroAssignPageEmpty8}{\showpaste} 1686\tab{5}\spadcommand{f : Float := 2/3} 1687\end{paste}\end{patch} 1688 1689\begin{patch}{ugIntroAssignPagePatch9} 1690\begin{paste}{ugIntroAssignPageFull9}{ugIntroAssignPageEmpty9} 1691\pastebutton{ugIntroAssignPageFull9}{\hidepaste} 1692\tab{5}\spadcommand{f : Float; f := 2/3\bound{fff }} 1693\indentrel{3}\begin{verbatim} 1694 (8) 0.6666666666_6666666667 1695 Type: Float 1696\end{verbatim} 1697\indentrel{-3}\end{paste}\end{patch} 1698 1699\begin{patch}{ugIntroAssignPageEmpty9} 1700\begin{paste}{ugIntroAssignPageEmpty9}{ugIntroAssignPagePatch9} 1701\pastebutton{ugIntroAssignPageEmpty9}{\showpaste} 1702\tab{5}\spadcommand{f : Float; f := 2/3\bound{fff }} 1703\end{paste}\end{patch} 1704 1705\begin{patch}{ugIntroAssignPagePatch10} 1706\begin{paste}{ugIntroAssignPageFull10}{ugIntroAssignPageEmpty10} 1707\pastebutton{ugIntroAssignPageFull10}{\hidepaste} 1708\tab{5}\spadcommand{q} 1709\indentrel{3}\begin{verbatim} 1710 (9) q 1711 Type: Variable(q) 1712\end{verbatim} 1713\indentrel{-3}\end{paste}\end{patch} 1714 1715\begin{patch}{ugIntroAssignPageEmpty10} 1716\begin{paste}{ugIntroAssignPageEmpty10}{ugIntroAssignPagePatch10} 1717\pastebutton{ugIntroAssignPageEmpty10}{\showpaste} 1718\tab{5}\spadcommand{q} 1719\end{paste}\end{patch} 1720 1721\begin{patch}{ugIntroAssignPagePatch11} 1722\begin{paste}{ugIntroAssignPageFull11}{ugIntroAssignPageEmpty11} 1723\pastebutton{ugIntroAssignPageFull11}{\hidepaste} 1724\tab{5}\spadcommand{[q, r]} 1725\indentrel{3}\begin{verbatim} 1726 (10) [q, r] 1727 Type: List(OrderedVariableList([q,r])) 1728\end{verbatim} 1729\indentrel{-3}\end{paste}\end{patch} 1730 1731\begin{patch}{ugIntroAssignPageEmpty11} 1732\begin{paste}{ugIntroAssignPageEmpty11}{ugIntroAssignPagePatch11} 1733\pastebutton{ugIntroAssignPageEmpty11}{\showpaste} 1734\tab{5}\spadcommand{[q, r]} 1735\end{paste}\end{patch} 1736 1737\begin{patch}{ugIntroAssignPagePatch12} 1738\begin{paste}{ugIntroAssignPageFull12}{ugIntroAssignPageEmpty12} 1739\pastebutton{ugIntroAssignPageFull12}{\hidepaste} 1740\tab{5}\spadcommand{f\free{fff }} 1741\indentrel{3}\begin{verbatim} 1742 (11) 0.6666666666_6666666667 1743 Type: Float 1744\end{verbatim} 1745\indentrel{-3}\end{paste}\end{patch} 1746 1747\begin{patch}{ugIntroAssignPageEmpty12} 1748\begin{paste}{ugIntroAssignPageEmpty12}{ugIntroAssignPagePatch12} 1749\pastebutton{ugIntroAssignPageEmpty12}{\showpaste} 1750\tab{5}\spadcommand{f\free{fff }} 1751\end{paste}\end{patch} 1752 1753\begin{patch}{ugIntroAssignPagePatch13} 1754\begin{paste}{ugIntroAssignPageFull13}{ugIntroAssignPageEmpty13} 1755\pastebutton{ugIntroAssignPageFull13}{\hidepaste} 1756\tab{5}\spadcommand{'f} 1757\indentrel{3}\begin{verbatim} 1758 (12) f 1759 Type: Variable(f) 1760\end{verbatim} 1761\indentrel{-3}\end{paste}\end{patch} 1762 1763\begin{patch}{ugIntroAssignPageEmpty13} 1764\begin{paste}{ugIntroAssignPageEmpty13}{ugIntroAssignPagePatch13} 1765\pastebutton{ugIntroAssignPageEmpty13}{\showpaste} 1766\tab{5}\spadcommand{'f} 1767\end{paste}\end{patch} 1768 1769\begin{patch}{ugIntroSeriesPagePatch1} 1770\begin{paste}{ugIntroSeriesPageFull1}{ugIntroSeriesPageEmpty1} 1771\pastebutton{ugIntroSeriesPageFull1}{\hidepaste} 1772\tab{5}\spadcommand{series(sin(a*x),x = 0)} 1773\indentrel{3}\begin{verbatim} 1774 (1) 1775 3 5 7 9 1776 a 3 a 5 a 7 a 9 1777 a x - -- x + --- x - ---- x + ------ x 1778 6 120 5040 362880 1779 + 1780 11 1781 a 11 12 1782 - -------- x + O(x ) 1783 39916800 1784 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 1785\end{verbatim} 1786\indentrel{-3}\end{paste}\end{patch} 1787 1788\begin{patch}{ugIntroSeriesPageEmpty1} 1789\begin{paste}{ugIntroSeriesPageEmpty1}{ugIntroSeriesPagePatch1} 1790\pastebutton{ugIntroSeriesPageEmpty1}{\showpaste} 1791\tab{5}\spadcommand{series(sin(a*x),x = 0)} 1792\end{paste}\end{patch} 1793 1794\begin{patch}{ugIntroSeriesPagePatch2} 1795\begin{paste}{ugIntroSeriesPageFull2}{ugIntroSeriesPageEmpty2} 1796\pastebutton{ugIntroSeriesPageFull2}{\hidepaste} 1797\tab{5}\spadcommand{series(sin(a*x),x = \%pi/4)} 1798\indentrel{3}\begin{verbatim} 1799 (2) 1800 a %pi a %pi %pi 1801 sin(-----) + a cos(-----)(x - ---) 1802 4 4 4 1803 + 1804 2 a %pi 3 a %pi 1805 a sin(-----) a cos(-----) 1806 4 %pi 2 4 %pi 3 1807 - ------------ (x - ---) - ------------ (x - ---) 1808 2 4 6 4 1809 + 1810 4 a %pi 5 a %pi 1811 a sin(-----) a cos(-----) 1812 4 %pi 4 4 %pi 5 1813 ------------ (x - ---) + ------------ (x - ---) 1814 24 4 120 4 1815 + 1816 6 a %pi 7 a %pi 1817 a sin(-----) a cos(-----) 1818 4 %pi 6 4 %pi 7 1819 - ------------ (x - ---) - ------------ (x - ---) 1820 720 4 5040 4 1821 + 1822 8 a %pi 9 a %pi 1823 a sin(-----) a cos(-----) 1824 4 %pi 8 4 %pi 9 1825 ------------ (x - ---) + ------------ (x - ---) 1826 40320 4 362880 4 1827 + 1828 10 a %pi 1829 a sin(-----) 1830 4 %pi 10 %pi 11 1831 - ------------- (x - ---) + O((x - ---) ) 1832 3628800 4 4 1833Type: UnivariatePuiseuxSeries(Expression(Integer),x,%pi/4) 1834\end{verbatim} 1835\indentrel{-3}\end{paste}\end{patch} 1836 1837\begin{patch}{ugIntroSeriesPageEmpty2} 1838\begin{paste}{ugIntroSeriesPageEmpty2}{ugIntroSeriesPagePatch2} 1839\pastebutton{ugIntroSeriesPageEmpty2}{\showpaste} 1840\tab{5}\spadcommand{series(sin(a*x),x = \%pi/4)} 1841\end{paste}\end{patch} 1842 1843\begin{patch}{ugIntroSeriesPagePatch3} 1844\begin{paste}{ugIntroSeriesPageFull3}{ugIntroSeriesPageEmpty3} 1845\pastebutton{ugIntroSeriesPageFull3}{\hidepaste} 1846\tab{5}\spadcommand{series(n +-> (-1)^((3*n - 4)/6)/factorial(n - 1/3),x = 0,4/3..,2)} 1847\indentrel{3}\begin{verbatim} 1848 4 10 1849 - -- 1850 3 1 3 5 1851 (3) x - - x + O(x ) 1852 6 1853 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 1854\end{verbatim} 1855\indentrel{-3}\end{paste}\end{patch} 1856 1857\begin{patch}{ugIntroSeriesPageEmpty3} 1858\begin{paste}{ugIntroSeriesPageEmpty3}{ugIntroSeriesPagePatch3} 1859\pastebutton{ugIntroSeriesPageEmpty3}{\showpaste} 1860\tab{5}\spadcommand{series(n +-> (-1)^((3*n - 4)/6)/factorial(n - 1/3),x = 0,4/3..,2)} 1861\end{paste}\end{patch} 1862 1863\begin{patch}{ugIntroSeriesPagePatch4} 1864\begin{paste}{ugIntroSeriesPageFull4}{ugIntroSeriesPageEmpty4} 1865\pastebutton{ugIntroSeriesPageFull4}{\hidepaste} 1866\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f }} 1867\indentrel{3}\begin{verbatim} 1868 (4) 1869 2 3 4 5 6 7 8 9 10 1870 1 + x + x + x + x + x + x + x + x + x + x 1871 + 1872 11 1873 O(x ) 1874 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 1875\end{verbatim} 1876\indentrel{-3}\end{paste}\end{patch} 1877 1878\begin{patch}{ugIntroSeriesPageEmpty4} 1879\begin{paste}{ugIntroSeriesPageEmpty4}{ugIntroSeriesPagePatch4} 1880\pastebutton{ugIntroSeriesPageEmpty4}{\showpaste} 1881\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f }} 1882\end{paste}\end{patch} 1883 1884\begin{patch}{ugIntroSeriesPagePatch5} 1885\begin{paste}{ugIntroSeriesPageFull5}{ugIntroSeriesPageEmpty5} 1886\pastebutton{ugIntroSeriesPageFull5}{\hidepaste} 1887\tab{5}\spadcommand{f ^ 2\free{f }} 1888\indentrel{3}\begin{verbatim} 1889 (5) 1890 2 3 4 5 6 7 1891 1 + 2 x + 3 x + 4 x + 5 x + 6 x + 7 x + 8 x 1892 + 1893 8 9 10 11 1894 9 x + 10 x + 11 x + O(x ) 1895 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 1896\end{verbatim} 1897\indentrel{-3}\end{paste}\end{patch} 1898 1899\begin{patch}{ugIntroSeriesPageEmpty5} 1900\begin{paste}{ugIntroSeriesPageEmpty5}{ugIntroSeriesPagePatch5} 1901\pastebutton{ugIntroSeriesPageEmpty5}{\showpaste} 1902\tab{5}\spadcommand{f ^ 2\free{f }} 1903\end{paste}\end{patch} 1904 1905\begin{patch}{ugIntroSeriesPagePatch6} 1906\begin{paste}{ugIntroSeriesPageFull6}{ugIntroSeriesPageEmpty6} 1907\pastebutton{ugIntroSeriesPageFull6}{\hidepaste} 1908\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f1 }} 1909\indentrel{3}\begin{verbatim} 1910 (6) 1911 2 3 4 5 6 7 8 9 10 1912 1 + x + x + x + x + x + x + x + x + x + x 1913 + 1914 11 1915 O(x ) 1916 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 1917\end{verbatim} 1918\indentrel{-3}\end{paste}\end{patch} 1919 1920\begin{patch}{ugIntroSeriesPageEmpty6} 1921\begin{paste}{ugIntroSeriesPageEmpty6}{ugIntroSeriesPagePatch6} 1922\pastebutton{ugIntroSeriesPageEmpty6}{\showpaste} 1923\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f1 }} 1924\end{paste}\end{patch} 1925 1926\begin{patch}{ugIntroSeriesPagePatch7} 1927\begin{paste}{ugIntroSeriesPageFull7}{ugIntroSeriesPageEmpty7} 1928\pastebutton{ugIntroSeriesPageFull7}{\hidepaste} 1929\tab{5}\spadcommand{g := log(f)\free{f1 }\bound{g }} 1930\indentrel{3}\begin{verbatim} 1931 (7) 1932 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1933 x + - x + - x + - x + - x + - x + - x + - x 1934 2 3 4 5 6 7 8 1935 + 1936 1 9 1 10 1 11 12 1937 - x + -- x + -- x + O(x ) 1938 9 10 11 1939 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 1940\end{verbatim} 1941\indentrel{-3}\end{paste}\end{patch} 1942 1943\begin{patch}{ugIntroSeriesPageEmpty7} 1944\begin{paste}{ugIntroSeriesPageEmpty7}{ugIntroSeriesPagePatch7} 1945\pastebutton{ugIntroSeriesPageEmpty7}{\showpaste} 1946\tab{5}\spadcommand{g := log(f)\free{f1 }\bound{g }} 1947\end{paste}\end{patch} 1948 1949\begin{patch}{ugIntroSeriesPagePatch8} 1950\begin{paste}{ugIntroSeriesPageFull8}{ugIntroSeriesPageEmpty8} 1951\pastebutton{ugIntroSeriesPageFull8}{\hidepaste} 1952\tab{5}\spadcommand{exp(g)\free{g }} 1953\indentrel{3}\begin{verbatim} 1954 (8) 1955 2 3 4 5 6 7 8 9 10 1956 1 + x + x + x + x + x + x + x + x + x + x 1957 + 1958 11 1959 O(x ) 1960 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 1961\end{verbatim} 1962\indentrel{-3}\end{paste}\end{patch} 1963 1964\begin{patch}{ugIntroSeriesPageEmpty8} 1965\begin{paste}{ugIntroSeriesPageEmpty8}{ugIntroSeriesPagePatch8} 1966\pastebutton{ugIntroSeriesPageEmpty8}{\showpaste} 1967\tab{5}\spadcommand{exp(g)\free{g }} 1968\end{paste}\end{patch} 1969 1970\begin{patch}{ugIntroSeriesPagePatch9} 1971\begin{paste}{ugIntroSeriesPageFull9}{ugIntroSeriesPageEmpty9} 1972\pastebutton{ugIntroSeriesPageFull9}{\hidepaste} 1973\tab{5}\spadcommand{f := taylor(exp(x))\bound{f2 }} 1974\indentrel{3}\begin{verbatim} 1975 (9) 1976 1 2 1 3 1 4 1 5 1 6 1977 1 + x + - x + - x + -- x + --- x + --- x 1978 2 6 24 120 720 1979 + 1980 1 7 1 8 1 9 1 10 11 1981 ---- x + ----- x + ------ x + ------- x + O(x ) 1982 5040 40320 362880 3628800 1983 Type: UnivariateTaylorSeries(Expression(Integer),x,0) 1984\end{verbatim} 1985\indentrel{-3}\end{paste}\end{patch} 1986 1987\begin{patch}{ugIntroSeriesPageEmpty9} 1988\begin{paste}{ugIntroSeriesPageEmpty9}{ugIntroSeriesPagePatch9} 1989\pastebutton{ugIntroSeriesPageEmpty9}{\showpaste} 1990\tab{5}\spadcommand{f := taylor(exp(x))\bound{f2 }} 1991\end{paste}\end{patch} 1992 1993\begin{patch}{ugIntroSeriesPagePatch10} 1994\begin{paste}{ugIntroSeriesPageFull10}{ugIntroSeriesPageEmpty10} 1995\pastebutton{ugIntroSeriesPageFull10}{\hidepaste} 1996\tab{5}\spadcommand{eval(f,1.0)\free{f2 }} 1997\indentrel{3}\begin{verbatim} 1998 (10) 1999 [1.0, 2.0, 2.5, 2.6666666666_666666667, 2000 2.7083333333_333333333, 2.7166666666_666666667, 2001 2.7180555555_555555556, 2.7182539682_53968254, 2002 2.7182787698_412698413, 2.7182815255_731922399, ...] 2003 Type: Stream(Expression(Float)) 2004\end{verbatim} 2005\indentrel{-3}\end{paste}\end{patch} 2006 2007\begin{patch}{ugIntroSeriesPageEmpty10} 2008\begin{paste}{ugIntroSeriesPageEmpty10}{ugIntroSeriesPagePatch10} 2009\pastebutton{ugIntroSeriesPageEmpty10}{\showpaste} 2010\tab{5}\spadcommand{eval(f,1.0)\free{f2 }} 2011\end{paste}\end{patch} 2012 2013\begin{patch}{ugIntroCallFunPagePatch1} 2014\begin{paste}{ugIntroCallFunPageFull1}{ugIntroCallFunPageEmpty1} 2015\pastebutton{ugIntroCallFunPageFull1}{\hidepaste} 2016\tab{5}\spadcommand{factor(120)} 2017\indentrel{3}\begin{verbatim} 2018 3 2019 (1) 2 3 5 2020 Type: Factored(Integer) 2021\end{verbatim} 2022\indentrel{-3}\end{paste}\end{patch} 2023 2024\begin{patch}{ugIntroCallFunPageEmpty1} 2025\begin{paste}{ugIntroCallFunPageEmpty1}{ugIntroCallFunPagePatch1} 2026\pastebutton{ugIntroCallFunPageEmpty1}{\showpaste} 2027\tab{5}\spadcommand{factor(120)} 2028\end{paste}\end{patch} 2029 2030\begin{patch}{ugIntroCallFunPagePatch2} 2031\begin{paste}{ugIntroCallFunPageFull2}{ugIntroCallFunPageEmpty2} 2032\pastebutton{ugIntroCallFunPageFull2}{\hidepaste} 2033\tab{5}\spadcommand{divide(125,7)} 2034\indentrel{3}\begin{verbatim} 2035 (2) [quotient = 17, remainder = 6] 2036 Type: Record(quotient: Integer,remainder: Integer) 2037\end{verbatim} 2038\indentrel{-3}\end{paste}\end{patch} 2039 2040\begin{patch}{ugIntroCallFunPageEmpty2} 2041\begin{paste}{ugIntroCallFunPageEmpty2}{ugIntroCallFunPagePatch2} 2042\pastebutton{ugIntroCallFunPageEmpty2}{\showpaste} 2043\tab{5}\spadcommand{divide(125,7)} 2044\end{paste}\end{patch} 2045 2046\begin{patch}{ugIntroCallFunPagePatch3} 2047\begin{paste}{ugIntroCallFunPageFull3}{ugIntroCallFunPageEmpty3} 2048\pastebutton{ugIntroCallFunPageFull3}{\hidepaste} 2049\tab{5}\spadcommand{quatern(3.4,5.6,2.9,0.1)} 2050\indentrel{3}\begin{verbatim} 2051 (3) 3.4 + 5.6 i + 2.9 j + 0.1 k 2052 Type: Quaternion(Float) 2053\end{verbatim} 2054\indentrel{-3}\end{paste}\end{patch} 2055 2056\begin{patch}{ugIntroCallFunPageEmpty3} 2057\begin{paste}{ugIntroCallFunPageEmpty3}{ugIntroCallFunPagePatch3} 2058\pastebutton{ugIntroCallFunPageEmpty3}{\showpaste} 2059\tab{5}\spadcommand{quatern(3.4,5.6,2.9,0.1)} 2060\end{paste}\end{patch} 2061 2062\begin{patch}{ugIntroCallFunPagePatch4} 2063\begin{paste}{ugIntroCallFunPageFull4}{ugIntroCallFunPageEmpty4} 2064\pastebutton{ugIntroCallFunPageFull4}{\hidepaste} 2065\tab{5}\spadcommand{factorial 10} 2066\indentrel{3}\begin{verbatim} 2067 (4) 3628800 2068 Type: PositiveInteger 2069\end{verbatim} 2070\indentrel{-3}\end{paste}\end{patch} 2071 2072\begin{patch}{ugIntroCallFunPageEmpty4} 2073\begin{paste}{ugIntroCallFunPageEmpty4}{ugIntroCallFunPagePatch4} 2074\pastebutton{ugIntroCallFunPageEmpty4}{\showpaste} 2075\tab{5}\spadcommand{factorial 10} 2076\end{paste}\end{patch} 2077 2078\begin{patch}{ugIntroCollectPagePatch1} 2079\begin{paste}{ugIntroCollectPageFull1}{ugIntroCollectPageEmpty1} 2080\pastebutton{ugIntroCollectPageFull1}{\hidepaste} 2081\tab{5}\spadcommand{u := [1,-7,11]\bound{u }} 2082\indentrel{3}\begin{verbatim} 2083 (1) [1, - 7, 11] 2084 Type: List(Integer) 2085\end{verbatim} 2086\indentrel{-3}\end{paste}\end{patch} 2087 2088\begin{patch}{ugIntroCollectPageEmpty1} 2089\begin{paste}{ugIntroCollectPageEmpty1}{ugIntroCollectPagePatch1} 2090\pastebutton{ugIntroCollectPageEmpty1}{\showpaste} 2091\tab{5}\spadcommand{u := [1,-7,11]\bound{u }} 2092\end{paste}\end{patch} 2093 2094\begin{patch}{ugIntroCollectPagePatch2} 2095\begin{paste}{ugIntroCollectPageFull2}{ugIntroCollectPageEmpty2} 2096\pastebutton{ugIntroCollectPageFull2}{\hidepaste} 2097\tab{5}\spadcommand{first rest rest u\free{u }} 2098\indentrel{3}\begin{verbatim} 2099 (2) 11 2100 Type: PositiveInteger 2101\end{verbatim} 2102\indentrel{-3}\end{paste}\end{patch} 2103 2104\begin{patch}{ugIntroCollectPageEmpty2} 2105\begin{paste}{ugIntroCollectPageEmpty2}{ugIntroCollectPagePatch2} 2106\pastebutton{ugIntroCollectPageEmpty2}{\showpaste} 2107\tab{5}\spadcommand{first rest rest u\free{u }} 2108\end{paste}\end{patch} 2109 2110\begin{patch}{ugIntroCollectPagePatch3} 2111\begin{paste}{ugIntroCollectPageFull3}{ugIntroCollectPageEmpty3} 2112\pastebutton{ugIntroCollectPageFull3}{\hidepaste} 2113\tab{5}\spadcommand{concat!(u,[9,1,3,-4]); u\free{u }\bound{u1 }} 2114\indentrel{3}\begin{verbatim} 2115 (3) [1, - 7, 11, 9, 1, 3, - 4] 2116 Type: List(Integer) 2117\end{verbatim} 2118\indentrel{-3}\end{paste}\end{patch} 2119 2120\begin{patch}{ugIntroCollectPageEmpty3} 2121\begin{paste}{ugIntroCollectPageEmpty3}{ugIntroCollectPagePatch3} 2122\pastebutton{ugIntroCollectPageEmpty3}{\showpaste} 2123\tab{5}\spadcommand{concat!(u,[9,1,3,-4]); u\free{u }\bound{u1 }} 2124\end{paste}\end{patch} 2125 2126\begin{patch}{ugIntroCollectPagePatch4} 2127\begin{paste}{ugIntroCollectPageFull4}{ugIntroCollectPageEmpty4} 2128\pastebutton{ugIntroCollectPageFull4}{\hidepaste} 2129\tab{5}\spadcommand{lastnode := rest(u,3)\free{u1 }\bound{u2 }} 2130\indentrel{3}\begin{verbatim} 2131 (4) [9, 1, 3, - 4] 2132 Type: List(Integer) 2133\end{verbatim} 2134\indentrel{-3}\end{paste}\end{patch} 2135 2136\begin{patch}{ugIntroCollectPageEmpty4} 2137\begin{paste}{ugIntroCollectPageEmpty4}{ugIntroCollectPagePatch4} 2138\pastebutton{ugIntroCollectPageEmpty4}{\showpaste} 2139\tab{5}\spadcommand{lastnode := rest(u,3)\free{u1 }\bound{u2 }} 2140\end{paste}\end{patch} 2141 2142\begin{patch}{ugIntroCollectPagePatch5} 2143\begin{paste}{ugIntroCollectPageFull5}{ugIntroCollectPageEmpty5} 2144\pastebutton{ugIntroCollectPageFull5}{\hidepaste} 2145\tab{5}\spadcommand{setrest!(lastnode,rest(u,2)); u\free{u2 }} 2146\indentrel{3}\begin{verbatim} 2147 _____ 2148 (5) [1, - 7, 11, 9] 2149 Type: List(Integer) 2150\end{verbatim} 2151\indentrel{-3}\end{paste}\end{patch} 2152 2153\begin{patch}{ugIntroCollectPageEmpty5} 2154\begin{paste}{ugIntroCollectPageEmpty5}{ugIntroCollectPagePatch5} 2155\pastebutton{ugIntroCollectPageEmpty5}{\showpaste} 2156\tab{5}\spadcommand{setrest!(lastnode,rest(u,2)); u\free{u2 }} 2157\end{paste}\end{patch} 2158 2159\begin{patch}{ugIntroCollectPagePatch6} 2160\begin{paste}{ugIntroCollectPageFull6}{ugIntroCollectPageEmpty6} 2161\pastebutton{ugIntroCollectPageFull6}{\hidepaste} 2162\tab{5}\spadcommand{[factor(i) for i in 2.. by 2]\bound{stream1 }} 2163\indentrel{3}\begin{verbatim} 2164 (6) 2165 2 3 2 4 2 2 2166 [2, 2 , 2 3, 2 , 2 5, 2 3, 2 7, 2 , 2 3 , 2 5, ...] 2167 Type: Stream(Factored(Integer)) 2168\end{verbatim} 2169\indentrel{-3}\end{paste}\end{patch} 2170 2171\begin{patch}{ugIntroCollectPageEmpty6} 2172\begin{paste}{ugIntroCollectPageEmpty6}{ugIntroCollectPagePatch6} 2173\pastebutton{ugIntroCollectPageEmpty6}{\showpaste} 2174\tab{5}\spadcommand{[factor(i) for i in 2.. by 2]\bound{stream1 }} 2175\end{paste}\end{patch} 2176 2177\begin{patch}{ugIntroCollectPagePatch7} 2178\begin{paste}{ugIntroCollectPageFull7}{ugIntroCollectPageEmpty7} 2179\pastebutton{ugIntroCollectPageFull7}{\hidepaste} 2180\tab{5}\spadcommand{\%.36\free{stream1 }} 2181\indentrel{3}\begin{verbatim} 2182 3 2 2183 (7) 2 3 2184 Type: Factored(Integer) 2185\end{verbatim} 2186\indentrel{-3}\end{paste}\end{patch} 2187 2188\begin{patch}{ugIntroCollectPageEmpty7} 2189\begin{paste}{ugIntroCollectPageEmpty7}{ugIntroCollectPagePatch7} 2190\pastebutton{ugIntroCollectPageEmpty7}{\showpaste} 2191\tab{5}\spadcommand{\%.36\free{stream1 }} 2192\end{paste}\end{patch} 2193 2194\begin{patch}{ugIntroCollectPagePatch8} 2195\begin{paste}{ugIntroCollectPageFull8}{ugIntroCollectPageEmpty8} 2196\pastebutton{ugIntroCollectPageFull8}{\hidepaste} 2197\tab{5}\spadcommand{a := oneDimensionalArray [1, -7, 3, 3/2]\bound{a }} 2198\indentrel{3}\begin{verbatim} 2199 3 2200 (8) [1, - 7, 3, -] 2201 2 2202 Type: OneDimensionalArray(Fraction(Integer)) 2203\end{verbatim} 2204\indentrel{-3}\end{paste}\end{patch} 2205 2206\begin{patch}{ugIntroCollectPageEmpty8} 2207\begin{paste}{ugIntroCollectPageEmpty8}{ugIntroCollectPagePatch8} 2208\pastebutton{ugIntroCollectPageEmpty8}{\showpaste} 2209\tab{5}\spadcommand{a := oneDimensionalArray [1, -7, 3, 3/2]\bound{a }} 2210\end{paste}\end{patch} 2211 2212\begin{patch}{ugIntroCollectPagePatch9} 2213\begin{paste}{ugIntroCollectPageFull9}{ugIntroCollectPageEmpty9} 2214\pastebutton{ugIntroCollectPageFull9}{\hidepaste} 2215\tab{5}\spadcommand{a.3 := 11; a\bound{a1 }\free{a }} 2216\indentrel{3}\begin{verbatim} 2217 3 2218 (9) [1, - 7, 11, -] 2219 2 2220 Type: OneDimensionalArray(Fraction(Integer)) 2221\end{verbatim} 2222\indentrel{-3}\end{paste}\end{patch} 2223 2224\begin{patch}{ugIntroCollectPageEmpty9} 2225\begin{paste}{ugIntroCollectPageEmpty9}{ugIntroCollectPagePatch9} 2226\pastebutton{ugIntroCollectPageEmpty9}{\showpaste} 2227\tab{5}\spadcommand{a.3 := 11; a\bound{a1 }\free{a }} 2228\end{paste}\end{patch} 2229 2230\begin{patch}{ugIntroCollectPagePatch10} 2231\begin{paste}{ugIntroCollectPageFull10}{ugIntroCollectPageEmpty10} 2232\pastebutton{ugIntroCollectPageFull10}{\hidepaste} 2233\tab{5}\spadcommand{concat!(a,oneDimensionalArray [1,-2])\free{a1 }} 2234\indentrel{3}\begin{verbatim} 2235error 2236\end{verbatim} 2237\indentrel{-3}\end{paste}\end{patch} 2238 2239\begin{patch}{ugIntroCollectPageEmpty10} 2240\begin{paste}{ugIntroCollectPageEmpty10}{ugIntroCollectPagePatch10} 2241\pastebutton{ugIntroCollectPageEmpty10}{\showpaste} 2242\tab{5}\spadcommand{concat!(a,oneDimensionalArray [1,-2])\free{a1 }} 2243\end{paste}\end{patch} 2244 2245\begin{patch}{ugIntroCollectPagePatch11} 2246\begin{paste}{ugIntroCollectPageFull11}{ugIntroCollectPageEmpty11} 2247\pastebutton{ugIntroCollectPageFull11}{\hidepaste} 2248\tab{5}\spadcommand{bits(32,true)} 2249\indentrel{3}\begin{verbatim} 2250 (10) "11111111111111111111111111111111" 2251 Type: Bits 2252\end{verbatim} 2253\indentrel{-3}\end{paste}\end{patch} 2254 2255\begin{patch}{ugIntroCollectPageEmpty11} 2256\begin{paste}{ugIntroCollectPageEmpty11}{ugIntroCollectPagePatch11} 2257\pastebutton{ugIntroCollectPageEmpty11}{\showpaste} 2258\tab{5}\spadcommand{bits(32,true)} 2259\end{paste}\end{patch} 2260 2261\begin{patch}{ugIntroCollectPagePatch12} 2262\begin{paste}{ugIntroCollectPageFull12}{ugIntroCollectPageEmpty12} 2263\pastebutton{ugIntroCollectPageFull12}{\hidepaste} 2264\tab{5}\spadcommand{f := flexibleArray [2, 7, -5]\bound{f }} 2265\indentrel{3}\begin{verbatim} 2266 (11) [2, 7, - 5] 2267 Type: FlexibleArray(Integer) 2268\end{verbatim} 2269\indentrel{-3}\end{paste}\end{patch} 2270 2271\begin{patch}{ugIntroCollectPageEmpty12} 2272\begin{paste}{ugIntroCollectPageEmpty12}{ugIntroCollectPagePatch12} 2273\pastebutton{ugIntroCollectPageEmpty12}{\showpaste} 2274\tab{5}\spadcommand{f := flexibleArray [2, 7, -5]\bound{f }} 2275\end{paste}\end{patch} 2276 2277\begin{patch}{ugIntroCollectPagePatch13} 2278\begin{paste}{ugIntroCollectPageFull13}{ugIntroCollectPageEmpty13} 2279\pastebutton{ugIntroCollectPageFull13}{\hidepaste} 2280\tab{5}\spadcommand{insert!(flexibleArray [11, -3],f,3)\free{f }} 2281\indentrel{3}\begin{verbatim} 2282 (12) [2, 7, 11, - 3, - 5] 2283 Type: FlexibleArray(Integer) 2284\end{verbatim} 2285\indentrel{-3}\end{paste}\end{patch} 2286 2287\begin{patch}{ugIntroCollectPageEmpty13} 2288\begin{paste}{ugIntroCollectPageEmpty13}{ugIntroCollectPagePatch13} 2289\pastebutton{ugIntroCollectPageEmpty13}{\showpaste} 2290\tab{5}\spadcommand{insert!(flexibleArray [11, -3],f,3)\free{f }} 2291\end{paste}\end{patch} 2292 2293\begin{patch}{ugIntroCollectPagePatch14} 2294\begin{paste}{ugIntroCollectPageFull14}{ugIntroCollectPageEmpty14} 2295\pastebutton{ugIntroCollectPageFull14}{\hidepaste} 2296\tab{5}\spadcommand{h := heap [-4,7,11,3,4,-7]\bound{h }} 2297\indentrel{3}\begin{verbatim} 2298 (13) [11, 7, - 4, 3, 4, - 7] 2299 Type: Heap(Integer) 2300\end{verbatim} 2301\indentrel{-3}\end{paste}\end{patch} 2302 2303\begin{patch}{ugIntroCollectPageEmpty14} 2304\begin{paste}{ugIntroCollectPageEmpty14}{ugIntroCollectPagePatch14} 2305\pastebutton{ugIntroCollectPageEmpty14}{\showpaste} 2306\tab{5}\spadcommand{h := heap [-4,7,11,3,4,-7]\bound{h }} 2307\end{paste}\end{patch} 2308 2309\begin{patch}{ugIntroCollectPagePatch15} 2310\begin{paste}{ugIntroCollectPageFull15}{ugIntroCollectPageEmpty15} 2311\pastebutton{ugIntroCollectPageFull15}{\hidepaste} 2312\tab{5}\spadcommand{[extract!(h) while not empty?(h)]\free{h }} 2313\indentrel{3}\begin{verbatim} 2314 (14) [11, 7, 4, 3, - 4, - 7] 2315 Type: List(Integer) 2316\end{verbatim} 2317\indentrel{-3}\end{paste}\end{patch} 2318 2319\begin{patch}{ugIntroCollectPageEmpty15} 2320\begin{paste}{ugIntroCollectPageEmpty15}{ugIntroCollectPagePatch15} 2321\pastebutton{ugIntroCollectPageEmpty15}{\showpaste} 2322\tab{5}\spadcommand{[extract!(h) while not empty?(h)]\free{h }} 2323\end{paste}\end{patch} 2324 2325\begin{patch}{ugIntroCollectPagePatch16} 2326\begin{paste}{ugIntroCollectPageFull16}{ugIntroCollectPageEmpty16} 2327\pastebutton{ugIntroCollectPageFull16}{\hidepaste} 2328\tab{5}\spadcommand{binarySearchTree [5,3,2,9,4,7,11]} 2329\indentrel{3}\begin{verbatim} 2330 (15) [[2, 3, 4], 5, [7, 9, 11]] 2331 Type: BinarySearchTree(PositiveInteger) 2332\end{verbatim} 2333\indentrel{-3}\end{paste}\end{patch} 2334 2335\begin{patch}{ugIntroCollectPageEmpty16} 2336\begin{paste}{ugIntroCollectPageEmpty16}{ugIntroCollectPagePatch16} 2337\pastebutton{ugIntroCollectPageEmpty16}{\showpaste} 2338\tab{5}\spadcommand{binarySearchTree [5,3,2,9,4,7,11]} 2339\end{paste}\end{patch} 2340 2341\begin{patch}{ugIntroCollectPagePatch17} 2342\begin{paste}{ugIntroCollectPageFull17}{ugIntroCollectPageEmpty17} 2343\pastebutton{ugIntroCollectPageFull17}{\hidepaste} 2344\tab{5}\spadcommand{modTree(8,[2,3,5,7])} 2345\indentrel{3}\begin{verbatim} 2346 (16) [0, 2, 3, 1] 2347 Type: List(Integer) 2348\end{verbatim} 2349\indentrel{-3}\end{paste}\end{patch} 2350 2351\begin{patch}{ugIntroCollectPageEmpty17} 2352\begin{paste}{ugIntroCollectPageEmpty17}{ugIntroCollectPagePatch17} 2353\pastebutton{ugIntroCollectPageEmpty17}{\showpaste} 2354\tab{5}\spadcommand{modTree(8,[2,3,5,7])} 2355\end{paste}\end{patch} 2356 2357\begin{patch}{ugIntroCollectPagePatch18} 2358\begin{paste}{ugIntroCollectPageFull18}{ugIntroCollectPageEmpty18} 2359\pastebutton{ugIntroCollectPageFull18}{\hidepaste} 2360\tab{5}\spadcommand{fs := set[1/3,4/5,-1/3,4/5]\bound{fs }} 2361\indentrel{3}\begin{verbatim} 2362 1 1 4 2363 (17) {- -, -, -} 2364 3 3 5 2365 Type: Set(Fraction(Integer)) 2366\end{verbatim} 2367\indentrel{-3}\end{paste}\end{patch} 2368 2369\begin{patch}{ugIntroCollectPageEmpty18} 2370\begin{paste}{ugIntroCollectPageEmpty18}{ugIntroCollectPagePatch18} 2371\pastebutton{ugIntroCollectPageEmpty18}{\showpaste} 2372\tab{5}\spadcommand{fs := set[1/3,4/5,-1/3,4/5]\bound{fs }} 2373\end{paste}\end{patch} 2374 2375\begin{patch}{ugIntroCollectPagePatch19} 2376\begin{paste}{ugIntroCollectPageFull19}{ugIntroCollectPageEmpty19} 2377\pastebutton{ugIntroCollectPageFull19}{\hidepaste} 2378\tab{5}\spadcommand{multiset [x rem 5 for x in primes(2,1000)]} 2379\indentrel{3}\begin{verbatim} 2380 (18) {47 : 2, 42 : 3, 0, 40 : 1, 38 : 4} 2381 Type: Multiset(Integer) 2382\end{verbatim} 2383\indentrel{-3}\end{paste}\end{patch} 2384 2385\begin{patch}{ugIntroCollectPageEmpty19} 2386\begin{paste}{ugIntroCollectPageEmpty19}{ugIntroCollectPagePatch19} 2387\pastebutton{ugIntroCollectPageEmpty19}{\showpaste} 2388\tab{5}\spadcommand{multiset [x rem 5 for x in primes(2,1000)]} 2389\end{paste}\end{patch} 2390 2391\begin{patch}{ugIntroCollectPagePatch20} 2392\begin{paste}{ugIntroCollectPageFull20}{ugIntroCollectPageEmpty20} 2393\pastebutton{ugIntroCollectPageFull20}{\hidepaste} 2394\tab{5}\spadcommand{t : Table(Integer,Integer) := empty()\bound{t }} 2395\indentrel{3}\begin{verbatim} 2396 (19) table() 2397 Type: Table(Integer,Integer) 2398\end{verbatim} 2399\indentrel{-3}\end{paste}\end{patch} 2400 2401\begin{patch}{ugIntroCollectPageEmpty20} 2402\begin{paste}{ugIntroCollectPageEmpty20}{ugIntroCollectPagePatch20} 2403\pastebutton{ugIntroCollectPageEmpty20}{\showpaste} 2404\tab{5}\spadcommand{t : Table(Integer,Integer) := empty()\bound{t }} 2405\end{paste}\end{patch} 2406 2407\begin{patch}{ugIntroCollectPagePatch21} 2408\begin{paste}{ugIntroCollectPageFull21}{ugIntroCollectPageEmpty21} 2409\pastebutton{ugIntroCollectPageFull21}{\hidepaste} 2410\tab{5}\spadcommand{howMany(k) == (n:=search(k,t); n case "failed" => 1; n+1)\bound{how }} 2411\indentrel{3}\begin{verbatim} 2412 Type: Void 2413\end{verbatim} 2414\indentrel{-3}\end{paste}\end{patch} 2415 2416\begin{patch}{ugIntroCollectPageEmpty21} 2417\begin{paste}{ugIntroCollectPageEmpty21}{ugIntroCollectPagePatch21} 2418\pastebutton{ugIntroCollectPageEmpty21}{\showpaste} 2419\tab{5}\spadcommand{howMany(k) == (n:=search(k,t); n case "failed" => 1; n+1)\bound{how }} 2420\end{paste}\end{patch} 2421 2422\begin{patch}{ugIntroCollectPagePatch22} 2423\begin{paste}{ugIntroCollectPageFull22}{ugIntroCollectPageEmpty22} 2424\pastebutton{ugIntroCollectPageFull22}{\hidepaste} 2425\tab{5}\spadcommand{for p in primes(2,1000) repeat (m:= p rem 5; t.m:= howMany(m)); t\free{how t }} 2426\indentrel{3}\begin{verbatim} 2427 (21) table(4 = 38,1 = 40,0 = 1,3 = 42,2 = 47) 2428 Type: Table(Integer,Integer) 2429\end{verbatim} 2430\indentrel{-3}\end{paste}\end{patch} 2431 2432\begin{patch}{ugIntroCollectPageEmpty22} 2433\begin{paste}{ugIntroCollectPageEmpty22}{ugIntroCollectPagePatch22} 2434\pastebutton{ugIntroCollectPageEmpty22}{\showpaste} 2435\tab{5}\spadcommand{for p in primes(2,1000) repeat (m:= p rem 5; t.m:= howMany(m)); t\free{how t }} 2436\end{paste}\end{patch} 2437 2438\begin{patch}{ugIntroCollectPagePatch23} 2439\begin{paste}{ugIntroCollectPageFull23}{ugIntroCollectPageEmpty23} 2440\pastebutton{ugIntroCollectPageFull23}{\hidepaste} 2441\tab{5}\spadcommand{daniel : Record(age : Integer, salary : Float)\bound{danieldec }} 2442\indentrel{3}\begin{verbatim} 2443 Type: Void 2444\end{verbatim} 2445\indentrel{-3}\end{paste}\end{patch} 2446 2447\begin{patch}{ugIntroCollectPageEmpty23} 2448\begin{paste}{ugIntroCollectPageEmpty23}{ugIntroCollectPagePatch23} 2449\pastebutton{ugIntroCollectPageEmpty23}{\showpaste} 2450\tab{5}\spadcommand{daniel : Record(age : Integer, salary : Float)\bound{danieldec }} 2451\end{paste}\end{patch} 2452 2453\begin{patch}{ugIntroCollectPagePatch24} 2454\begin{paste}{ugIntroCollectPageFull24}{ugIntroCollectPageEmpty24} 2455\pastebutton{ugIntroCollectPageFull24}{\hidepaste} 2456\tab{5}\spadcommand{daniel := [28, 32005.12]\free{danieldec }\bound{daniel }} 2457\indentrel{3}\begin{verbatim} 2458 (23) [age = 28, salary = 32005.12] 2459 Type: Record(age: Integer,salary: Float) 2460\end{verbatim} 2461\indentrel{-3}\end{paste}\end{patch} 2462 2463\begin{patch}{ugIntroCollectPageEmpty24} 2464\begin{paste}{ugIntroCollectPageEmpty24}{ugIntroCollectPagePatch24} 2465\pastebutton{ugIntroCollectPageEmpty24}{\showpaste} 2466\tab{5}\spadcommand{daniel := [28, 32005.12]\free{danieldec }\bound{daniel }} 2467\end{paste}\end{patch} 2468 2469\begin{patch}{ugIntroCollectPagePatch25} 2470\begin{paste}{ugIntroCollectPageFull25}{ugIntroCollectPageEmpty25} 2471\pastebutton{ugIntroCollectPageFull25}{\hidepaste} 2472\tab{5}\spadcommand{daniel.salary := 35000; daniel\free{daniel }} 2473\indentrel{3}\begin{verbatim} 2474 (24) [age = 28, salary = 35000.0] 2475 Type: Record(age: Integer,salary: Float) 2476\end{verbatim} 2477\indentrel{-3}\end{paste}\end{patch} 2478 2479\begin{patch}{ugIntroCollectPageEmpty25} 2480\begin{paste}{ugIntroCollectPageEmpty25}{ugIntroCollectPagePatch25} 2481\pastebutton{ugIntroCollectPageEmpty25}{\showpaste} 2482\tab{5}\spadcommand{daniel.salary := 35000; daniel\free{daniel }} 2483\end{paste}\end{patch} 2484 2485\begin{patch}{ugIntroCollectPagePatch26} 2486\begin{paste}{ugIntroCollectPageFull26}{ugIntroCollectPageEmpty26} 2487\pastebutton{ugIntroCollectPageFull26}{\hidepaste} 2488\tab{5}\spadcommand{dog: Union(licenseNumber: Integer, name: String)\bound{xint }} 2489\indentrel{3}\begin{verbatim} 2490 Type: Void 2491\end{verbatim} 2492\indentrel{-3}\end{paste}\end{patch} 2493 2494\begin{patch}{ugIntroCollectPageEmpty26} 2495\begin{paste}{ugIntroCollectPageEmpty26}{ugIntroCollectPagePatch26} 2496\pastebutton{ugIntroCollectPageEmpty26}{\showpaste} 2497\tab{5}\spadcommand{dog: Union(licenseNumber: Integer, name: String)\bound{xint }} 2498\end{paste}\end{patch} 2499 2500\begin{patch}{ugIntroCollectPagePatch27} 2501\begin{paste}{ugIntroCollectPageFull27}{ugIntroCollectPageEmpty27} 2502\pastebutton{ugIntroCollectPageFull27}{\hidepaste} 2503\tab{5}\spadcommand{dog := "Whisper"\free{xint }} 2504\indentrel{3}\begin{verbatim} 2505 (26) "Whisper" 2506 Type: Union(name: String,...) 2507\end{verbatim} 2508\indentrel{-3}\end{paste}\end{patch} 2509 2510\begin{patch}{ugIntroCollectPageEmpty27} 2511\begin{paste}{ugIntroCollectPageEmpty27}{ugIntroCollectPagePatch27} 2512\pastebutton{ugIntroCollectPageEmpty27}{\showpaste} 2513\tab{5}\spadcommand{dog := "Whisper"\free{xint }} 2514\end{paste}\end{patch} 2515 2516\begin{patch}{ugIntroNumbersPagePatch1} 2517\begin{paste}{ugIntroNumbersPageFull1}{ugIntroNumbersPageEmpty1} 2518\pastebutton{ugIntroNumbersPageFull1}{\hidepaste} 2519\tab{5}\spadcommand{11^13 * 13^11 * 17^7 - 19^5 * 23^3} 2520\indentrel{3}\begin{verbatim} 2521 (1) 25387751112538918594666224484237298 2522 Type: PositiveInteger 2523\end{verbatim} 2524\indentrel{-3}\end{paste}\end{patch} 2525 2526\begin{patch}{ugIntroNumbersPageEmpty1} 2527\begin{paste}{ugIntroNumbersPageEmpty1}{ugIntroNumbersPagePatch1} 2528\pastebutton{ugIntroNumbersPageEmpty1}{\showpaste} 2529\tab{5}\spadcommand{11^13 * 13^11 * 17^7 - 19^5 * 23^3} 2530\end{paste}\end{patch} 2531 2532\begin{patch}{ugIntroNumbersPagePatch2} 2533\begin{paste}{ugIntroNumbersPageFull2}{ugIntroNumbersPageEmpty2} 2534\pastebutton{ugIntroNumbersPageFull2}{\hidepaste} 2535\tab{5}\spadcommand{factor 643238070748569023720594412551704344145570763243\bound{ex1 }} 2536\indentrel{3}\begin{verbatim} 2537 13 11 7 5 3 2 2538 (2) 11 13 17 19 23 29 2539 Type: Factored(Integer) 2540\end{verbatim} 2541\indentrel{-3}\end{paste}\end{patch} 2542 2543\begin{patch}{ugIntroNumbersPageEmpty2} 2544\begin{paste}{ugIntroNumbersPageEmpty2}{ugIntroNumbersPagePatch2} 2545\pastebutton{ugIntroNumbersPageEmpty2}{\showpaste} 2546\tab{5}\spadcommand{factor 643238070748569023720594412551704344145570763243\bound{ex1 }} 2547\end{paste}\end{patch} 2548 2549\begin{patch}{ugIntroNumbersPagePatch3} 2550\begin{paste}{ugIntroNumbersPageFull3}{ugIntroNumbersPageEmpty3} 2551\pastebutton{ugIntroNumbersPageFull3}{\hidepaste} 2552\tab{5}\spadcommand{\% * 12\free{ex1 }} 2553\indentrel{3}\begin{verbatim} 2554 2 13 11 7 5 3 2 2555 (3) 2 3 11 13 17 19 23 29 2556 Type: Factored(Integer) 2557\end{verbatim} 2558\indentrel{-3}\end{paste}\end{patch} 2559 2560\begin{patch}{ugIntroNumbersPageEmpty3} 2561\begin{paste}{ugIntroNumbersPageEmpty3}{ugIntroNumbersPagePatch3} 2562\pastebutton{ugIntroNumbersPageEmpty3}{\showpaste} 2563\tab{5}\spadcommand{\% * 12\free{ex1 }} 2564\end{paste}\end{patch} 2565 2566\begin{patch}{ugIntroNumbersPagePatch4} 2567\begin{paste}{ugIntroNumbersPageFull4}{ugIntroNumbersPageEmpty4} 2568\pastebutton{ugIntroNumbersPageFull4}{\hidepaste} 2569\tab{5}\spadcommand{radix(25937424601,11)} 2570\indentrel{3}\begin{verbatim} 2571 (4) 10000000000 2572 Type: RadixExpansion(11) 2573\end{verbatim} 2574\indentrel{-3}\end{paste}\end{patch} 2575 2576\begin{patch}{ugIntroNumbersPageEmpty4} 2577\begin{paste}{ugIntroNumbersPageEmpty4}{ugIntroNumbersPagePatch4} 2578\pastebutton{ugIntroNumbersPageEmpty4}{\showpaste} 2579\tab{5}\spadcommand{radix(25937424601,11)} 2580\end{paste}\end{patch} 2581 2582\begin{patch}{ugIntroNumbersPagePatch5} 2583\begin{paste}{ugIntroNumbersPageFull5}{ugIntroNumbersPageEmpty5} 2584\pastebutton{ugIntroNumbersPageFull5}{\hidepaste} 2585\tab{5}\spadcommand{roman(1992)} 2586\indentrel{3}\begin{verbatim} 2587 (5) MCMXCII 2588 Type: RomanNumeral 2589\end{verbatim} 2590\indentrel{-3}\end{paste}\end{patch} 2591 2592\begin{patch}{ugIntroNumbersPageEmpty5} 2593\begin{paste}{ugIntroNumbersPageEmpty5}{ugIntroNumbersPagePatch5} 2594\pastebutton{ugIntroNumbersPageEmpty5}{\showpaste} 2595\tab{5}\spadcommand{roman(1992)} 2596\end{paste}\end{patch} 2597 2598\begin{patch}{ugIntroNumbersPagePatch6} 2599\begin{paste}{ugIntroNumbersPageFull6}{ugIntroNumbersPageEmpty6} 2600\pastebutton{ugIntroNumbersPageFull6}{\hidepaste} 2601\tab{5}\spadcommand{r := 10 + 9/2 + 8/3 + 7/4 + 6/5 + 5/6 + 4/7 + 3/8 + 2/9\bound{r }} 2602\indentrel{3}\begin{verbatim} 2603 55739 2604 (6) ----- 2605 2520 2606 Type: Fraction(Integer) 2607\end{verbatim} 2608\indentrel{-3}\end{paste}\end{patch} 2609 2610\begin{patch}{ugIntroNumbersPageEmpty6} 2611\begin{paste}{ugIntroNumbersPageEmpty6}{ugIntroNumbersPagePatch6} 2612\pastebutton{ugIntroNumbersPageEmpty6}{\showpaste} 2613\tab{5}\spadcommand{r := 10 + 9/2 + 8/3 + 7/4 + 6/5 + 5/6 + 4/7 + 3/8 + 2/9\bound{r }} 2614\end{paste}\end{patch} 2615 2616\begin{patch}{ugIntroNumbersPagePatch7} 2617\begin{paste}{ugIntroNumbersPageFull7}{ugIntroNumbersPageEmpty7} 2618\pastebutton{ugIntroNumbersPageFull7}{\hidepaste} 2619\tab{5}\spadcommand{map(factor,r)\free{r }} 2620\indentrel{3}\begin{verbatim} 2621 139 401 2622 (7) ------- 2623 3 2 2624 2 3 5 7 2625 Type: Fraction(Factored(Integer)) 2626\end{verbatim} 2627\indentrel{-3}\end{paste}\end{patch} 2628 2629\begin{patch}{ugIntroNumbersPageEmpty7} 2630\begin{paste}{ugIntroNumbersPageEmpty7}{ugIntroNumbersPagePatch7} 2631\pastebutton{ugIntroNumbersPageEmpty7}{\showpaste} 2632\tab{5}\spadcommand{map(factor,r)\free{r }} 2633\end{paste}\end{patch} 2634 2635\begin{patch}{ugIntroNumbersPagePatch8} 2636\begin{paste}{ugIntroNumbersPageFull8}{ugIntroNumbersPageEmpty8} 2637\pastebutton{ugIntroNumbersPageFull8}{\hidepaste} 2638\tab{5}\spadcommand{11@SingleInteger} 2639\indentrel{3}\begin{verbatim} 2640 (8) 11 2641 Type: SingleInteger 2642\end{verbatim} 2643\indentrel{-3}\end{paste}\end{patch} 2644 2645\begin{patch}{ugIntroNumbersPageEmpty8} 2646\begin{paste}{ugIntroNumbersPageEmpty8}{ugIntroNumbersPagePatch8} 2647\pastebutton{ugIntroNumbersPageEmpty8}{\showpaste} 2648\tab{5}\spadcommand{11@SingleInteger} 2649\end{paste}\end{patch} 2650 2651\begin{patch}{ugIntroNumbersPagePatch9} 2652\begin{paste}{ugIntroNumbersPageFull9}{ugIntroNumbersPageEmpty9} 2653\pastebutton{ugIntroNumbersPageFull9}{\hidepaste} 2654\tab{5}\spadcommand{123.21@DoubleFloat} 2655\indentrel{3}\begin{verbatim} 2656 (9) 123.21000000000001 2657 Type: DoubleFloat 2658\end{verbatim} 2659\indentrel{-3}\end{paste}\end{patch} 2660 2661\begin{patch}{ugIntroNumbersPageEmpty9} 2662\begin{paste}{ugIntroNumbersPageEmpty9}{ugIntroNumbersPagePatch9} 2663\pastebutton{ugIntroNumbersPageEmpty9}{\showpaste} 2664\tab{5}\spadcommand{123.21@DoubleFloat} 2665\end{paste}\end{patch} 2666 2667\begin{patch}{ugIntroNumbersPagePatch10} 2668\begin{paste}{ugIntroNumbersPageFull10}{ugIntroNumbersPageEmpty10} 2669\pastebutton{ugIntroNumbersPageFull10}{\hidepaste} 2670\tab{5}\spadcommand{r :: Float\free{r }} 2671\indentrel{3}\begin{verbatim} 2672 (10) 22.1186507936_50793651 2673 Type: Float 2674\end{verbatim} 2675\indentrel{-3}\end{paste}\end{patch} 2676 2677\begin{patch}{ugIntroNumbersPageEmpty10} 2678\begin{paste}{ugIntroNumbersPageEmpty10}{ugIntroNumbersPagePatch10} 2679\pastebutton{ugIntroNumbersPageEmpty10}{\showpaste} 2680\tab{5}\spadcommand{r :: Float\free{r }} 2681\end{paste}\end{patch} 2682 2683\begin{patch}{ugIntroNumbersPagePatch11} 2684\begin{paste}{ugIntroNumbersPageFull11}{ugIntroNumbersPageEmpty11} 2685\pastebutton{ugIntroNumbersPageFull11}{\hidepaste} 2686\tab{5}\spadcommand{digits(22)\bound{fewerdigits }} 2687\indentrel{3}\begin{verbatim} 2688 (11) 20 2689 Type: PositiveInteger 2690\end{verbatim} 2691\indentrel{-3}\end{paste}\end{patch} 2692 2693\begin{patch}{ugIntroNumbersPageEmpty11} 2694\begin{paste}{ugIntroNumbersPageEmpty11}{ugIntroNumbersPagePatch11} 2695\pastebutton{ugIntroNumbersPageEmpty11}{\showpaste} 2696\tab{5}\spadcommand{digits(22)\bound{fewerdigits }} 2697\end{paste}\end{patch} 2698 2699\begin{patch}{ugIntroNumbersPagePatch12} 2700\begin{paste}{ugIntroNumbersPageFull12}{ugIntroNumbersPageEmpty12} 2701\pastebutton{ugIntroNumbersPageFull12}{\hidepaste} 2702\tab{5}\spadcommand{exp(\%pi * sqrt 163.0)\free{fewerdigits }} 2703\indentrel{3}\begin{verbatim} 2704 (12) 26253741_2640768744.0 2705 Type: Float 2706\end{verbatim} 2707\indentrel{-3}\end{paste}\end{patch} 2708 2709\begin{patch}{ugIntroNumbersPageEmpty12} 2710\begin{paste}{ugIntroNumbersPageEmpty12}{ugIntroNumbersPagePatch12} 2711\pastebutton{ugIntroNumbersPageEmpty12}{\showpaste} 2712\tab{5}\spadcommand{exp(\%pi * sqrt 163.0)\free{fewerdigits }} 2713\end{paste}\end{patch} 2714 2715\begin{patch}{ugIntroNumbersPagePatch13} 2716\begin{paste}{ugIntroNumbersPageFull13}{ugIntroNumbersPageEmpty13} 2717\pastebutton{ugIntroNumbersPageFull13}{\hidepaste} 2718\tab{5}\spadcommand{digits(40); exp(\%pi * sqrt 163.0)\free{moredigits }} 2719\indentrel{3}\begin{verbatim} 2720 (13) 26253741_2640768743.9999999999_9925007259_76 2721 Type: Float 2722\end{verbatim} 2723\indentrel{-3}\end{paste}\end{patch} 2724 2725\begin{patch}{ugIntroNumbersPageEmpty13} 2726\begin{paste}{ugIntroNumbersPageEmpty13}{ugIntroNumbersPagePatch13} 2727\pastebutton{ugIntroNumbersPageEmpty13}{\showpaste} 2728\tab{5}\spadcommand{digits(40); exp(\%pi * sqrt 163.0)\free{moredigits }} 2729\end{paste}\end{patch} 2730 2731\begin{patch}{ugIntroNumbersPagePatch14} 2732\begin{paste}{ugIntroNumbersPageFull14}{ugIntroNumbersPageEmpty14} 2733\pastebutton{ugIntroNumbersPageFull14}{\hidepaste} 2734\tab{5}\spadcommand{(2/3 + \%i)^3\bound{gaussint }} 2735\indentrel{3}\begin{verbatim} 2736 46 1 2737 (14) - -- + - %i 2738 27 3 2739 Type: Complex(Fraction(Integer)) 2740\end{verbatim} 2741\indentrel{-3}\end{paste}\end{patch} 2742 2743\begin{patch}{ugIntroNumbersPageEmpty14} 2744\begin{paste}{ugIntroNumbersPageEmpty14}{ugIntroNumbersPagePatch14} 2745\pastebutton{ugIntroNumbersPageEmpty14}{\showpaste} 2746\tab{5}\spadcommand{(2/3 + \%i)^3\bound{gaussint }} 2747\end{paste}\end{patch} 2748 2749\begin{patch}{ugIntroNumbersPagePatch15} 2750\begin{paste}{ugIntroNumbersPageFull15}{ugIntroNumbersPageEmpty15} 2751\pastebutton{ugIntroNumbersPageFull15}{\hidepaste} 2752\tab{5}\spadcommand{conjugate \%\free{gaussint }} 2753\indentrel{3}\begin{verbatim} 2754 46 1 2755 (15) - -- - - %i 2756 27 3 2757 Type: Complex(Fraction(Integer)) 2758\end{verbatim} 2759\indentrel{-3}\end{paste}\end{patch} 2760 2761\begin{patch}{ugIntroNumbersPageEmpty15} 2762\begin{paste}{ugIntroNumbersPageEmpty15}{ugIntroNumbersPagePatch15} 2763\pastebutton{ugIntroNumbersPageEmpty15}{\showpaste} 2764\tab{5}\spadcommand{conjugate \%\free{gaussint }} 2765\end{paste}\end{patch} 2766 2767\begin{patch}{ugIntroNumbersPagePatch16} 2768\begin{paste}{ugIntroNumbersPageFull16}{ugIntroNumbersPageEmpty16} 2769\pastebutton{ugIntroNumbersPageFull16}{\hidepaste} 2770\tab{5}\spadcommand{factor(89 - 23 * \%i)} 2771\indentrel{3}\begin{verbatim} 2772 2 2 2773 (16) - (1 + %i)(2 + %i) (3 + 2 %i) 2774 Type: Factored(Complex(Integer)) 2775\end{verbatim} 2776\indentrel{-3}\end{paste}\end{patch} 2777 2778\begin{patch}{ugIntroNumbersPageEmpty16} 2779\begin{paste}{ugIntroNumbersPageEmpty16}{ugIntroNumbersPagePatch16} 2780\pastebutton{ugIntroNumbersPageEmpty16}{\showpaste} 2781\tab{5}\spadcommand{factor(89 - 23 * \%i)} 2782\end{paste}\end{patch} 2783 2784\begin{patch}{ugIntroNumbersPagePatch17} 2785\begin{paste}{ugIntroNumbersPageFull17}{ugIntroNumbersPageEmpty17} 2786\pastebutton{ugIntroNumbersPageFull17}{\hidepaste} 2787\tab{5}\spadcommand{exp(\%pi/4.0 * \%i)} 2788\indentrel{3}\begin{verbatim} 2789 (17) 2790 0.7071067811_8654752440_0844362104_8490392849 2791 + 2792 0.7071067811_8654752440_0844362104_8490392848 %i 2793 Type: Complex(Float) 2794\end{verbatim} 2795\indentrel{-3}\end{paste}\end{patch} 2796 2797\begin{patch}{ugIntroNumbersPageEmpty17} 2798\begin{paste}{ugIntroNumbersPageEmpty17}{ugIntroNumbersPagePatch17} 2799\pastebutton{ugIntroNumbersPageEmpty17}{\showpaste} 2800\tab{5}\spadcommand{exp(\%pi/4.0 * \%i)} 2801\end{paste}\end{patch} 2802 2803\begin{patch}{ugIntroNumbersPagePatch18} 2804\begin{paste}{ugIntroNumbersPageFull18}{ugIntroNumbersPageEmpty18} 2805\pastebutton{ugIntroNumbersPageFull18}{\hidepaste} 2806\tab{5}\spadcommand{decimal(1/352)} 2807\indentrel{3}\begin{verbatim} 2808 __ 2809 (18) 0.0028409 2810 Type: DecimalExpansion 2811\end{verbatim} 2812\indentrel{-3}\end{paste}\end{patch} 2813 2814\begin{patch}{ugIntroNumbersPageEmpty18} 2815\begin{paste}{ugIntroNumbersPageEmpty18}{ugIntroNumbersPagePatch18} 2816\pastebutton{ugIntroNumbersPageEmpty18}{\showpaste} 2817\tab{5}\spadcommand{decimal(1/352)} 2818\end{paste}\end{patch} 2819 2820\begin{patch}{ugIntroNumbersPagePatch19} 2821\begin{paste}{ugIntroNumbersPageFull19}{ugIntroNumbersPageEmpty19} 2822\pastebutton{ugIntroNumbersPageFull19}{\hidepaste} 2823\tab{5}\spadcommand{continuedFraction(6543/210)} 2824\indentrel{3}\begin{verbatim} 2825 1 | 1 | 1 | 1 | 2826 (19) 31 + +---+ + +---+ + +---+ + +---+ 2827 | 6 | 2 | 1 | 3 2828 Type: ContinuedFraction(Integer) 2829\end{verbatim} 2830\indentrel{-3}\end{paste}\end{patch} 2831 2832\begin{patch}{ugIntroNumbersPageEmpty19} 2833\begin{paste}{ugIntroNumbersPageEmpty19}{ugIntroNumbersPagePatch19} 2834\pastebutton{ugIntroNumbersPageEmpty19}{\showpaste} 2835\tab{5}\spadcommand{continuedFraction(6543/210)} 2836\end{paste}\end{patch} 2837 2838\begin{patch}{ugIntroNumbersPagePatch20} 2839\begin{paste}{ugIntroNumbersPageFull20}{ugIntroNumbersPageEmpty20} 2840\pastebutton{ugIntroNumbersPageFull20}{\hidepaste} 2841\tab{5}\spadcommand{partialFraction(1,factorial(10))\bound{partfrac }} 2842\indentrel{3}\begin{verbatim} 2843 159 23 12 1 2844 (20) --- - -- - -- + - 2845 8 4 2 7 2846 2 3 5 2847 Type: PartialFraction(Integer) 2848\end{verbatim} 2849\indentrel{-3}\end{paste}\end{patch} 2850 2851\begin{patch}{ugIntroNumbersPageEmpty20} 2852\begin{paste}{ugIntroNumbersPageEmpty20}{ugIntroNumbersPagePatch20} 2853\pastebutton{ugIntroNumbersPageEmpty20}{\showpaste} 2854\tab{5}\spadcommand{partialFraction(1,factorial(10))\bound{partfrac }} 2855\end{paste}\end{patch} 2856 2857\begin{patch}{ugIntroNumbersPagePatch21} 2858\begin{paste}{ugIntroNumbersPageFull21}{ugIntroNumbersPageEmpty21} 2859\pastebutton{ugIntroNumbersPageFull21}{\hidepaste} 2860\tab{5}\spadcommand{padicFraction(\%)\free{partfrac }} 2861\indentrel{3}\begin{verbatim} 2862 (21) 2863 1 1 1 1 1 1 2 1 2 2 2 1 2864 - + -- + -- + -- + -- + -- - -- - -- - -- - - - -- + - 2865 2 4 5 6 7 8 2 3 4 5 2 7 2866 2 2 2 2 2 3 3 3 5 2867 Type: PartialFraction(Integer) 2868\end{verbatim} 2869\indentrel{-3}\end{paste}\end{patch} 2870 2871\begin{patch}{ugIntroNumbersPageEmpty21} 2872\begin{paste}{ugIntroNumbersPageEmpty21}{ugIntroNumbersPagePatch21} 2873\pastebutton{ugIntroNumbersPageEmpty21}{\showpaste} 2874\tab{5}\spadcommand{padicFraction(\%)\free{partfrac }} 2875\end{paste}\end{patch} 2876 2877\begin{patch}{ugIntroNumbersPagePatch22} 2878\begin{paste}{ugIntroNumbersPageFull22}{ugIntroNumbersPageEmpty22} 2879\pastebutton{ugIntroNumbersPageFull22}{\hidepaste} 2880\tab{5}\spadcommand{radix(4/7, 8)\bound{rad }} 2881\indentrel{3}\begin{verbatim} 2882 _ 2883 (22) 0.4 2884 Type: RadixExpansion(8) 2885\end{verbatim} 2886\indentrel{-3}\end{paste}\end{patch} 2887 2888\begin{patch}{ugIntroNumbersPageEmpty22} 2889\begin{paste}{ugIntroNumbersPageEmpty22}{ugIntroNumbersPagePatch22} 2890\pastebutton{ugIntroNumbersPageEmpty22}{\showpaste} 2891\tab{5}\spadcommand{radix(4/7, 8)\bound{rad }} 2892\end{paste}\end{patch} 2893 2894\begin{patch}{ugIntroNumbersPagePatch23} 2895\begin{paste}{ugIntroNumbersPageFull23}{ugIntroNumbersPageEmpty23} 2896\pastebutton{ugIntroNumbersPageFull23}{\hidepaste} 2897\tab{5}\spadcommand{\% + 2/3*\%i\free{rad }} 2898\indentrel{3}\begin{verbatim} 2899 4 2 2900 (23) - + - %i 2901 7 3 2902 Type: Complex(Fraction(Integer)) 2903\end{verbatim} 2904\indentrel{-3}\end{paste}\end{patch} 2905 2906\begin{patch}{ugIntroNumbersPageEmpty23} 2907\begin{paste}{ugIntroNumbersPageEmpty23}{ugIntroNumbersPagePatch23} 2908\pastebutton{ugIntroNumbersPageEmpty23}{\showpaste} 2909\tab{5}\spadcommand{\% + 2/3*\%i\free{rad }} 2910\end{paste}\end{patch} 2911 2912\begin{patch}{ugIntroNumbersPagePatch24} 2913\begin{paste}{ugIntroNumbersPageFull24}{ugIntroNumbersPageEmpty24} 2914\pastebutton{ugIntroNumbersPageFull24}{\hidepaste} 2915\tab{5}\spadcommand{(5 + sqrt 63 + sqrt 847)^(1/3)} 2916\indentrel{3}\begin{verbatim} 2917 +-----------+ 2918 3| +-+ 2919 (24) \|14 \|7 + 5 2920 Type: AlgebraicNumber 2921\end{verbatim} 2922\indentrel{-3}\end{paste}\end{patch} 2923 2924\begin{patch}{ugIntroNumbersPageEmpty24} 2925\begin{paste}{ugIntroNumbersPageEmpty24}{ugIntroNumbersPagePatch24} 2926\pastebutton{ugIntroNumbersPageEmpty24}{\showpaste} 2927\tab{5}\spadcommand{(5 + sqrt 63 + sqrt 847)^(1/3)} 2928\end{paste}\end{patch} 2929 2930\begin{patch}{ugIntroNumbersPagePatch25} 2931\begin{paste}{ugIntroNumbersPageFull25}{ugIntroNumbersPageEmpty25} 2932\pastebutton{ugIntroNumbersPageFull25}{\hidepaste} 2933\tab{5}\spadcommand{x : PrimeField 7 := 5\bound{x }} 2934\indentrel{3}\begin{verbatim} 2935 (25) 5 2936 Type: PrimeField(7) 2937\end{verbatim} 2938\indentrel{-3}\end{paste}\end{patch} 2939 2940\begin{patch}{ugIntroNumbersPageEmpty25} 2941\begin{paste}{ugIntroNumbersPageEmpty25}{ugIntroNumbersPagePatch25} 2942\pastebutton{ugIntroNumbersPageEmpty25}{\showpaste} 2943\tab{5}\spadcommand{x : PrimeField 7 := 5\bound{x }} 2944\end{paste}\end{patch} 2945 2946\begin{patch}{ugIntroNumbersPagePatch26} 2947\begin{paste}{ugIntroNumbersPageFull26}{ugIntroNumbersPageEmpty26} 2948\pastebutton{ugIntroNumbersPageFull26}{\hidepaste} 2949\tab{5}\spadcommand{x^3\free{x }} 2950\indentrel{3}\begin{verbatim} 2951 (26) 6 2952 Type: PrimeField(7) 2953\end{verbatim} 2954\indentrel{-3}\end{paste}\end{patch} 2955 2956\begin{patch}{ugIntroNumbersPageEmpty26} 2957\begin{paste}{ugIntroNumbersPageEmpty26}{ugIntroNumbersPagePatch26} 2958\pastebutton{ugIntroNumbersPageEmpty26}{\showpaste} 2959\tab{5}\spadcommand{x^3\free{x }} 2960\end{paste}\end{patch} 2961 2962\begin{patch}{ugIntroNumbersPagePatch27} 2963\begin{paste}{ugIntroNumbersPageFull27}{ugIntroNumbersPageEmpty27} 2964\pastebutton{ugIntroNumbersPageFull27}{\hidepaste} 2965\tab{5}\spadcommand{1/x\free{x }} 2966\indentrel{3}\begin{verbatim} 2967 (27) 3 2968 Type: PrimeField(7) 2969\end{verbatim} 2970\indentrel{-3}\end{paste}\end{patch} 2971 2972\begin{patch}{ugIntroNumbersPageEmpty27} 2973\begin{paste}{ugIntroNumbersPageEmpty27}{ugIntroNumbersPagePatch27} 2974\pastebutton{ugIntroNumbersPageEmpty27}{\showpaste} 2975\tab{5}\spadcommand{1/x\free{x }} 2976\end{paste}\end{patch} 2977 2978\begin{patch}{ugIntroNumbersPagePatch28} 2979\begin{paste}{ugIntroNumbersPageFull28}{ugIntroNumbersPageEmpty28} 2980\pastebutton{ugIntroNumbersPageFull28}{\hidepaste} 2981\tab{5}\spadcommand{y : IntegerMod 6 := 5\bound{y }} 2982\indentrel{3}\begin{verbatim} 2983 (28) 5 2984 Type: IntegerMod(6) 2985\end{verbatim} 2986\indentrel{-3}\end{paste}\end{patch} 2987 2988\begin{patch}{ugIntroNumbersPageEmpty28} 2989\begin{paste}{ugIntroNumbersPageEmpty28}{ugIntroNumbersPagePatch28} 2990\pastebutton{ugIntroNumbersPageEmpty28}{\showpaste} 2991\tab{5}\spadcommand{y : IntegerMod 6 := 5\bound{y }} 2992\end{paste}\end{patch} 2993 2994\begin{patch}{ugIntroNumbersPagePatch29} 2995\begin{paste}{ugIntroNumbersPageFull29}{ugIntroNumbersPageEmpty29} 2996\pastebutton{ugIntroNumbersPageFull29}{\hidepaste} 2997\tab{5}\spadcommand{y^3\free{y }} 2998\indentrel{3}\begin{verbatim} 2999 (29) 5 3000 Type: IntegerMod(6) 3001\end{verbatim} 3002\indentrel{-3}\end{paste}\end{patch} 3003 3004\begin{patch}{ugIntroNumbersPageEmpty29} 3005\begin{paste}{ugIntroNumbersPageEmpty29}{ugIntroNumbersPagePatch29} 3006\pastebutton{ugIntroNumbersPageEmpty29}{\showpaste} 3007\tab{5}\spadcommand{y^3\free{y }} 3008\end{paste}\end{patch} 3009 3010\begin{patch}{ugIntroNumbersPagePatch30} 3011\begin{paste}{ugIntroNumbersPageFull30}{ugIntroNumbersPageEmpty30} 3012\pastebutton{ugIntroNumbersPageFull30}{\hidepaste} 3013\tab{5}\spadcommand{1/y\free{y }} 3014\indentrel{3}\begin{verbatim} 3015error 3016\end{verbatim} 3017\indentrel{-3}\end{paste}\end{patch} 3018 3019\begin{patch}{ugIntroNumbersPageEmpty30} 3020\begin{paste}{ugIntroNumbersPageEmpty30}{ugIntroNumbersPagePatch30} 3021\pastebutton{ugIntroNumbersPageEmpty30}{\showpaste} 3022\tab{5}\spadcommand{1/y\free{y }} 3023\end{paste}\end{patch} 3024 3025\begin{patch}{ugIntroNumbersPagePatch31} 3026\begin{paste}{ugIntroNumbersPageFull31}{ugIntroNumbersPageEmpty31} 3027\pastebutton{ugIntroNumbersPageFull31}{\hidepaste} 3028\tab{5}\spadcommand{a := rootOf(a^5 + a^3 + a^2 + 3,a)\bound{a }} 3029\indentrel{3}\begin{verbatim} 3030 (30) a 3031 Type: Expression(Integer) 3032\end{verbatim} 3033\indentrel{-3}\end{paste}\end{patch} 3034 3035\begin{patch}{ugIntroNumbersPageEmpty31} 3036\begin{paste}{ugIntroNumbersPageEmpty31}{ugIntroNumbersPagePatch31} 3037\pastebutton{ugIntroNumbersPageEmpty31}{\showpaste} 3038\tab{5}\spadcommand{a := rootOf(a^5 + a^3 + a^2 + 3,a)\bound{a }} 3039\end{paste}\end{patch} 3040 3041\begin{patch}{ugIntroNumbersPagePatch32} 3042\begin{paste}{ugIntroNumbersPageFull32}{ugIntroNumbersPageEmpty32} 3043\pastebutton{ugIntroNumbersPageFull32}{\hidepaste} 3044\tab{5}\spadcommand{(a + 1)^10\free{a }} 3045\indentrel{3}\begin{verbatim} 3046 4 3 2 3047 (31) - 85 a - 264 a - 378 a - 458 a - 287 3048 Type: Expression(Integer) 3049\end{verbatim} 3050\indentrel{-3}\end{paste}\end{patch} 3051 3052\begin{patch}{ugIntroNumbersPageEmpty32} 3053\begin{paste}{ugIntroNumbersPageEmpty32}{ugIntroNumbersPagePatch32} 3054\pastebutton{ugIntroNumbersPageEmpty32}{\showpaste} 3055\tab{5}\spadcommand{(a + 1)^10\free{a }} 3056\end{paste}\end{patch} 3057 3058\begin{patch}{ugIntroNumbersPagePatch33} 3059\begin{paste}{ugIntroNumbersPageFull33}{ugIntroNumbersPageEmpty33} 3060\pastebutton{ugIntroNumbersPageFull33}{\hidepaste} 3061\tab{5}\spadcommand{b := rootOf(b^4 + a,b)\bound{b }\free{a }} 3062\indentrel{3}\begin{verbatim} 3063 (32) b 3064 Type: Expression(Integer) 3065\end{verbatim} 3066\indentrel{-3}\end{paste}\end{patch} 3067 3068\begin{patch}{ugIntroNumbersPageEmpty33} 3069\begin{paste}{ugIntroNumbersPageEmpty33}{ugIntroNumbersPagePatch33} 3070\pastebutton{ugIntroNumbersPageEmpty33}{\showpaste} 3071\tab{5}\spadcommand{b := rootOf(b^4 + a,b)\bound{b }\free{a }} 3072\end{paste}\end{patch} 3073 3074\begin{patch}{ugIntroNumbersPagePatch34} 3075\begin{paste}{ugIntroNumbersPageFull34}{ugIntroNumbersPageEmpty34} 3076\pastebutton{ugIntroNumbersPageFull34}{\hidepaste} 3077\tab{5}\spadcommand{2/(b - 1)\free{b }\bound{check }} 3078\indentrel{3}\begin{verbatim} 3079 2 3080 (33) ----- 3081 b - 1 3082 Type: Expression(Integer) 3083\end{verbatim} 3084\indentrel{-3}\end{paste}\end{patch} 3085 3086\begin{patch}{ugIntroNumbersPageEmpty34} 3087\begin{paste}{ugIntroNumbersPageEmpty34}{ugIntroNumbersPagePatch34} 3088\pastebutton{ugIntroNumbersPageEmpty34}{\showpaste} 3089\tab{5}\spadcommand{2/(b - 1)\free{b }\bound{check }} 3090\end{paste}\end{patch} 3091 3092\begin{patch}{ugIntroNumbersPagePatch35} 3093\begin{paste}{ugIntroNumbersPageFull35}{ugIntroNumbersPageEmpty35} 3094\pastebutton{ugIntroNumbersPageFull35}{\hidepaste} 3095\tab{5}\spadcommand{ratDenom(\%)\free{check }\bound{check1 }} 3096\indentrel{3}\begin{verbatim} 3097 (34) 3098 4 3 2 3 3099 (a - a + 2 a - a + 1)b 3100 + 3101 4 3 2 2 3102 (a - a + 2 a - a + 1)b 3103 + 3104 4 3 2 4 3 2 3105 (a - a + 2 a - a + 1)b + a - a + 2 a - a + 1 3106 Type: Expression(Integer) 3107\end{verbatim} 3108\indentrel{-3}\end{paste}\end{patch} 3109 3110\begin{patch}{ugIntroNumbersPageEmpty35} 3111\begin{paste}{ugIntroNumbersPageEmpty35}{ugIntroNumbersPagePatch35} 3112\pastebutton{ugIntroNumbersPageEmpty35}{\showpaste} 3113\tab{5}\spadcommand{ratDenom(\%)\free{check }\bound{check1 }} 3114\end{paste}\end{patch} 3115 3116\begin{patch}{ugIntroNumbersPagePatch36} 3117\begin{paste}{ugIntroNumbersPageFull36}{ugIntroNumbersPageEmpty36} 3118\pastebutton{ugIntroNumbersPageFull36}{\hidepaste} 3119\tab{5}\spadcommand{2/\%+1\free{check1 }\bound{check2 }} 3120\indentrel{3}\begin{verbatim} 3121 (35) 3122 4 3 2 3 3123 (a - a + 2 a - a + 1)b 3124 + 3125 4 3 2 2 3126 (a - a + 2 a - a + 1)b 3127 + 3128 4 3 2 4 3 2 3129 (a - a + 2 a - a + 1)b + a - a + 2 a - a + 3 3130 / 3131 4 3 2 3 3132 (a - a + 2 a - a + 1)b 3133 + 3134 4 3 2 2 3135 (a - a + 2 a - a + 1)b 3136 + 3137 4 3 2 4 3 2 3138 (a - a + 2 a - a + 1)b + a - a + 2 a - a + 1 3139 Type: Expression(Integer) 3140\end{verbatim} 3141\indentrel{-3}\end{paste}\end{patch} 3142 3143\begin{patch}{ugIntroNumbersPageEmpty36} 3144\begin{paste}{ugIntroNumbersPageEmpty36}{ugIntroNumbersPagePatch36} 3145\pastebutton{ugIntroNumbersPageEmpty36}{\showpaste} 3146\tab{5}\spadcommand{2/\%+1\free{check1 }\bound{check2 }} 3147\end{paste}\end{patch} 3148 3149\begin{patch}{ugIntroNumbersPagePatch37} 3150\begin{paste}{ugIntroNumbersPageFull37}{ugIntroNumbersPageEmpty37} 3151\pastebutton{ugIntroNumbersPageFull37}{\hidepaste} 3152\tab{5}\spadcommand{ratDenom(\%)\free{check2 }} 3153\indentrel{3}\begin{verbatim} 3154 (36) b 3155 Type: Expression(Integer) 3156\end{verbatim} 3157\indentrel{-3}\end{paste}\end{patch} 3158 3159\begin{patch}{ugIntroNumbersPageEmpty37} 3160\begin{paste}{ugIntroNumbersPageEmpty37}{ugIntroNumbersPagePatch37} 3161\pastebutton{ugIntroNumbersPageEmpty37}{\showpaste} 3162\tab{5}\spadcommand{ratDenom(\%)\free{check2 }} 3163\end{paste}\end{patch} 3164 3165\begin{patch}{ugIntroNumbersPagePatch38} 3166\begin{paste}{ugIntroNumbersPageFull38}{ugIntroNumbersPageEmpty38} 3167\pastebutton{ugIntroNumbersPageFull38}{\hidepaste} 3168\tab{5}\spadcommand{q:=quatern(1,2,3,4)*quatern(5,6,7,8) - quatern(5,6,7,8)*quatern(1,2,3,4)} 3169\indentrel{3}\begin{verbatim} 3170 (37) - 8 i + 16 j - 8 k 3171 Type: Quaternion(Integer) 3172\end{verbatim} 3173\indentrel{-3}\end{paste}\end{patch} 3174 3175\begin{patch}{ugIntroNumbersPageEmpty38} 3176\begin{paste}{ugIntroNumbersPageEmpty38}{ugIntroNumbersPagePatch38} 3177\pastebutton{ugIntroNumbersPageEmpty38}{\showpaste} 3178\tab{5}\spadcommand{q:=quatern(1,2,3,4)*quatern(5,6,7,8) - quatern(5,6,7,8)*quatern(1,2,3,4)} 3179\end{paste}\end{patch} 3180 3181\begin{patch}{ugIntroDiffEqnsPagePatch1} 3182\begin{paste}{ugIntroDiffEqnsPageFull1}{ugIntroDiffEqnsPageEmpty1} 3183\pastebutton{ugIntroDiffEqnsPageFull1}{\hidepaste} 3184\tab{5}\spadcommand{y := operator 'y\bound{y }} 3185\indentrel{3}\begin{verbatim} 3186 (1) y 3187 Type: BasicOperator 3188\end{verbatim} 3189\indentrel{-3}\end{paste}\end{patch} 3190 3191\begin{patch}{ugIntroDiffEqnsPageEmpty1} 3192\begin{paste}{ugIntroDiffEqnsPageEmpty1}{ugIntroDiffEqnsPagePatch1} 3193\pastebutton{ugIntroDiffEqnsPageEmpty1}{\showpaste} 3194\tab{5}\spadcommand{y := operator 'y\bound{y }} 3195\end{paste}\end{patch} 3196 3197\begin{patch}{ugIntroDiffEqnsPagePatch2} 3198\begin{paste}{ugIntroDiffEqnsPageFull2}{ugIntroDiffEqnsPageEmpty2} 3199\pastebutton{ugIntroDiffEqnsPageFull2}{\hidepaste} 3200\tab{5}\spadcommand{deq := x^3 * D(y x, x, 3) + x^2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x^4\bound{e3 }\free{y }} 3201\indentrel{3}\begin{verbatim} 3202 3 ,,, 2 ,, , 4 3203 (2) x y (x) + x y (x) - 2 x y (x) + 2 y(x) = 2 x 3204 3205 Type: Equation(Expression(Integer)) 3206\end{verbatim} 3207\indentrel{-3}\end{paste}\end{patch} 3208 3209\begin{patch}{ugIntroDiffEqnsPageEmpty2} 3210\begin{paste}{ugIntroDiffEqnsPageEmpty2}{ugIntroDiffEqnsPagePatch2} 3211\pastebutton{ugIntroDiffEqnsPageEmpty2}{\showpaste} 3212\tab{5}\spadcommand{deq := x^3 * D(y x, x, 3) + x^2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x^4\bound{e3 }\free{y }} 3213\end{paste}\end{patch} 3214 3215\begin{patch}{ugIntroDiffEqnsPagePatch3} 3216\begin{paste}{ugIntroDiffEqnsPageFull3}{ugIntroDiffEqnsPageEmpty3} 3217\pastebutton{ugIntroDiffEqnsPageFull3}{\hidepaste} 3218\tab{5}\spadcommand{solve(deq, y, x)\free{e3 }\free{y }} 3219\indentrel{3}\begin{verbatim} 3220 (3) 3221 5 3 2 3222 x - 10 x + 20 x + 4 3223 [particular = ----------------------, 3224 15 x 3225 3 2 3 3 2 3226 2 x - 3 x + 1 x - 1 x - 3 x - 1 3227 basis = [---------------, ------, -------------]] 3228 x x x 3229Type: Union(Record(particular: Expression(Integer),basis: List(Expression(Integer))),...) 3230\end{verbatim} 3231\indentrel{-3}\end{paste}\end{patch} 3232 3233\begin{patch}{ugIntroDiffEqnsPageEmpty3} 3234\begin{paste}{ugIntroDiffEqnsPageEmpty3}{ugIntroDiffEqnsPagePatch3} 3235\pastebutton{ugIntroDiffEqnsPageEmpty3}{\showpaste} 3236\tab{5}\spadcommand{solve(deq, y, x)\free{e3 }\free{y }} 3237\end{paste}\end{patch} 3238 3239\begin{patch}{ugIntroDiffEqnsPagePatch4} 3240\begin{paste}{ugIntroDiffEqnsPageFull4}{ugIntroDiffEqnsPageEmpty4} 3241\pastebutton{ugIntroDiffEqnsPageFull4}{\hidepaste} 3242\tab{5}\spadcommand{deq := (x^2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0\bound{e5 }\free{y }} 3243\indentrel{3}\begin{verbatim} 3244 2 ,, , 3245 (4) (x + 1)y (x) + 3 x y (x) + y(x) = 0 3246 3247 Type: Equation(Expression(Integer)) 3248\end{verbatim} 3249\indentrel{-3}\end{paste}\end{patch} 3250 3251\begin{patch}{ugIntroDiffEqnsPageEmpty4} 3252\begin{paste}{ugIntroDiffEqnsPageEmpty4}{ugIntroDiffEqnsPagePatch4} 3253\pastebutton{ugIntroDiffEqnsPageEmpty4}{\showpaste} 3254\tab{5}\spadcommand{deq := (x^2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0\bound{e5 }\free{y }} 3255\end{paste}\end{patch} 3256 3257\begin{patch}{ugIntroDiffEqnsPagePatch5} 3258\begin{paste}{ugIntroDiffEqnsPageFull5}{ugIntroDiffEqnsPageEmpty5} 3259\pastebutton{ugIntroDiffEqnsPageFull5}{\hidepaste} 3260\tab{5}\spadcommand{solve(deq, y, x)\free{e5 }\free{y }} 3261\indentrel{3}\begin{verbatim} 3262 (5) 3263 [particular = 0, 3264 +------+ 3265 | 2 3266 1 log(\|x + 1 - x) 3267 basis = [---------, ------------------]] 3268 +------+ +------+ 3269 | 2 | 2 3270 \|x + 1 \|x + 1 3271Type: Union(Record(particular: Expression(Integer),basis: List(Expression(Integer))),...) 3272\end{verbatim} 3273\indentrel{-3}\end{paste}\end{patch} 3274 3275\begin{patch}{ugIntroDiffEqnsPageEmpty5} 3276\begin{paste}{ugIntroDiffEqnsPageEmpty5}{ugIntroDiffEqnsPagePatch5} 3277\pastebutton{ugIntroDiffEqnsPageEmpty5}{\showpaste} 3278\tab{5}\spadcommand{solve(deq, y, x)\free{e5 }\free{y }} 3279\end{paste}\end{patch} 3280 3281\begin{patch}{ugIntroDiffEqnsPagePatch6} 3282\begin{paste}{ugIntroDiffEqnsPageFull6}{ugIntroDiffEqnsPageEmpty6} 3283\pastebutton{ugIntroDiffEqnsPageFull6}{\hidepaste} 3284\tab{5}\spadcommand{eq := 2*x^3 * D(y x,x,2) + 3*x^2 * D(y x,x) - 2 * y x\bound{eq }\free{y }} 3285\indentrel{3}\begin{verbatim} 3286 3 ,, 2 , 3287 (6) 2 x y (x) + 3 x y (x) - 2 y(x) 3288 3289 Type: Expression(Integer) 3290\end{verbatim} 3291\indentrel{-3}\end{paste}\end{patch} 3292 3293\begin{patch}{ugIntroDiffEqnsPageEmpty6} 3294\begin{paste}{ugIntroDiffEqnsPageEmpty6}{ugIntroDiffEqnsPagePatch6} 3295\pastebutton{ugIntroDiffEqnsPageEmpty6}{\showpaste} 3296\tab{5}\spadcommand{eq := 2*x^3 * D(y x,x,2) + 3*x^2 * D(y x,x) - 2 * y x\bound{eq }\free{y }} 3297\end{paste}\end{patch} 3298 3299\begin{patch}{ugIntroDiffEqnsPagePatch7} 3300\begin{paste}{ugIntroDiffEqnsPageFull7}{ugIntroDiffEqnsPageEmpty7} 3301\pastebutton{ugIntroDiffEqnsPageFull7}{\hidepaste} 3302\tab{5}\spadcommand{solve(eq,y,x).basis\free{eq }} 3303\indentrel{3}\begin{verbatim} 3304 2 2 3305 - ---- ---- 3306 +-+ +-+ 3307 \|x \|x 3308 (7) [%e , %e ] 3309 Type: List(Expression(Integer)) 3310\end{verbatim} 3311\indentrel{-3}\end{paste}\end{patch} 3312 3313\begin{patch}{ugIntroDiffEqnsPageEmpty7} 3314\begin{paste}{ugIntroDiffEqnsPageEmpty7}{ugIntroDiffEqnsPagePatch7} 3315\pastebutton{ugIntroDiffEqnsPageEmpty7}{\showpaste} 3316\tab{5}\spadcommand{solve(eq,y,x).basis\free{eq }} 3317\end{paste}\end{patch} 3318 3319\begin{patch}{ugIntroDiffEqnsPagePatch8} 3320\begin{paste}{ugIntroDiffEqnsPageFull8}{ugIntroDiffEqnsPageEmpty8} 3321\pastebutton{ugIntroDiffEqnsPageFull8}{\hidepaste} 3322\tab{5}\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)\bound{deqi }\free{y }} 3323\indentrel{3}\begin{verbatim} 3324 , y(x) 3325 (8) y (x) = ----------------- 3326 y(x)log(y(x)) + x 3327 Type: Equation(Expression(Integer)) 3328\end{verbatim} 3329\indentrel{-3}\end{paste}\end{patch} 3330 3331\begin{patch}{ugIntroDiffEqnsPageEmpty8} 3332\begin{paste}{ugIntroDiffEqnsPageEmpty8}{ugIntroDiffEqnsPagePatch8} 3333\pastebutton{ugIntroDiffEqnsPageEmpty8}{\showpaste} 3334\tab{5}\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)\bound{deqi }\free{y }} 3335\end{paste}\end{patch} 3336 3337\begin{patch}{ugIntroDiffEqnsPagePatch9} 3338\begin{paste}{ugIntroDiffEqnsPageFull9}{ugIntroDiffEqnsPageEmpty9} 3339\pastebutton{ugIntroDiffEqnsPageFull9}{\hidepaste} 3340\tab{5}\spadcommand{solve(deq, y, x)\free{deqi y }} 3341\indentrel{3}\begin{verbatim} 3342 2 3343 y(x)log(y(x)) - 2 x 3344 (9) -------------------- 3345 2 y(x) 3346 Type: Union(Expression(Integer),...) 3347\end{verbatim} 3348\indentrel{-3}\end{paste}\end{patch} 3349 3350\begin{patch}{ugIntroDiffEqnsPageEmpty9} 3351\begin{paste}{ugIntroDiffEqnsPageEmpty9}{ugIntroDiffEqnsPagePatch9} 3352\pastebutton{ugIntroDiffEqnsPageEmpty9}{\showpaste} 3353\tab{5}\spadcommand{solve(deq, y, x)\free{deqi y }} 3354\end{paste}\end{patch} 3355 3356\begin{patch}{ugIntroDiffEqnsPagePatch10} 3357\begin{paste}{ugIntroDiffEqnsPageFull10}{ugIntroDiffEqnsPageEmpty10} 3358\pastebutton{ugIntroDiffEqnsPageFull10}{\hidepaste} 3359\tab{5}\spadcommand{x := operator 'x\bound{x }} 3360\indentrel{3}\begin{verbatim} 3361 (10) x 3362 Type: BasicOperator 3363\end{verbatim} 3364\indentrel{-3}\end{paste}\end{patch} 3365 3366\begin{patch}{ugIntroDiffEqnsPageEmpty10} 3367\begin{paste}{ugIntroDiffEqnsPageEmpty10}{ugIntroDiffEqnsPagePatch10} 3368\pastebutton{ugIntroDiffEqnsPageEmpty10}{\showpaste} 3369\tab{5}\spadcommand{x := operator 'x\bound{x }} 3370\end{paste}\end{patch} 3371 3372\begin{patch}{ugIntroDiffEqnsPagePatch11} 3373\begin{paste}{ugIntroDiffEqnsPageFull11}{ugIntroDiffEqnsPageEmpty11} 3374\pastebutton{ugIntroDiffEqnsPageFull11}{\hidepaste} 3375\tab{5}\spadcommand{eq1 := D(x(t), t) = 1 + x(t)^2\free{x }\free{y }\bound{eq1 }} 3376\indentrel{3}\begin{verbatim} 3377 , 2 3378 (11) x (t) = x(t) + 1 3379 3380 Type: Equation(Expression(Integer)) 3381\end{verbatim} 3382\indentrel{-3}\end{paste}\end{patch} 3383 3384\begin{patch}{ugIntroDiffEqnsPageEmpty11} 3385\begin{paste}{ugIntroDiffEqnsPageEmpty11}{ugIntroDiffEqnsPagePatch11} 3386\pastebutton{ugIntroDiffEqnsPageEmpty11}{\showpaste} 3387\tab{5}\spadcommand{eq1 := D(x(t), t) = 1 + x(t)^2\free{x }\free{y }\bound{eq1 }} 3388\end{paste}\end{patch} 3389 3390\begin{patch}{ugIntroDiffEqnsPagePatch12} 3391\begin{paste}{ugIntroDiffEqnsPageFull12}{ugIntroDiffEqnsPageEmpty12} 3392\pastebutton{ugIntroDiffEqnsPageFull12}{\hidepaste} 3393\tab{5}\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)\free{x }\free{y }\bound{eq2 }} 3394\indentrel{3}\begin{verbatim} 3395 , 3396 (12) y (t) = x(t)y(t) 3397 3398 Type: Equation(Expression(Integer)) 3399\end{verbatim} 3400\indentrel{-3}\end{paste}\end{patch} 3401 3402\begin{patch}{ugIntroDiffEqnsPageEmpty12} 3403\begin{paste}{ugIntroDiffEqnsPageEmpty12}{ugIntroDiffEqnsPagePatch12} 3404\pastebutton{ugIntroDiffEqnsPageEmpty12}{\showpaste} 3405\tab{5}\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)\free{x }\free{y }\bound{eq2 }} 3406\end{paste}\end{patch} 3407 3408\begin{patch}{ugIntroDiffEqnsPagePatch13} 3409\begin{paste}{ugIntroDiffEqnsPageFull13}{ugIntroDiffEqnsPageEmpty13} 3410\pastebutton{ugIntroDiffEqnsPageFull13}{\hidepaste} 3411\tab{5}\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])\free{x }\free{y }\free{eq1 }\free{eq2 }} 3412\indentrel{3}\begin{verbatim} 3413 (13) 3414 1 3 2 5 17 7 62 9 11 3415 [t + - t + -- t + --- t + ---- t + O(t ), 3416 3 15 315 2835 3417 3418 1 2 5 4 61 6 277 8 50521 10 3419 1 + - t + -- t + --- t + ---- t + ------- t 3420 2 24 720 8064 3628800 3421 + 3422 11 3423 O(t ) 3424 ] 3425Type: List(UnivariateTaylorSeries(Expression(Integer),t,0)) 3426\end{verbatim} 3427\indentrel{-3}\end{paste}\end{patch} 3428 3429\begin{patch}{ugIntroDiffEqnsPageEmpty13} 3430\begin{paste}{ugIntroDiffEqnsPageEmpty13}{ugIntroDiffEqnsPagePatch13} 3431\pastebutton{ugIntroDiffEqnsPageEmpty13}{\showpaste} 3432\tab{5}\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])\free{x }\free{y }\free{eq1 }\free{eq2 }} 3433\end{paste}\end{patch} 3434 3435\begin{patch}{ugIntroGraphicsPagePatch1} 3436\begin{paste}{ugIntroGraphicsPageFull1}{ugIntroGraphicsPageEmpty1} 3437\pastebutton{ugIntroGraphicsPageFull1}{\hidepaste} 3438\tab{5}\spadgraph{draw(cos(5*t/8), t=0..16*\%pi, coordinates==polar)} 3439\center{\unixcommand{\inputimage{\env{FRICAS}/share/viewports/ugIntroGraphicsPage1.VIEW/image}}{\env{FRICAS}/bin/viewAlone\space{1} \env{FRICAS}/share/viewports/ugIntroGraphicsPage1}} 3440\end{paste}\end{patch} 3441 3442\begin{patch}{ugIntroGraphicsPageEmpty1} 3443\begin{paste}{ugIntroGraphicsPageEmpty1}{ugIntroGraphicsPagePatch1} 3444\pastebutton{ugIntroGraphicsPageEmpty1}{\showpaste} 3445\tab{5}\spadgraph{draw(cos(5*t/8), t=0..16*\%pi, coordinates==polar)} 3446\end{paste}\end{patch} 3447 3448\begin{patch}{ugIntroGraphicsPagePatch2} 3449\begin{paste}{ugIntroGraphicsPageFull2}{ugIntroGraphicsPageEmpty2} 3450\pastebutton{ugIntroGraphicsPageFull2}{\hidepaste} 3451\tab{5}\spadgraph{draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction == (x,y) +-> argument atan complex(x,y))} 3452\center{\unixcommand{\inputimage{\env{FRICAS}/share/viewports/ugIntroGraphicsPage2.VIEW/image}}{\env{FRICAS}/bin/viewAlone\space{1} \env{FRICAS}/share/viewports/ugIntroGraphicsPage2}} 3453\end{paste}\end{patch} 3454 3455\begin{patch}{ugIntroGraphicsPageEmpty2} 3456\begin{paste}{ugIntroGraphicsPageEmpty2}{ugIntroGraphicsPagePatch2} 3457\pastebutton{ugIntroGraphicsPageEmpty2}{\showpaste} 3458\tab{5}\spadgraph{draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction == (x,y) +-> argument atan complex(x,y))} 3459\end{paste}\end{patch} 3460 3461\begin{patch}{ugIntroCommentsPagePatch1} 3462\begin{paste}{ugIntroCommentsPageFull1}{ugIntroCommentsPageEmpty1} 3463\pastebutton{ugIntroCommentsPageFull1}{\hidepaste} 3464\tab{5}\spadcommand{2 + 3 -- this is rather simple, no?} 3465\indentrel{3}\begin{verbatim} 3466 (1) 5 3467 Type: PositiveInteger 3468\end{verbatim} 3469\indentrel{-3}\end{paste}\end{patch} 3470 3471\begin{patch}{ugIntroCommentsPageEmpty1} 3472\begin{paste}{ugIntroCommentsPageEmpty1}{ugIntroCommentsPagePatch1} 3473\pastebutton{ugIntroCommentsPageEmpty1}{\showpaste} 3474\tab{5}\spadcommand{2 + 3 -- this is rather simple, no?} 3475\end{paste}\end{patch} 3476 3477\begin{patch}{ugIntroSolutionPagePatch1} 3478\begin{paste}{ugIntroSolutionPageFull1}{ugIntroSolutionPageEmpty1} 3479\pastebutton{ugIntroSolutionPageFull1}{\hidepaste} 3480\tab{5}\spadcommand{S(t) == [x^2-2*y^2 - t,x*y-y-5*x + 5]\bound{S1 }} 3481\indentrel{3}\begin{verbatim} 3482 Type: Void 3483\end{verbatim} 3484\indentrel{-3}\end{paste}\end{patch} 3485 3486\begin{patch}{ugIntroSolutionPageEmpty1} 3487\begin{paste}{ugIntroSolutionPageEmpty1}{ugIntroSolutionPagePatch1} 3488\pastebutton{ugIntroSolutionPageEmpty1}{\showpaste} 3489\tab{5}\spadcommand{S(t) == [x^2-2*y^2 - t,x*y-y-5*x + 5]\bound{S1 }} 3490\end{paste}\end{patch} 3491 3492\begin{patch}{ugIntroSolutionPagePatch2} 3493\begin{paste}{ugIntroSolutionPageFull2}{ugIntroSolutionPageEmpty2} 3494\pastebutton{ugIntroSolutionPageFull2}{\hidepaste} 3495\tab{5}\spadcommand{solve(S(19),1/10^20)\free{S1 }} 3496\indentrel{3}\begin{verbatim} 3497 (2) 3498 80336736493669365924189585 3499 [[y = 5, x = - --------------------------], 3500 9671406556917033397649408 3501 80336736493669365924189585 3502 [y = 5, x = --------------------------]] 3503 9671406556917033397649408 3504Type: List(List(Equation(Polynomial(Fraction(Integer))))) 3505\end{verbatim} 3506\indentrel{-3}\end{paste}\end{patch} 3507 3508\begin{patch}{ugIntroSolutionPageEmpty2} 3509\begin{paste}{ugIntroSolutionPageEmpty2}{ugIntroSolutionPagePatch2} 3510\pastebutton{ugIntroSolutionPageEmpty2}{\showpaste} 3511\tab{5}\spadcommand{solve(S(19),1/10^20)\free{S1 }} 3512\end{paste}\end{patch} 3513 3514\begin{patch}{ugIntroSolutionPagePatch3} 3515\begin{paste}{ugIntroSolutionPageFull3}{ugIntroSolutionPageEmpty3} 3516\pastebutton{ugIntroSolutionPageFull3}{\hidepaste} 3517\tab{5}\spadcommand{complexSolve(S(19),10.e-20)\free{S1 }} 3518\indentrel{3}\begin{verbatim} 3519 (3) 3520 [ 3521 [y = 5.0, 3522 3523 x 3524 = 3525 8.3066238629_1807485258_4262744905_6951556981_516 3526 9148184_0582865006_639146088 3527 ] 3528 , 3529 3530 [y = 5.0, 3531 x = - 8.3066238629_1807485258_4262744905_695155698] 3532 , 3533 [y = - 3.0 %i, x = 1.0], [y = 3.0 %i, x = 1.0]] 3534 Type: List(List(Equation(Polynomial(Complex(Float))))) 3535\end{verbatim} 3536\indentrel{-3}\end{paste}\end{patch} 3537 3538\begin{patch}{ugIntroSolutionPageEmpty3} 3539\begin{paste}{ugIntroSolutionPageEmpty3}{ugIntroSolutionPagePatch3} 3540\pastebutton{ugIntroSolutionPageEmpty3}{\showpaste} 3541\tab{5}\spadcommand{complexSolve(S(19),10.e-20)\free{S1 }} 3542\end{paste}\end{patch} 3543 3544\begin{patch}{ugIntroSolutionPagePatch4} 3545\begin{paste}{ugIntroSolutionPageFull4}{ugIntroSolutionPageEmpty4} 3546\pastebutton{ugIntroSolutionPageFull4}{\hidepaste} 3547\tab{5}\spadcommand{radicalSolve(S(a),[x,y])\free{S1 }} 3548\indentrel{3}\begin{verbatim} 3549 (4) 3550 +------+ +------+ 3551 [[x = - \|a + 50 , y = 5], [x = \|a + 50 , y = 5], 3552 +-------+ +-------+ 3553 \|- a + 1 \|- a + 1 3554 [x = 1, y = ----------], [x = 1, y = - ----------]] 3555 +-+ +-+ 3556 \|2 \|2 3557 Type: List(List(Equation(Expression(Integer)))) 3558\end{verbatim} 3559\indentrel{-3}\end{paste}\end{patch} 3560 3561\begin{patch}{ugIntroSolutionPageEmpty4} 3562\begin{paste}{ugIntroSolutionPageEmpty4}{ugIntroSolutionPagePatch4} 3563\pastebutton{ugIntroSolutionPageEmpty4}{\showpaste} 3564\tab{5}\spadcommand{radicalSolve(S(a),[x,y])\free{S1 }} 3565\end{paste}\end{patch} 3566 3567\begin{patch}{ugIntroSolutionPagePatch5} 3568\begin{paste}{ugIntroSolutionPageFull5}{ugIntroSolutionPageEmpty5} 3569\pastebutton{ugIntroSolutionPageFull5}{\hidepaste} 3570\tab{5}\spadcommand{eqns := [x^2 - y + z,x^2*z + x^4 - b*y, y^2 *z - a - b*x]\bound{e }} 3571\indentrel{3}\begin{verbatim} 3572 2 2 4 2 3573 (5) [z - y + x , x z - b y + x , y z - b x - a] 3574 Type: List(Polynomial(Integer)) 3575\end{verbatim} 3576\indentrel{-3}\end{paste}\end{patch} 3577 3578\begin{patch}{ugIntroSolutionPageEmpty5} 3579\begin{paste}{ugIntroSolutionPageEmpty5}{ugIntroSolutionPagePatch5} 3580\pastebutton{ugIntroSolutionPageEmpty5}{\showpaste} 3581\tab{5}\spadcommand{eqns := [x^2 - y + z,x^2*z + x^4 - b*y, y^2 *z - a - b*x]\bound{e }} 3582\end{paste}\end{patch} 3583 3584\begin{patch}{ugIntroSolutionPagePatch6} 3585\begin{paste}{ugIntroSolutionPageFull6}{ugIntroSolutionPageEmpty6} 3586\pastebutton{ugIntroSolutionPageFull6}{\hidepaste} 3587\tab{5}\spadcommand{solve(eqns,[x,y,z])\free{e }} 3588\indentrel{3}\begin{verbatim} 3589 (6) 3590 2 3591 a a 3592 [[x = - -, y = 0, z = - --], 3593 b 2 3594 b 3595 3596 3 2 2 3597 z + 2 b z + b z - a 3598 [x = ---------------------, y = z + b, 3599 b 3600 3601 6 5 2 4 3 3 3602 z + 4 b z + 6 b z + (4 b - 2 a)z 3603 + 3604 4 2 2 3 2 3605 (b - 4 a b)z - 2 a b z - b + a 3606 = 3607 0 3608 ] 3609 ] 3610Type: List(List(Equation(Fraction(Polynomial(Integer))))) 3611\end{verbatim} 3612\indentrel{-3}\end{paste}\end{patch} 3613 3614\begin{patch}{ugIntroSolutionPageEmpty6} 3615\begin{paste}{ugIntroSolutionPageEmpty6}{ugIntroSolutionPagePatch6} 3616\pastebutton{ugIntroSolutionPageEmpty6}{\showpaste} 3617\tab{5}\spadcommand{solve(eqns,[x,y,z])\free{e }} 3618\end{paste}\end{patch} 3619 3620\begin{patch}{ugIntroPreviousPagePatch1} 3621\begin{paste}{ugIntroPreviousPageFull1}{ugIntroPreviousPageEmpty1} 3622\pastebutton{ugIntroPreviousPageFull1}{\hidepaste} 3623\tab{5}\spadcommand{10 ^ 10\bound{prev }} 3624\indentrel{3}\begin{verbatim} 3625 (1) 10000000000 3626 Type: PositiveInteger 3627\end{verbatim} 3628\indentrel{-3}\end{paste}\end{patch} 3629 3630\begin{patch}{ugIntroPreviousPageEmpty1} 3631\begin{paste}{ugIntroPreviousPageEmpty1}{ugIntroPreviousPagePatch1} 3632\pastebutton{ugIntroPreviousPageEmpty1}{\showpaste} 3633\tab{5}\spadcommand{10 ^ 10\bound{prev }} 3634\end{paste}\end{patch} 3635 3636\begin{patch}{ugIntroPreviousPagePatch2} 3637\begin{paste}{ugIntroPreviousPageFull2}{ugIntroPreviousPageEmpty2} 3638\pastebutton{ugIntroPreviousPageFull2}{\hidepaste} 3639\tab{5}\spadcommand{\% - 1\free{prev }\bound{prev1 }} 3640\indentrel{3}\begin{verbatim} 3641 (2) 9999999999 3642 Type: PositiveInteger 3643\end{verbatim} 3644\indentrel{-3}\end{paste}\end{patch} 3645 3646\begin{patch}{ugIntroPreviousPageEmpty2} 3647\begin{paste}{ugIntroPreviousPageEmpty2}{ugIntroPreviousPagePatch2} 3648\pastebutton{ugIntroPreviousPageEmpty2}{\showpaste} 3649\tab{5}\spadcommand{\% - 1\free{prev }\bound{prev1 }} 3650\end{paste}\end{patch} 3651 3652\begin{patch}{ugIntroPreviousPagePatch3} 3653\begin{paste}{ugIntroPreviousPageFull3}{ugIntroPreviousPageEmpty3} 3654\pastebutton{ugIntroPreviousPageFull3}{\hidepaste} 3655\tab{5}\spadcommand{\%\%(-1)\free{prev1 }\bound{prev2 }} 3656\indentrel{3}\begin{verbatim} 3657 (3) 9999999999 3658 Type: PositiveInteger 3659\end{verbatim} 3660\indentrel{-3}\end{paste}\end{patch} 3661 3662\begin{patch}{ugIntroPreviousPageEmpty3} 3663\begin{paste}{ugIntroPreviousPageEmpty3}{ugIntroPreviousPagePatch3} 3664\pastebutton{ugIntroPreviousPageEmpty3}{\showpaste} 3665\tab{5}\spadcommand{\%\%(-1)\free{prev1 }\bound{prev2 }} 3666\end{paste}\end{patch} 3667 3668\begin{patch}{ugIntroPreviousPagePatch4} 3669\begin{paste}{ugIntroPreviousPageFull4}{ugIntroPreviousPageEmpty4} 3670\pastebutton{ugIntroPreviousPageFull4}{\hidepaste} 3671\tab{5}\spadcommand{\%\%(1)\free{prev2 }} 3672\indentrel{3}\begin{verbatim} 3673 (4) 10000000000 3674 Type: PositiveInteger 3675\end{verbatim} 3676\indentrel{-3}\end{paste}\end{patch} 3677 3678\begin{patch}{ugIntroPreviousPageEmpty4} 3679\begin{paste}{ugIntroPreviousPageEmpty4}{ugIntroPreviousPagePatch4} 3680\pastebutton{ugIntroPreviousPageEmpty4}{\showpaste} 3681\tab{5}\spadcommand{\%\%(1)\free{prev2 }} 3682\end{paste}\end{patch} 3683 3684