\begin{patch}{ugIntroCalcDerivPagePatch1} \begin{paste}{ugIntroCalcDerivPageFull1}{ugIntroCalcDerivPageEmpty1} \pastebutton{ugIntroCalcDerivPageFull1}{\hidepaste} \tab{5}\spadcommand{f := exp exp x\bound{f }} \indentrel{3}\begin{verbatim} x %e (1) %e Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPageEmpty1} \begin{paste}{ugIntroCalcDerivPageEmpty1}{ugIntroCalcDerivPagePatch1} \pastebutton{ugIntroCalcDerivPageEmpty1}{\showpaste} \tab{5}\spadcommand{f := exp exp x\bound{f }} \end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPagePatch2} \begin{paste}{ugIntroCalcDerivPageFull2}{ugIntroCalcDerivPageEmpty2} \pastebutton{ugIntroCalcDerivPageFull2}{\hidepaste} \tab{5}\spadcommand{D(f, x)\free{f }} \indentrel{3}\begin{verbatim} x x %e (2) %e %e Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPageEmpty2} \begin{paste}{ugIntroCalcDerivPageEmpty2}{ugIntroCalcDerivPagePatch2} \pastebutton{ugIntroCalcDerivPageEmpty2}{\showpaste} \tab{5}\spadcommand{D(f, x)\free{f }} \end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPagePatch3} \begin{paste}{ugIntroCalcDerivPageFull3}{ugIntroCalcDerivPageEmpty3} \pastebutton{ugIntroCalcDerivPageFull3}{\hidepaste} \tab{5}\spadcommand{D(f, x, 4)\free{f }} \indentrel{3}\begin{verbatim} x x 4 x 3 x 2 x %e (3) ((%e ) + 6(%e ) + 7(%e ) + %e )%e Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPageEmpty3} \begin{paste}{ugIntroCalcDerivPageEmpty3}{ugIntroCalcDerivPagePatch3} \pastebutton{ugIntroCalcDerivPageEmpty3}{\showpaste} \tab{5}\spadcommand{D(f, x, 4)\free{f }} \end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPagePatch4} \begin{paste}{ugIntroCalcDerivPageFull4}{ugIntroCalcDerivPageEmpty4} \pastebutton{ugIntroCalcDerivPageFull4}{\hidepaste} \tab{5}\spadcommand{g := sin(x**2 + y)\bound{g }} \indentrel{3}\begin{verbatim} 2 (4) sin(y + x ) Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPageEmpty4} \begin{paste}{ugIntroCalcDerivPageEmpty4}{ugIntroCalcDerivPagePatch4} \pastebutton{ugIntroCalcDerivPageEmpty4}{\showpaste} \tab{5}\spadcommand{g := sin(x**2 + y)\bound{g }} \end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPagePatch5} \begin{paste}{ugIntroCalcDerivPageFull5}{ugIntroCalcDerivPageEmpty5} \pastebutton{ugIntroCalcDerivPageFull5}{\hidepaste} \tab{5}\spadcommand{D(g, y)\free{g }} \indentrel{3}\begin{verbatim} 2 (5) cos(y + x ) Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPageEmpty5} \begin{paste}{ugIntroCalcDerivPageEmpty5}{ugIntroCalcDerivPagePatch5} \pastebutton{ugIntroCalcDerivPageEmpty5}{\showpaste} \tab{5}\spadcommand{D(g, y)\free{g }} \end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPagePatch6} \begin{paste}{ugIntroCalcDerivPageFull6}{ugIntroCalcDerivPageEmpty6} \pastebutton{ugIntroCalcDerivPageFull6}{\hidepaste} \tab{5}\spadcommand{D(g, [y, y, x, x])\free{g }} \indentrel{3}\begin{verbatim} 2 2 2 (6) 4x sin(y + x ) - 2cos(y + x ) Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPageEmpty6} \begin{paste}{ugIntroCalcDerivPageEmpty6}{ugIntroCalcDerivPagePatch6} \pastebutton{ugIntroCalcDerivPageEmpty6}{\showpaste} \tab{5}\spadcommand{D(g, [y, y, x, x])\free{g }} \end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPagePatch7} \begin{paste}{ugIntroCalcDerivPageFull7}{ugIntroCalcDerivPageEmpty7} \pastebutton{ugIntroCalcDerivPageFull7}{\hidepaste} \tab{5}\spadcommand{D(F,x)} \indentrel{3}\begin{verbatim} (7) 0 Type: Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPageEmpty7} \begin{paste}{ugIntroCalcDerivPageEmpty7}{ugIntroCalcDerivPagePatch7} \pastebutton{ugIntroCalcDerivPageEmpty7}{\showpaste} \tab{5}\spadcommand{D(F,x)} \end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPagePatch8} \begin{paste}{ugIntroCalcDerivPageFull8}{ugIntroCalcDerivPageEmpty8} \pastebutton{ugIntroCalcDerivPageFull8}{\hidepaste} \tab{5}\spadcommand{F := operator 'F; x := operator 'x; y := operator 'y\bound{F x y }} \indentrel{3}\begin{verbatim} (8) y Type: BasicOperator \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPageEmpty8} \begin{paste}{ugIntroCalcDerivPageEmpty8}{ugIntroCalcDerivPagePatch8} \pastebutton{ugIntroCalcDerivPageEmpty8}{\showpaste} \tab{5}\spadcommand{F := operator 'F; x := operator 'x; y := operator 'y\bound{F x y }} \end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPagePatch9} \begin{paste}{ugIntroCalcDerivPageFull9}{ugIntroCalcDerivPageEmpty9} \pastebutton{ugIntroCalcDerivPageFull9}{\hidepaste} \tab{5}\spadcommand{a := F(x z, y z, z**2) + x y(z+1)\bound{a }\free{F }\free{x }\free{y }} \indentrel{3}\begin{verbatim} 2 (9) x(y(z + 1)) + F(x(z),y(z),z ) Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPageEmpty9} \begin{paste}{ugIntroCalcDerivPageEmpty9}{ugIntroCalcDerivPagePatch9} \pastebutton{ugIntroCalcDerivPageEmpty9}{\showpaste} \tab{5}\spadcommand{a := F(x z, y z, z**2) + x y(z+1)\bound{a }\free{F }\free{x }\free{y }} \end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPagePatch10} \begin{paste}{ugIntroCalcDerivPageFull10}{ugIntroCalcDerivPageEmpty10} \pastebutton{ugIntroCalcDerivPageFull10}{\hidepaste} \tab{5}\spadcommand{dadz := D(a, z)\bound{da }\free{a }} \indentrel{3}\begin{verbatim} (10) 2 , 2 2zF (x(z),y(z),z ) + y (z)F (x(z),y(z),z ) ,3 ,2 + , 2 , , x (z)F (x(z),y(z),z ) + x (y(z + 1))y (z + 1) ,1 Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPageEmpty10} \begin{paste}{ugIntroCalcDerivPageEmpty10}{ugIntroCalcDerivPagePatch10} \pastebutton{ugIntroCalcDerivPageEmpty10}{\showpaste} \tab{5}\spadcommand{dadz := D(a, z)\bound{da }\free{a }} \end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPagePatch11} \begin{paste}{ugIntroCalcDerivPageFull11}{ugIntroCalcDerivPageEmpty11} \pastebutton{ugIntroCalcDerivPageFull11}{\hidepaste} \tab{5}\spadcommand{eval(eval(dadz, 'x, z +-> exp z), 'y, z +-> log(z+1))\free{da }} \indentrel{3}\begin{verbatim} (11) 2 z 2 (2z + 2z)F (%e ,log(z + 1),z ) ,3 + z 2 F (%e ,log(z + 1),z ) ,2 + z z 2 (z + 1)%e F (%e ,log(z + 1),z ) + z + 1 ,1 / z + 1 Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPageEmpty11} \begin{paste}{ugIntroCalcDerivPageEmpty11}{ugIntroCalcDerivPagePatch11} \pastebutton{ugIntroCalcDerivPageEmpty11}{\showpaste} \tab{5}\spadcommand{eval(eval(dadz, 'x, z +-> exp z), 'y, z +-> log(z+1))\free{da }} \end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPagePatch12} \begin{paste}{ugIntroCalcDerivPageFull12}{ugIntroCalcDerivPageEmpty12} \pastebutton{ugIntroCalcDerivPageFull12}{\hidepaste} \tab{5}\spadcommand{eval(eval(a, 'x, z +-> exp z), 'y, z +-> log(z+1))\free{a }\bound{eva }} \indentrel{3}\begin{verbatim} z 2 (12) F(%e ,log(z + 1),z ) + z + 2 Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPageEmpty12} \begin{paste}{ugIntroCalcDerivPageEmpty12}{ugIntroCalcDerivPagePatch12} \pastebutton{ugIntroCalcDerivPageEmpty12}{\showpaste} \tab{5}\spadcommand{eval(eval(a, 'x, z +-> exp z), 'y, z +-> log(z+1))\free{a }\bound{eva }} \end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPagePatch13} \begin{paste}{ugIntroCalcDerivPageFull13}{ugIntroCalcDerivPageEmpty13} \pastebutton{ugIntroCalcDerivPageFull13}{\hidepaste} \tab{5}\spadcommand{D(\%, z)\free{eva }} \indentrel{3}\begin{verbatim} (13) 2 z 2 (2z + 2z)F (%e ,log(z + 1),z ) ,3 + z 2 F (%e ,log(z + 1),z ) ,2 + z z 2 (z + 1)%e F (%e ,log(z + 1),z ) + z + 1 ,1 / z + 1 Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcDerivPageEmpty13} \begin{paste}{ugIntroCalcDerivPageEmpty13}{ugIntroCalcDerivPagePatch13} \pastebutton{ugIntroCalcDerivPageEmpty13}{\showpaste} \tab{5}\spadcommand{D(\%, z)\free{eva }} \end{paste}\end{patch} \begin{patch}{ugIntroVariablesPagePatch1} \begin{paste}{ugIntroVariablesPageFull1}{ugIntroVariablesPageEmpty1} \pastebutton{ugIntroVariablesPageFull1}{\hidepaste} \tab{5}\spadcommand{(x**2 - x*y**3 +3*y)**2} \indentrel{3}\begin{verbatim} 2 6 4 3 3 2 2 4 (1) x y - 6x y - 2x y + 9y + 6x y + x Type: Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroVariablesPageEmpty1} \begin{paste}{ugIntroVariablesPageEmpty1}{ugIntroVariablesPagePatch1} \pastebutton{ugIntroVariablesPageEmpty1}{\showpaste} \tab{5}\spadcommand{(x**2 - x*y**3 +3*y)**2} \end{paste}\end{patch} \begin{patch}{ugIntroVariablesPagePatch2} \begin{paste}{ugIntroVariablesPageFull2}{ugIntroVariablesPageEmpty2} \pastebutton{ugIntroVariablesPageFull2}{\hidepaste} \tab{5}\spadcommand{p: UP(x,INT) := (3*x-1)**2 * (2*x + 8)} \indentrel{3}\begin{verbatim} 3 2 (2) 18x + 60x - 46x + 8 Type: UnivariatePolynomial(x,Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroVariablesPageEmpty2} \begin{paste}{ugIntroVariablesPageEmpty2}{ugIntroVariablesPagePatch2} \pastebutton{ugIntroVariablesPageEmpty2}{\showpaste} \tab{5}\spadcommand{p: UP(x,INT) := (3*x-1)**2 * (2*x + 8)} \end{paste}\end{patch} \begin{patch}{ugIntroVariablesPagePatch3} \begin{paste}{ugIntroVariablesPageFull3}{ugIntroVariablesPageEmpty3} \pastebutton{ugIntroVariablesPageFull3}{\hidepaste} \tab{5}\spadcommand{m: MPOLY([x,y],INT) := (x**2-x*y**3+3*y)**2\bound{m }} \indentrel{3}\begin{verbatim} 4 3 3 6 2 4 2 (3) x - 2y x + (y + 6y)x - 6y x + 9y Type: MultivariatePolynomial([x,y],Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroVariablesPageEmpty3} \begin{paste}{ugIntroVariablesPageEmpty3}{ugIntroVariablesPagePatch3} \pastebutton{ugIntroVariablesPageEmpty3}{\showpaste} \tab{5}\spadcommand{m: MPOLY([x,y],INT) := (x**2-x*y**3+3*y)**2\bound{m }} \end{paste}\end{patch} \begin{patch}{ugIntroVariablesPagePatch4} \begin{paste}{ugIntroVariablesPageFull4}{ugIntroVariablesPageEmpty4} \pastebutton{ugIntroVariablesPageFull4}{\hidepaste} \tab{5}\spadcommand{m :: MPOLY([y,x],INT)\free{m }} \indentrel{3}\begin{verbatim} 2 6 4 3 3 2 2 4 (4) x y - 6x y - 2x y + 9y + 6x y + x Type: MultivariatePolynomial([y,x],Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroVariablesPageEmpty4} \begin{paste}{ugIntroVariablesPageEmpty4}{ugIntroVariablesPagePatch4} \pastebutton{ugIntroVariablesPageEmpty4}{\showpaste} \tab{5}\spadcommand{m :: MPOLY([y,x],INT)\free{m }} \end{paste}\end{patch} \begin{patch}{ugIntroVariablesPagePatch5} \begin{paste}{ugIntroVariablesPageFull5}{ugIntroVariablesPageEmpty5} \pastebutton{ugIntroVariablesPageFull5}{\hidepaste} \tab{5}\spadcommand{m :: DMP([y,x],INT)\free{m }} \indentrel{3}\begin{verbatim} 6 2 4 3 3 2 2 4 (5) y x - 6y x - 2y x + 9y + 6y x + x Type: DistributedMultivariatePolynomial([y,x],Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroVariablesPageEmpty5} \begin{paste}{ugIntroVariablesPageEmpty5}{ugIntroVariablesPagePatch5} \pastebutton{ugIntroVariablesPageEmpty5}{\showpaste} \tab{5}\spadcommand{m :: DMP([y,x],INT)\free{m }} \end{paste}\end{patch} \begin{patch}{ugIntroVariablesPagePatch6} \begin{paste}{ugIntroVariablesPageFull6}{ugIntroVariablesPageEmpty6} \pastebutton{ugIntroVariablesPageFull6}{\hidepaste} \tab{5}\spadcommand{m :: HDMP([y,x],INT)\free{m }} \indentrel{3}\begin{verbatim} 6 2 3 3 4 4 2 2 (6) y x - 2y x - 6y x + x + 6y x + 9y Type: HomogeneousDistributedMultivariatePolynomial([y,x],Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroVariablesPageEmpty6} \begin{paste}{ugIntroVariablesPageEmpty6}{ugIntroVariablesPagePatch6} \pastebutton{ugIntroVariablesPageEmpty6}{\showpaste} \tab{5}\spadcommand{m :: HDMP([y,x],INT)\free{m }} \end{paste}\end{patch} \begin{patch}{ugIntroIntegratePagePatch1} \begin{paste}{ugIntroIntegratePageFull1}{ugIntroIntegratePageEmpty1} \pastebutton{ugIntroIntegratePageFull1}{\hidepaste} \tab{5}\spadcommand{integrate((x**2+2*x+1)/((x+1)**6+1),x)} \indentrel{3}\begin{verbatim} 3 2 atan(x + 3x + 3x + 1) (1) ����������������������� 3 Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroIntegratePageEmpty1} \begin{paste}{ugIntroIntegratePageEmpty1}{ugIntroIntegratePagePatch1} \pastebutton{ugIntroIntegratePageEmpty1}{\showpaste} \tab{5}\spadcommand{integrate((x**2+2*x+1)/((x+1)**6+1),x)} \end{paste}\end{patch} \begin{patch}{ugIntroIntegratePagePatch2} \begin{paste}{ugIntroIntegratePageFull2}{ugIntroIntegratePageEmpty2} \pastebutton{ugIntroIntegratePageFull2}{\hidepaste} \tab{5}\spadcommand{integrate(1/(x**2 + a),x)} \indentrel{3}\begin{verbatim} 2 ���Ŀ (x - a)\�- a + 2a x �Ŀ log(���������������������) x\�a 2 atan(�����) x + a a (2) [��������������������������,�����������] ���Ŀ �Ŀ 2\�- a \�a Type: Union(List Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroIntegratePageEmpty2} \begin{paste}{ugIntroIntegratePageEmpty2}{ugIntroIntegratePagePatch2} \pastebutton{ugIntroIntegratePageEmpty2}{\showpaste} \tab{5}\spadcommand{integrate(1/(x**2 + a),x)} \end{paste}\end{patch} \begin{patch}{ugIntroIntegratePagePatch3} \begin{paste}{ugIntroIntegratePageFull3}{ugIntroIntegratePageEmpty3} \pastebutton{ugIntroIntegratePageFull3}{\hidepaste} \tab{5}\spadcommand{complexIntegrate(1/(x**2 + a),x)} \indentrel{3}\begin{verbatim} ���Ŀ ���Ŀ x\�- a + a x\�- a - a log(�����������) - log(�����������) ���Ŀ ���Ŀ \�- a \�- a (3) ����������������������������������� ���Ŀ 2\�- a Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroIntegratePageEmpty3} \begin{paste}{ugIntroIntegratePageEmpty3}{ugIntroIntegratePagePatch3} \pastebutton{ugIntroIntegratePageEmpty3}{\showpaste} \tab{5}\spadcommand{complexIntegrate(1/(x**2 + a),x)} \end{paste}\end{patch} \begin{patch}{ugIntroIntegratePagePatch4} \begin{paste}{ugIntroIntegratePageFull4}{ugIntroIntegratePageEmpty4} \pastebutton{ugIntroIntegratePageFull4}{\hidepaste} \tab{5}\spadcommand{integrate(x**3 / (a+b*x)**(1/3),x)} \indentrel{3}\begin{verbatim} (4) 3 3 2 2 2 3 3�������Ŀ2 (120b x - 135a b x + 162a b x - 243a )\�b x + a ��������������������������������������������������� 4 440b Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroIntegratePageEmpty4} \begin{paste}{ugIntroIntegratePageEmpty4}{ugIntroIntegratePagePatch4} \pastebutton{ugIntroIntegratePageEmpty4}{\showpaste} \tab{5}\spadcommand{integrate(x**3 / (a+b*x)**(1/3),x)} \end{paste}\end{patch} \begin{patch}{ugIntroIntegratePagePatch5} \begin{paste}{ugIntroIntegratePageFull5}{ugIntroIntegratePageEmpty5} \pastebutton{ugIntroIntegratePageFull5}{\hidepaste} \tab{5}\spadcommand{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)} \indentrel{3}\begin{verbatim} (5) - 2 2 �Ŀ 2b x \�3 * 3�Ŀ3�������Ŀ2 3�Ŀ2 3�������Ŀ log(\�a \�b x + a + \�a \�b x + a + a) + 2 2 �Ŀ 3�Ŀ2 3�������Ŀ 4b x \�3 log(\�a \�b x + a - a) + �Ŀ3�Ŀ2 3�������Ŀ �Ŀ 2 2 2\�3 \�a \�b x + a + a\�3 12b x atan(����������������������������) 3a + �Ŀ3�Ŀ3�������Ŀ2 (12b x - 9a)\�3 \�a \�b x + a / 2 2 �Ŀ3�Ŀ 18a x \�3 \�a Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroIntegratePageEmpty5} \begin{paste}{ugIntroIntegratePageEmpty5}{ugIntroIntegratePagePatch5} \pastebutton{ugIntroIntegratePageEmpty5}{\showpaste} \tab{5}\spadcommand{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)} \end{paste}\end{patch} \begin{patch}{ugIntroIntegratePagePatch6} \begin{paste}{ugIntroIntegratePageFull6}{ugIntroIntegratePageEmpty6} \pastebutton{ugIntroIntegratePageFull6}{\hidepaste} \tab{5}\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)} \indentrel{3}\begin{verbatim} x ��������Ŀ ڿ log(\�b + %S a + 1) (6) � �������������������� d%S �� %S Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroIntegratePageEmpty6} \begin{paste}{ugIntroIntegratePageEmpty6}{ugIntroIntegratePagePatch6} \pastebutton{ugIntroIntegratePageEmpty6}{\showpaste} \tab{5}\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)} \end{paste}\end{patch} \begin{patch}{ugIntroIntegratePagePatch7} \begin{paste}{ugIntroIntegratePageFull7}{ugIntroIntegratePageEmpty7} \pastebutton{ugIntroIntegratePageFull7}{\hidepaste} \tab{5}\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b)) / (sqrt(x+b) * (x + cosh(1+sqrt(x + b)))), x)} \indentrel{3}\begin{verbatim} (7) �����Ŀ - 2cosh(\�x + b + 1) - 2x 2log(���������������������������������������) �����Ŀ �����Ŀ sinh(\�x + b + 1) - cosh(\�x + b + 1) + �����Ŀ - 2\�x + b Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroIntegratePageEmpty7} \begin{paste}{ugIntroIntegratePageEmpty7}{ugIntroIntegratePagePatch7} \pastebutton{ugIntroIntegratePageEmpty7}{\showpaste} \tab{5}\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b)) / (sqrt(x+b) * (x + cosh(1+sqrt(x + b)))), x)} \end{paste}\end{patch} \begin{patch}{ugIntroIntegratePagePatch8} \begin{paste}{ugIntroIntegratePageFull8}{ugIntroIntegratePageEmpty8} \pastebutton{ugIntroIntegratePageFull8}{\hidepaste} \tab{5}\spadcommand{integrate(tan(atan(x)/3),x)} \indentrel{3}\begin{verbatim} (8) atan(x) 2 atan(x) 2 8log(3tan(�������) - 1) - 3tan(�������) 3 3 + atan(x) 18x tan(�������) 3 / 18 Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroIntegratePageEmpty8} \begin{paste}{ugIntroIntegratePageEmpty8}{ugIntroIntegratePagePatch8} \pastebutton{ugIntroIntegratePageEmpty8}{\showpaste} \tab{5}\spadcommand{integrate(tan(atan(x)/3),x)} \end{paste}\end{patch} \begin{patch}{ugIntroIntegratePagePatch9} \begin{paste}{ugIntroIntegratePageFull9}{ugIntroIntegratePageEmpty9} \pastebutton{ugIntroIntegratePageFull9}{\hidepaste} \tab{5}\spadcommand{integrate((x + 1) / (x*(x + log x) ** (3/2)), x)} \indentrel{3}\begin{verbatim} ����������Ŀ 2\�log(x) + x (9) - �������������� log(x) + x Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroIntegratePageEmpty9} \begin{paste}{ugIntroIntegratePageEmpty9}{ugIntroIntegratePagePatch9} \pastebutton{ugIntroIntegratePageEmpty9}{\showpaste} \tab{5}\spadcommand{integrate((x + 1) / (x*(x + log x) ** (3/2)), x)} \end{paste}\end{patch} \begin{patch}{ugIntroIntegratePagePatch10} \begin{paste}{ugIntroIntegratePageFull10}{ugIntroIntegratePageEmpty10} \pastebutton{ugIntroIntegratePageFull10}{\hidepaste} \tab{5}\spadcommand{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)} \indentrel{3}\begin{verbatim} ���Ŀ erf(x) - 1 ���Ŀ (erf(x) - 1)\�%pi log(����������) - 2\�%pi erf(x) + 1 (10) ������������������������������������������� 8erf(x) - 8 Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroIntegratePageEmpty10} \begin{paste}{ugIntroIntegratePageEmpty10}{ugIntroIntegratePagePatch10} \pastebutton{ugIntroIntegratePageEmpty10}{\showpaste} \tab{5}\spadcommand{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)} \end{paste}\end{patch} \begin{patch}{ugIntroCalcLimitsPagePatch1} \begin{paste}{ugIntroCalcLimitsPageFull1}{ugIntroCalcLimitsPageEmpty1} \pastebutton{ugIntroCalcLimitsPageFull1}{\hidepaste} \tab{5}\spadcommand{g := csc(a*x) / csch(b*x)\bound{g }} \indentrel{3}\begin{verbatim} csc(a x) (1) ��������� csch(b x) Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcLimitsPageEmpty1} \begin{paste}{ugIntroCalcLimitsPageEmpty1}{ugIntroCalcLimitsPagePatch1} \pastebutton{ugIntroCalcLimitsPageEmpty1}{\showpaste} \tab{5}\spadcommand{g := csc(a*x) / csch(b*x)\bound{g }} \end{paste}\end{patch} \begin{patch}{ugIntroCalcLimitsPagePatch2} \begin{paste}{ugIntroCalcLimitsPageFull2}{ugIntroCalcLimitsPageEmpty2} \pastebutton{ugIntroCalcLimitsPageFull2}{\hidepaste} \tab{5}\spadcommand{limit(g,x=0)\free{g }} \indentrel{3}\begin{verbatim} b (2) � a Type: Union(OrderedCompletion Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcLimitsPageEmpty2} \begin{paste}{ugIntroCalcLimitsPageEmpty2}{ugIntroCalcLimitsPagePatch2} \pastebutton{ugIntroCalcLimitsPageEmpty2}{\showpaste} \tab{5}\spadcommand{limit(g,x=0)\free{g }} \end{paste}\end{patch} \begin{patch}{ugIntroCalcLimitsPagePatch3} \begin{paste}{ugIntroCalcLimitsPageFull3}{ugIntroCalcLimitsPageEmpty3} \pastebutton{ugIntroCalcLimitsPageFull3}{\hidepaste} \tab{5}\spadcommand{h := (1 + k/x)**x\bound{h }} \indentrel{3}\begin{verbatim} x + k x (3) (�����) x Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcLimitsPageEmpty3} \begin{paste}{ugIntroCalcLimitsPageEmpty3}{ugIntroCalcLimitsPagePatch3} \pastebutton{ugIntroCalcLimitsPageEmpty3}{\showpaste} \tab{5}\spadcommand{h := (1 + k/x)**x\bound{h }} \end{paste}\end{patch} \begin{patch}{ugIntroCalcLimitsPagePatch4} \begin{paste}{ugIntroCalcLimitsPageFull4}{ugIntroCalcLimitsPageEmpty4} \pastebutton{ugIntroCalcLimitsPageFull4}{\hidepaste} \tab{5}\spadcommand{limit(h,x=\%plusInfinity)\free{h }} \indentrel{3}\begin{verbatim} k (4) %e Type: Union(OrderedCompletion Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcLimitsPageEmpty4} \begin{paste}{ugIntroCalcLimitsPageEmpty4}{ugIntroCalcLimitsPagePatch4} \pastebutton{ugIntroCalcLimitsPageEmpty4}{\showpaste} \tab{5}\spadcommand{limit(h,x=\%plusInfinity)\free{h }} \end{paste}\end{patch} \begin{patch}{ugIntroCalcLimitsPagePatch5} \begin{paste}{ugIntroCalcLimitsPageFull5}{ugIntroCalcLimitsPageEmpty5} \pastebutton{ugIntroCalcLimitsPageFull5}{\hidepaste} \tab{5}\spadcommand{limit(sqrt(y**2)/y,y = 0)} \indentrel{3}\begin{verbatim} (5) [leftHandLimit= - 1,rightHandLimit= 1] Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcLimitsPageEmpty5} \begin{paste}{ugIntroCalcLimitsPageEmpty5}{ugIntroCalcLimitsPagePatch5} \pastebutton{ugIntroCalcLimitsPageEmpty5}{\showpaste} \tab{5}\spadcommand{limit(sqrt(y**2)/y,y = 0)} \end{paste}\end{patch} \begin{patch}{ugIntroCalcLimitsPagePatch6} \begin{paste}{ugIntroCalcLimitsPageFull6}{ugIntroCalcLimitsPageEmpty6} \pastebutton{ugIntroCalcLimitsPageFull6}{\hidepaste} \tab{5}\spadcommand{limit(exp(-1/x**2),x = 0)} \indentrel{3}\begin{verbatim} (6) 0 Type: Union(OrderedCompletion Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcLimitsPageEmpty6} \begin{paste}{ugIntroCalcLimitsPageEmpty6}{ugIntroCalcLimitsPagePatch6} \pastebutton{ugIntroCalcLimitsPageEmpty6}{\showpaste} \tab{5}\spadcommand{limit(exp(-1/x**2),x = 0)} \end{paste}\end{patch} \begin{patch}{ugIntroCalcLimitsPagePatch7} \begin{paste}{ugIntroCalcLimitsPageFull7}{ugIntroCalcLimitsPageEmpty7} \pastebutton{ugIntroCalcLimitsPageFull7}{\hidepaste} \tab{5}\spadcommand{complexLimit(exp(-1/x**2),x = 0)} \indentrel{3}\begin{verbatim} (7) "failed" Type: Union("failed",...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCalcLimitsPageEmpty7} \begin{paste}{ugIntroCalcLimitsPageEmpty7}{ugIntroCalcLimitsPagePatch7} \pastebutton{ugIntroCalcLimitsPageEmpty7}{\showpaste} \tab{5}\spadcommand{complexLimit(exp(-1/x**2),x = 0)} \end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPagePatch1} \begin{paste}{ugIntroSysCmmandsPageFull1}{ugIntroSysCmmandsPageEmpty1} \pastebutton{ugIntroSysCmmandsPageFull1}{\hidepaste} \tab{5}\spadcommand{)what operations integrate} \indentrel{3}\begin{verbatim} \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPageEmpty1} \begin{paste}{ugIntroSysCmmandsPageEmpty1}{ugIntroSysCmmandsPagePatch1} \pastebutton{ugIntroSysCmmandsPageEmpty1}{\showpaste} \tab{5}\spadcommand{)what operations integrate} \end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPagePatch2} \begin{paste}{ugIntroSysCmmandsPageFull2}{ugIntroSysCmmandsPageEmpty2} \pastebutton{ugIntroSysCmmandsPageFull2}{\hidepaste} \tab{5}\spadcommand{f(0) == 1; g(0) == 1\bound{u1 }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPageEmpty2} \begin{paste}{ugIntroSysCmmandsPageEmpty2}{ugIntroSysCmmandsPagePatch2} \pastebutton{ugIntroSysCmmandsPageEmpty2}{\showpaste} \tab{5}\spadcommand{f(0) == 1; g(0) == 1\bound{u1 }} \end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPagePatch3} \begin{paste}{ugIntroSysCmmandsPageFull3}{ugIntroSysCmmandsPageEmpty3} \pastebutton{ugIntroSysCmmandsPageFull3}{\hidepaste} \tab{5}\spadcommand{f(n) == e/2*f(n-1) - x*g(n-1)\bound{u2 }\free{u1 }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPageEmpty3} \begin{paste}{ugIntroSysCmmandsPageEmpty3}{ugIntroSysCmmandsPagePatch3} \pastebutton{ugIntroSysCmmandsPageEmpty3}{\showpaste} \tab{5}\spadcommand{f(n) == e/2*f(n-1) - x*g(n-1)\bound{u2 }\free{u1 }} \end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPagePatch4} \begin{paste}{ugIntroSysCmmandsPageFull4}{ugIntroSysCmmandsPageEmpty4} \pastebutton{ugIntroSysCmmandsPageFull4}{\hidepaste} \tab{5}\spadcommand{g(n) == -x*f(n-1) + d/3*g(n-1)\bound{u3 }\free{u2 }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPageEmpty4} \begin{paste}{ugIntroSysCmmandsPageEmpty4}{ugIntroSysCmmandsPagePatch4} \pastebutton{ugIntroSysCmmandsPageEmpty4}{\showpaste} \tab{5}\spadcommand{g(n) == -x*f(n-1) + d/3*g(n-1)\bound{u3 }\free{u2 }} \end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPagePatch5} \begin{paste}{ugIntroSysCmmandsPageFull5}{ugIntroSysCmmandsPageEmpty5} \pastebutton{ugIntroSysCmmandsPageFull5}{\hidepaste} \tab{5}\spadcommand{f(3)\bound{u4 }\free{u3 }} \indentrel{3}\begin{verbatim} (4) 3 1 2 1 2 1 1 2 1 3 - x + (e + � d)x + (- � e - � d e - � d )x + � e 3 4 6 9 8 Type: Polynomial Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPageEmpty5} \begin{paste}{ugIntroSysCmmandsPageEmpty5}{ugIntroSysCmmandsPagePatch5} \pastebutton{ugIntroSysCmmandsPageEmpty5}{\showpaste} \tab{5}\spadcommand{f(3)\bound{u4 }\free{u3 }} \end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPagePatch6} \begin{paste}{ugIntroSysCmmandsPageFull6}{ugIntroSysCmmandsPageEmpty6} \pastebutton{ugIntroSysCmmandsPageFull6}{\hidepaste} \tab{5}\spadcommand{)undo 2\bound{u5 }\free{u4 }} \indentrel{3}\begin{verbatim} \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPageEmpty6} \begin{paste}{ugIntroSysCmmandsPageEmpty6}{ugIntroSysCmmandsPagePatch6} \pastebutton{ugIntroSysCmmandsPageEmpty6}{\showpaste} \tab{5}\spadcommand{)undo 2\bound{u5 }\free{u4 }} \end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPagePatch7} \begin{paste}{ugIntroSysCmmandsPageFull7}{ugIntroSysCmmandsPageEmpty7} \pastebutton{ugIntroSysCmmandsPageFull7}{\hidepaste} \tab{5}\spadcommand{f(n) == d/3*f(n-1) - x*g(n-1)\bound{u6 }\free{u5 }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPageEmpty7} \begin{paste}{ugIntroSysCmmandsPageEmpty7}{ugIntroSysCmmandsPagePatch7} \pastebutton{ugIntroSysCmmandsPageEmpty7}{\showpaste} \tab{5}\spadcommand{f(n) == d/3*f(n-1) - x*g(n-1)\bound{u6 }\free{u5 }} \end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPagePatch8} \begin{paste}{ugIntroSysCmmandsPageFull8}{ugIntroSysCmmandsPageEmpty8} \pastebutton{ugIntroSysCmmandsPageFull8}{\hidepaste} \tab{5}\spadcommand{)undo )redo\bound{u7 }\free{u6 }} \indentrel{3}\begin{verbatim} \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPageEmpty8} \begin{paste}{ugIntroSysCmmandsPageEmpty8}{ugIntroSysCmmandsPagePatch8} \pastebutton{ugIntroSysCmmandsPageEmpty8}{\showpaste} \tab{5}\spadcommand{)undo )redo\bound{u7 }\free{u6 }} \end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPagePatch9} \begin{paste}{ugIntroSysCmmandsPageFull9}{ugIntroSysCmmandsPageEmpty9} \pastebutton{ugIntroSysCmmandsPageFull9}{\hidepaste} \tab{5}\spadcommand{)undo 4\bound{u8 }\free{u7 }} \indentrel{3}\begin{verbatim} \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPageEmpty9} \begin{paste}{ugIntroSysCmmandsPageEmpty9}{ugIntroSysCmmandsPagePatch9} \pastebutton{ugIntroSysCmmandsPageEmpty9}{\showpaste} \tab{5}\spadcommand{)undo 4\bound{u8 }\free{u7 }} \end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPagePatch10} \begin{paste}{ugIntroSysCmmandsPageFull10}{ugIntroSysCmmandsPageEmpty10} \pastebutton{ugIntroSysCmmandsPageFull10}{\hidepaste} \tab{5}\spadcommand{f(3)\bound{u9 }\free{u8 }} \indentrel{3}\begin{verbatim} (6) 3 1 2 1 2 1 1 2 1 3 - x + (e + � d)x + (- � e - � d e - � d )x + � e 3 4 6 9 8 Type: Polynomial Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSysCmmandsPageEmpty10} \begin{paste}{ugIntroSysCmmandsPageEmpty10}{ugIntroSysCmmandsPagePatch10} \pastebutton{ugIntroSysCmmandsPageEmpty10}{\showpaste} \tab{5}\spadcommand{f(3)\bound{u9 }\free{u8 }} \end{paste}\end{patch} \begin{patch}{ugIntroArithmeticPagePatch1} \begin{paste}{ugIntroArithmeticPageFull1}{ugIntroArithmeticPageEmpty1} \pastebutton{ugIntroArithmeticPageFull1}{\hidepaste} \tab{5}\spadcommand{1 + 2 - 3 / 4 * 3 ** 2 - 1} \indentrel{3}\begin{verbatim} 19 (1) - �� 4 Type: Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroArithmeticPageEmpty1} \begin{paste}{ugIntroArithmeticPageEmpty1}{ugIntroArithmeticPagePatch1} \pastebutton{ugIntroArithmeticPageEmpty1}{\showpaste} \tab{5}\spadcommand{1 + 2 - 3 / 4 * 3 ** 2 - 1} \end{paste}\end{patch} \begin{patch}{ugIntroArithmeticPagePatch2} \begin{paste}{ugIntroArithmeticPageFull2}{ugIntroArithmeticPageEmpty2} \pastebutton{ugIntroArithmeticPageFull2}{\hidepaste} \tab{5}\spadcommand{((1 + 2) - ((3 / 4) * (3 ** 2))) - 1} \indentrel{3}\begin{verbatim} 19 (2) - �� 4 Type: Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroArithmeticPageEmpty2} \begin{paste}{ugIntroArithmeticPageEmpty2}{ugIntroArithmeticPagePatch2} \pastebutton{ugIntroArithmeticPageEmpty2}{\showpaste} \tab{5}\spadcommand{((1 + 2) - ((3 / 4) * (3 ** 2))) - 1} \end{paste}\end{patch} \begin{patch}{ugIntroArithmeticPagePatch3} \begin{paste}{ugIntroArithmeticPageFull3}{ugIntroArithmeticPageEmpty3} \pastebutton{ugIntroArithmeticPageFull3}{\hidepaste} \tab{5}\spadcommand{1 + 2 - 3/ (4 * 3 ** (2 - 1))} \indentrel{3}\begin{verbatim} 11 (3) �� 4 Type: Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroArithmeticPageEmpty3} \begin{paste}{ugIntroArithmeticPageEmpty3}{ugIntroArithmeticPagePatch3} \pastebutton{ugIntroArithmeticPageEmpty3}{\showpaste} \tab{5}\spadcommand{1 + 2 - 3/ (4 * 3 ** (2 - 1))} \end{paste}\end{patch} \begin{patch}{ugIntroConversionPagePatch1} \begin{paste}{ugIntroConversionPageFull1}{ugIntroConversionPageEmpty1} \pastebutton{ugIntroConversionPageFull1}{\hidepaste} \tab{5}\spadcommand{p := r**2 + 2/3\bound{p }} \indentrel{3}\begin{verbatim} 2 2 (1) r + � 3 Type: Polynomial Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroConversionPageEmpty1} \begin{paste}{ugIntroConversionPageEmpty1}{ugIntroConversionPagePatch1} \pastebutton{ugIntroConversionPageEmpty1}{\showpaste} \tab{5}\spadcommand{p := r**2 + 2/3\bound{p }} \end{paste}\end{patch} \begin{patch}{ugIntroConversionPagePatch2} \begin{paste}{ugIntroConversionPageFull2}{ugIntroConversionPageEmpty2} \pastebutton{ugIntroConversionPageFull2}{\hidepaste} \tab{5}\spadcommand{p :: Fraction Polynomial Integer\free{p }} \indentrel{3}\begin{verbatim} 2 3r + 2 (2) ������� 3 Type: Fraction Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroConversionPageEmpty2} \begin{paste}{ugIntroConversionPageEmpty2}{ugIntroConversionPagePatch2} \pastebutton{ugIntroConversionPageEmpty2}{\showpaste} \tab{5}\spadcommand{p :: Fraction Polynomial Integer\free{p }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch1} \begin{paste}{ugIntroYouPageFull1}{ugIntroYouPageEmpty1} \pastebutton{ugIntroYouPageFull1}{\hidepaste} \tab{5}\spadcommand{fact(0) == 1\bound{fact }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty1} \begin{paste}{ugIntroYouPageEmpty1}{ugIntroYouPagePatch1} \pastebutton{ugIntroYouPageEmpty1}{\showpaste} \tab{5}\spadcommand{fact(0) == 1\bound{fact }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch2} \begin{paste}{ugIntroYouPageFull2}{ugIntroYouPageEmpty2} \pastebutton{ugIntroYouPageFull2}{\hidepaste} \tab{5}\spadcommand{fact(n) == n*fact(n-1)\bound{facta }\free{fact }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty2} \begin{paste}{ugIntroYouPageEmpty2}{ugIntroYouPagePatch2} \pastebutton{ugIntroYouPageEmpty2}{\showpaste} \tab{5}\spadcommand{fact(n) == n*fact(n-1)\bound{facta }\free{fact }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch3} \begin{paste}{ugIntroYouPageFull3}{ugIntroYouPageEmpty3} \pastebutton{ugIntroYouPageFull3}{\hidepaste} \tab{5}\spadcommand{fact(50)\free{facta }} \indentrel{3}\begin{verbatim} (3) 304140932017133780436126081660647688443776415689605120_ 00000000000 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty3} \begin{paste}{ugIntroYouPageEmpty3}{ugIntroYouPagePatch3} \pastebutton{ugIntroYouPageEmpty3}{\showpaste} \tab{5}\spadcommand{fact(50)\free{facta }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch4} \begin{paste}{ugIntroYouPageFull4}{ugIntroYouPageEmpty4} \pastebutton{ugIntroYouPageFull4}{\hidepaste} \tab{5}\spadcommand{fac(n) == if n < 3 then n else n * fac(n - 1)\bound{fac }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty4} \begin{paste}{ugIntroYouPageEmpty4}{ugIntroYouPagePatch4} \pastebutton{ugIntroYouPageEmpty4}{\showpaste} \tab{5}\spadcommand{fac(n) == if n < 3 then n else n * fac(n - 1)\bound{fac }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch5} \begin{paste}{ugIntroYouPageFull5}{ugIntroYouPageEmpty5} \pastebutton{ugIntroYouPageFull5}{\hidepaste} \tab{5}\spadcommand{fac(50)\free{fac }} \indentrel{3}\begin{verbatim} (5) 304140932017133780436126081660647688443776415689605120_ 00000000000 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty5} \begin{paste}{ugIntroYouPageEmpty5}{ugIntroYouPagePatch5} \pastebutton{ugIntroYouPageEmpty5}{\showpaste} \tab{5}\spadcommand{fac(50)\free{fac }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch6} \begin{paste}{ugIntroYouPageFull6}{ugIntroYouPageEmpty6} \pastebutton{ugIntroYouPageFull6}{\hidepaste} \tab{5}\spadcommand{fa(n) == (a := 1; for i in 2..n repeat a := a*i; a)\bound{fa }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty6} \begin{paste}{ugIntroYouPageEmpty6}{ugIntroYouPagePatch6} \pastebutton{ugIntroYouPageEmpty6}{\showpaste} \tab{5}\spadcommand{fa(n) == (a := 1; for i in 2..n repeat a := a*i; a)\bound{fa }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch7} \begin{paste}{ugIntroYouPageFull7}{ugIntroYouPageEmpty7} \pastebutton{ugIntroYouPageFull7}{\hidepaste} \tab{5}\spadcommand{fa(50)\free{fa }} \indentrel{3}\begin{verbatim} (7) 304140932017133780436126081660647688443776415689605120_ 00000000000 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty7} \begin{paste}{ugIntroYouPageEmpty7}{ugIntroYouPagePatch7} \pastebutton{ugIntroYouPageEmpty7}{\showpaste} \tab{5}\spadcommand{fa(50)\free{fa }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch8} \begin{paste}{ugIntroYouPageFull8}{ugIntroYouPageEmpty8} \pastebutton{ugIntroYouPageFull8}{\hidepaste} \tab{5}\spadcommand{f(n) == reduce(*,[i for i in 2..n])\bound{f }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty8} \begin{paste}{ugIntroYouPageEmpty8}{ugIntroYouPagePatch8} \pastebutton{ugIntroYouPageEmpty8}{\showpaste} \tab{5}\spadcommand{f(n) == reduce(*,[i for i in 2..n])\bound{f }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch9} \begin{paste}{ugIntroYouPageFull9}{ugIntroYouPageEmpty9} \pastebutton{ugIntroYouPageFull9}{\hidepaste} \tab{5}\spadcommand{f(50)\free{f }} \indentrel{3}\begin{verbatim} (9) 304140932017133780436126081660647688443776415689605120_ 00000000000 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty9} \begin{paste}{ugIntroYouPageEmpty9}{ugIntroYouPagePatch9} \pastebutton{ugIntroYouPageEmpty9}{\showpaste} \tab{5}\spadcommand{f(50)\free{f }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch10} \begin{paste}{ugIntroYouPageFull10}{ugIntroYouPageEmpty10} \pastebutton{ugIntroYouPageFull10}{\hidepaste} \tab{5}\spadcommand{factorial(50)} \indentrel{3}\begin{verbatim} (10) 304140932017133780436126081660647688443776415689605120_ 00000000000 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty10} \begin{paste}{ugIntroYouPageEmpty10}{ugIntroYouPagePatch10} \pastebutton{ugIntroYouPageEmpty10}{\showpaste} \tab{5}\spadcommand{factorial(50)} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch11} \begin{paste}{ugIntroYouPageFull11}{ugIntroYouPageEmpty11} \pastebutton{ugIntroYouPageFull11}{\hidepaste} \tab{5}\spadcommand{permMat(n, i, j) == m := diagonalMatrix [(if i = k or j = k then 0 else 1) for k in 1..n] m(i,j) := 1 m(j,i) := 1 m \bound{permMat }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty11} \begin{paste}{ugIntroYouPageEmpty11}{ugIntroYouPagePatch11} \pastebutton{ugIntroYouPageEmpty11}{\showpaste} \tab{5}\spadcommand{permMat(n, i, j) == m := diagonalMatrix [(if i = k or j = k then 0 else 1) for k in 1..n] m(i,j) := 1 m(j,i) := 1 m \bound{permMat }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch12} \begin{paste}{ugIntroYouPageFull12}{ugIntroYouPageEmpty12} \pastebutton{ugIntroYouPageFull12}{\hidepaste} \tab{5}\spadcommand{p := permMat(4,2,3)\free{permMat }\bound{p }} \indentrel{3}\begin{verbatim} �1 0 0 0� � � �0 0 1 0� (12) � � �0 1 0 0� � � �0 0 0 1� Type: Matrix Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty12} \begin{paste}{ugIntroYouPageEmpty12}{ugIntroYouPagePatch12} \pastebutton{ugIntroYouPageEmpty12}{\showpaste} \tab{5}\spadcommand{p := permMat(4,2,3)\free{permMat }\bound{p }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch13} \begin{paste}{ugIntroYouPageFull13}{ugIntroYouPageEmpty13} \pastebutton{ugIntroYouPageFull13}{\hidepaste} \tab{5}\spadcommand{m := matrix [[4*i + j for j in 1..4] for i in 0..3]\bound{m }} \indentrel{3}\begin{verbatim} �1 2 3 4 � � � �5 6 7 8 � (13) � � �9 10 11 12� � � �13 14 15 16� Type: Matrix Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty13} \begin{paste}{ugIntroYouPageEmpty13}{ugIntroYouPagePatch13} \pastebutton{ugIntroYouPageEmpty13}{\showpaste} \tab{5}\spadcommand{m := matrix [[4*i + j for j in 1..4] for i in 0..3]\bound{m }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch14} \begin{paste}{ugIntroYouPageFull14}{ugIntroYouPageEmpty14} \pastebutton{ugIntroYouPageFull14}{\hidepaste} \tab{5}\spadcommand{permMat(4,2,3) * m\free{p m }} \indentrel{3}\begin{verbatim} �1 2 3 4 � � � �9 10 11 12� (14) � � �5 6 7 8 � � � �13 14 15 16� Type: Matrix Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty14} \begin{paste}{ugIntroYouPageEmpty14}{ugIntroYouPagePatch14} \pastebutton{ugIntroYouPageEmpty14}{\showpaste} \tab{5}\spadcommand{permMat(4,2,3) * m\free{p m }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch15} \begin{paste}{ugIntroYouPageFull15}{ugIntroYouPageEmpty15} \pastebutton{ugIntroYouPageFull15}{\hidepaste} \tab{5}\spadcommand{t : (Float -> Float, Float) -> Float\bound{tdecl }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty15} \begin{paste}{ugIntroYouPageEmpty15}{ugIntroYouPagePatch15} \pastebutton{ugIntroYouPageEmpty15}{\showpaste} \tab{5}\spadcommand{t : (Float -> Float, Float) -> Float\bound{tdecl }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch16} \begin{paste}{ugIntroYouPageFull16}{ugIntroYouPageEmpty16} \pastebutton{ugIntroYouPageFull16}{\hidepaste} \tab{5}\spadcommand{t(fun, x) == fun(x)**2 + sin(x)**2\free{tdecl }\bound{t }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty16} \begin{paste}{ugIntroYouPageEmpty16}{ugIntroYouPagePatch16} \pastebutton{ugIntroYouPageEmpty16}{\showpaste} \tab{5}\spadcommand{t(fun, x) == fun(x)**2 + sin(x)**2\free{tdecl }\bound{t }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch17} \begin{paste}{ugIntroYouPageFull17}{ugIntroYouPageEmpty17} \pastebutton{ugIntroYouPageFull17}{\hidepaste} \tab{5}\spadcommand{t(cos, 5.2058)\free{t }} \indentrel{3}\begin{verbatim} (17) 1.0 Type: Float \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty17} \begin{paste}{ugIntroYouPageEmpty17}{ugIntroYouPagePatch17} \pastebutton{ugIntroYouPageEmpty17}{\showpaste} \tab{5}\spadcommand{t(cos, 5.2058)\free{t }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch18} \begin{paste}{ugIntroYouPageFull18}{ugIntroYouPageEmpty18} \pastebutton{ugIntroYouPageFull18}{\hidepaste} \tab{5}\spadcommand{cosinv(y) == cos(1/y)\bound{cosinv }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty18} \begin{paste}{ugIntroYouPageEmpty18}{ugIntroYouPagePatch18} \pastebutton{ugIntroYouPageEmpty18}{\showpaste} \tab{5}\spadcommand{cosinv(y) == cos(1/y)\bound{cosinv }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch19} \begin{paste}{ugIntroYouPageFull19}{ugIntroYouPageEmpty19} \pastebutton{ugIntroYouPageFull19}{\hidepaste} \tab{5}\spadcommand{t(cosinv, 5.2058)\free{t }\free{cosinv }} \indentrel{3}\begin{verbatim} (19) 1.7392237241 800516493 Type: Float \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty19} \begin{paste}{ugIntroYouPageEmpty19}{ugIntroYouPagePatch19} \pastebutton{ugIntroYouPageEmpty19}{\showpaste} \tab{5}\spadcommand{t(cosinv, 5.2058)\free{t }\free{cosinv }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch20} \begin{paste}{ugIntroYouPageFull20}{ugIntroYouPageEmpty20} \pastebutton{ugIntroYouPageFull20}{\hidepaste} \tab{5}\spadcommand{groupSqrt := rule(sqrt(a) * sqrt(b) == sqrt(a*b))\bound{g }} \indentrel{3}\begin{verbatim} �Ŀ �Ŀ ���Ŀ (20) %BD\�a \�b == %BD\�a b Type: RewriteRule(Integer,Integer,Expression Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty20} \begin{paste}{ugIntroYouPageEmpty20}{ugIntroYouPagePatch20} \pastebutton{ugIntroYouPageEmpty20}{\showpaste} \tab{5}\spadcommand{groupSqrt := rule(sqrt(a) * sqrt(b) == sqrt(a*b))\bound{g }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch21} \begin{paste}{ugIntroYouPageFull21}{ugIntroYouPageEmpty21} \pastebutton{ugIntroYouPageFull21}{\hidepaste} \tab{5}\spadcommand{a := (sqrt(x) + sqrt(y) + sqrt(z))**4\bound{sxy }} \indentrel{3}\begin{verbatim} (21) �Ŀ �Ŀ �Ŀ ((4z + 4y + 12x)\�y + (4z + 12y + 4x)\�x )\�z + �Ŀ �Ŀ 2 2 (12z + 4y + 4x)\�x \�y + z + (6y + 6x)z + y + 6x y + 2 x Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty21} \begin{paste}{ugIntroYouPageEmpty21}{ugIntroYouPagePatch21} \pastebutton{ugIntroYouPageEmpty21}{\showpaste} \tab{5}\spadcommand{a := (sqrt(x) + sqrt(y) + sqrt(z))**4\bound{sxy }} \end{paste}\end{patch} \begin{patch}{ugIntroYouPagePatch22} \begin{paste}{ugIntroYouPageFull22}{ugIntroYouPageEmpty22} \pastebutton{ugIntroYouPageFull22}{\hidepaste} \tab{5}\spadcommand{groupSqrt a\free{sxy }\free{g }} \indentrel{3}\begin{verbatim} (22) ���Ŀ ���Ŀ (4z + 4y + 12x)\�y z + (4z + 12y + 4x)\�x z + ���Ŀ 2 2 (12z + 4y + 4x)\�x y + z + (6y + 6x)z + y + 6x y + 2 x Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroYouPageEmpty22} \begin{paste}{ugIntroYouPageEmpty22}{ugIntroYouPagePatch22} \pastebutton{ugIntroYouPageEmpty22}{\showpaste} \tab{5}\spadcommand{groupSqrt a\free{sxy }\free{g }} \end{paste}\end{patch} \begin{patch}{ugIntroTypesPagePatch1} \begin{paste}{ugIntroTypesPageFull1}{ugIntroTypesPageEmpty1} \pastebutton{ugIntroTypesPageFull1}{\hidepaste} \tab{5}\spadcommand{8} \indentrel{3}\begin{verbatim} (1) 8 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroTypesPageEmpty1} \begin{paste}{ugIntroTypesPageEmpty1}{ugIntroTypesPagePatch1} \pastebutton{ugIntroTypesPageEmpty1}{\showpaste} \tab{5}\spadcommand{8} \end{paste}\end{patch} \begin{patch}{ugIntroTypesPagePatch2} \begin{paste}{ugIntroTypesPageFull2}{ugIntroTypesPageEmpty2} \pastebutton{ugIntroTypesPageFull2}{\hidepaste} \tab{5}\spadcommand{-8} \indentrel{3}\begin{verbatim} (2) - 8 Type: Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroTypesPageEmpty2} \begin{paste}{ugIntroTypesPageEmpty2}{ugIntroTypesPagePatch2} \pastebutton{ugIntroTypesPageEmpty2}{\showpaste} \tab{5}\spadcommand{-8} \end{paste}\end{patch} \begin{patch}{ugIntroTypesPagePatch3} \begin{paste}{ugIntroTypesPageFull3}{ugIntroTypesPageEmpty3} \pastebutton{ugIntroTypesPageFull3}{\hidepaste} \tab{5}\spadcommand{x**8} \indentrel{3}\begin{verbatim} 8 (3) x Type: Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroTypesPageEmpty3} \begin{paste}{ugIntroTypesPageEmpty3}{ugIntroTypesPagePatch3} \pastebutton{ugIntroTypesPageEmpty3}{\showpaste} \tab{5}\spadcommand{x**8} \end{paste}\end{patch} \begin{patch}{ugIntroTypesPagePatch4} \begin{paste}{ugIntroTypesPageFull4}{ugIntroTypesPageEmpty4} \pastebutton{ugIntroTypesPageFull4}{\hidepaste} \tab{5}\spadcommand{x**(-8)} \indentrel{3}\begin{verbatim} 1 (4) �� 8 x Type: Fraction Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroTypesPageEmpty4} \begin{paste}{ugIntroTypesPageEmpty4}{ugIntroTypesPagePatch4} \pastebutton{ugIntroTypesPageEmpty4}{\showpaste} \tab{5}\spadcommand{x**(-8)} \end{paste}\end{patch} \begin{patch}{ugIntroTwoDimPagePatch1} \begin{paste}{ugIntroTwoDimPageFull1}{ugIntroTwoDimPageEmpty1} \pastebutton{ugIntroTwoDimPageFull1}{\hidepaste} \tab{5}\spadcommand{m := matrix([[1,2], [3,4]])\bound{m }} \indentrel{3}\begin{verbatim} �1 2� (1) � � �3 4� Type: Matrix Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroTwoDimPageEmpty1} \begin{paste}{ugIntroTwoDimPageEmpty1}{ugIntroTwoDimPagePatch1} \pastebutton{ugIntroTwoDimPageEmpty1}{\showpaste} \tab{5}\spadcommand{m := matrix([[1,2], [3,4]])\bound{m }} \end{paste}\end{patch} \begin{patch}{ugIntroTwoDimPagePatch2} \begin{paste}{ugIntroTwoDimPageFull2}{ugIntroTwoDimPageEmpty2} \pastebutton{ugIntroTwoDimPageFull2}{\hidepaste} \tab{5}\spadcommand{matrix([[1/(i + j - x) for i in 1..4] for j in 1..4])\bound{hilb }} \indentrel{3}\begin{verbatim} � 1 1 1 1 � �- ����� - ����� - ����� - ����ij � x - 2 x - 3 x - 4 x - 5� � � � 1 1 1 1 � �- ����� - ����� - ����� - ����ij � x - 3 x - 4 x - 5 x - 6� (2) � � � 1 1 1 1 � �- ����� - ����� - ����� - ����ij � x - 4 x - 5 x - 6 x - 7� � � � 1 1 1 1 � �- ����� - ����� - ����� - ����ij � x - 5 x - 6 x - 7 x - 8� Type: Matrix Fraction Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroTwoDimPageEmpty2} \begin{paste}{ugIntroTwoDimPageEmpty2}{ugIntroTwoDimPagePatch2} \pastebutton{ugIntroTwoDimPageEmpty2}{\showpaste} \tab{5}\spadcommand{matrix([[1/(i + j - x) for i in 1..4] for j in 1..4])\bound{hilb }} \end{paste}\end{patch} \begin{patch}{ugIntroTwoDimPagePatch3} \begin{paste}{ugIntroTwoDimPageFull3}{ugIntroTwoDimPageEmpty3} \pastebutton{ugIntroTwoDimPageFull3}{\hidepaste} \tab{5}\spadcommand{vm := matrix [[1,1,1], [x,y,z], [x*x,y*y,z*z]]\bound{vm }} \indentrel{3}\begin{verbatim} �1 1 1 � � � (3) �x y z � � � � 2 2 2� �x y z � Type: Matrix Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroTwoDimPageEmpty3} \begin{paste}{ugIntroTwoDimPageEmpty3}{ugIntroTwoDimPagePatch3} \pastebutton{ugIntroTwoDimPageEmpty3}{\showpaste} \tab{5}\spadcommand{vm := matrix [[1,1,1], [x,y,z], [x*x,y*y,z*z]]\bound{vm }} \end{paste}\end{patch} \begin{patch}{ugIntroTwoDimPagePatch4} \begin{paste}{ugIntroTwoDimPageFull4}{ugIntroTwoDimPageEmpty4} \pastebutton{ugIntroTwoDimPageFull4}{\hidepaste} \tab{5}\spadcommand{vm(3,3)\free{vm }} \indentrel{3}\begin{verbatim} 2 (4) z Type: Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroTwoDimPageEmpty4} \begin{paste}{ugIntroTwoDimPageEmpty4}{ugIntroTwoDimPagePatch4} \pastebutton{ugIntroTwoDimPageEmpty4}{\showpaste} \tab{5}\spadcommand{vm(3,3)\free{vm }} \end{paste}\end{patch} \begin{patch}{ugIntroTwoDimPagePatch5} \begin{paste}{ugIntroTwoDimPageFull5}{ugIntroTwoDimPageEmpty5} \pastebutton{ugIntroTwoDimPageFull5}{\hidepaste} \tab{5}\spadcommand{column(vm,2)\free{vm }} \indentrel{3}\begin{verbatim} 2 (5) [1,y,y ] Type: Vector Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroTwoDimPageEmpty5} \begin{paste}{ugIntroTwoDimPageEmpty5}{ugIntroTwoDimPagePatch5} \pastebutton{ugIntroTwoDimPageEmpty5}{\showpaste} \tab{5}\spadcommand{column(vm,2)\free{vm }} \end{paste}\end{patch} \begin{patch}{ugIntroTwoDimPagePatch6} \begin{paste}{ugIntroTwoDimPageFull6}{ugIntroTwoDimPageEmpty6} \pastebutton{ugIntroTwoDimPageFull6}{\hidepaste} \tab{5}\spadcommand{vm * vm\free{vm }} \indentrel{3}\begin{verbatim} (6) � 2 2 2 � � x + x + 1 y + y + 1 z + z + 1 � � � � 2 2 2 3 � � x z + x y + x y z + y + x z + y z + x � � � � 2 2 2 2 2 2 3 2 4 2 2� �x z + x y + x y z + y + x z + y z + x � Type: Matrix Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroTwoDimPageEmpty6} \begin{paste}{ugIntroTwoDimPageEmpty6}{ugIntroTwoDimPagePatch6} \pastebutton{ugIntroTwoDimPageEmpty6}{\showpaste} \tab{5}\spadcommand{vm * vm\free{vm }} \end{paste}\end{patch} \begin{patch}{ugIntroTwoDimPagePatch7} \begin{paste}{ugIntroTwoDimPageFull7}{ugIntroTwoDimPageEmpty7} \pastebutton{ugIntroTwoDimPageFull7}{\hidepaste} \tab{5}\spadcommand{factor determinant vm\free{vm }\bound{d }} \indentrel{3}\begin{verbatim} (7) (y - x)(z - y)(z - x) Type: Factored Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroTwoDimPageEmpty7} \begin{paste}{ugIntroTwoDimPageEmpty7}{ugIntroTwoDimPagePatch7} \pastebutton{ugIntroTwoDimPageEmpty7}{\showpaste} \tab{5}\spadcommand{factor determinant vm\free{vm }\bound{d }} \end{paste}\end{patch} \begin{patch}{ugIntroAssignPagePatch1} \begin{paste}{ugIntroAssignPageFull1}{ugIntroAssignPageEmpty1} \pastebutton{ugIntroAssignPageFull1}{\hidepaste} \tab{5}\spadcommand{(x - y*z)**2} \indentrel{3}\begin{verbatim} 2 2 2 (1) y z - 2x y z + x Type: Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroAssignPageEmpty1} \begin{paste}{ugIntroAssignPageEmpty1}{ugIntroAssignPagePatch1} \pastebutton{ugIntroAssignPageEmpty1}{\showpaste} \tab{5}\spadcommand{(x - y*z)**2} \end{paste}\end{patch} \begin{patch}{ugIntroAssignPagePatch2} \begin{paste}{ugIntroAssignPageFull2}{ugIntroAssignPageEmpty2} \pastebutton{ugIntroAssignPageFull2}{\hidepaste} \tab{5}\spadcommand{x := 4} \indentrel{3}\begin{verbatim} (2) 4 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroAssignPageEmpty2} \begin{paste}{ugIntroAssignPageEmpty2}{ugIntroAssignPagePatch2} \pastebutton{ugIntroAssignPageEmpty2}{\showpaste} \tab{5}\spadcommand{x := 4} \end{paste}\end{patch} \begin{patch}{ugIntroAssignPagePatch3} \begin{paste}{ugIntroAssignPageFull3}{ugIntroAssignPageEmpty3} \pastebutton{ugIntroAssignPageFull3}{\hidepaste} \tab{5}\spadcommand{x := z + 3/5} \indentrel{3}\begin{verbatim} 3 (3) z + � 5 Type: Polynomial Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroAssignPageEmpty3} \begin{paste}{ugIntroAssignPageEmpty3}{ugIntroAssignPagePatch3} \pastebutton{ugIntroAssignPageEmpty3}{\showpaste} \tab{5}\spadcommand{x := z + 3/5} \end{paste}\end{patch} \begin{patch}{ugIntroAssignPagePatch4} \begin{paste}{ugIntroAssignPageFull4}{ugIntroAssignPageEmpty4} \pastebutton{ugIntroAssignPageFull4}{\hidepaste} \tab{5}\spadcommand{y : Integer\bound{y }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroAssignPageEmpty4} \begin{paste}{ugIntroAssignPageEmpty4}{ugIntroAssignPagePatch4} \pastebutton{ugIntroAssignPageEmpty4}{\showpaste} \tab{5}\spadcommand{y : Integer\bound{y }} \end{paste}\end{patch} \begin{patch}{ugIntroAssignPagePatch5} \begin{paste}{ugIntroAssignPageFull5}{ugIntroAssignPageEmpty5} \pastebutton{ugIntroAssignPageFull5}{\hidepaste} \tab{5}\spadcommand{y := 89\bound{y1 }\free{y }} \indentrel{3}\begin{verbatim} (5) 89 Type: Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroAssignPageEmpty5} \begin{paste}{ugIntroAssignPageEmpty5}{ugIntroAssignPagePatch5} \pastebutton{ugIntroAssignPageEmpty5}{\showpaste} \tab{5}\spadcommand{y := 89\bound{y1 }\free{y }} \end{paste}\end{patch} \begin{patch}{ugIntroAssignPagePatch6} \begin{paste}{ugIntroAssignPageFull6}{ugIntroAssignPageEmpty6} \pastebutton{ugIntroAssignPageFull6}{\hidepaste} \tab{5}\spadcommand{y := sin \%pi} \indentrel{3}\begin{verbatim} (6) 0 Type: Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroAssignPageEmpty6} \begin{paste}{ugIntroAssignPageEmpty6}{ugIntroAssignPagePatch6} \pastebutton{ugIntroAssignPageEmpty6}{\showpaste} \tab{5}\spadcommand{y := sin \%pi} \end{paste}\end{patch} \begin{patch}{ugIntroAssignPagePatch7} \begin{paste}{ugIntroAssignPageFull7}{ugIntroAssignPageEmpty7} \pastebutton{ugIntroAssignPageFull7}{\hidepaste} \tab{5}\spadcommand{y := 2/3} \indentrel{3}\begin{verbatim} 2 � 3 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroAssignPageEmpty7} \begin{paste}{ugIntroAssignPageEmpty7}{ugIntroAssignPagePatch7} \pastebutton{ugIntroAssignPageEmpty7}{\showpaste} \tab{5}\spadcommand{y := 2/3} \end{paste}\end{patch} \begin{patch}{ugIntroAssignPagePatch8} \begin{paste}{ugIntroAssignPageFull8}{ugIntroAssignPageEmpty8} \pastebutton{ugIntroAssignPageFull8}{\hidepaste} \tab{5}\spadcommand{f : Float := 2/3} \indentrel{3}\begin{verbatim} (7) 0.6666666666 6666666667 Type: Float \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroAssignPageEmpty8} \begin{paste}{ugIntroAssignPageEmpty8}{ugIntroAssignPagePatch8} \pastebutton{ugIntroAssignPageEmpty8}{\showpaste} \tab{5}\spadcommand{f : Float := 2/3} \end{paste}\end{patch} \begin{patch}{ugIntroAssignPagePatch9} \begin{paste}{ugIntroAssignPageFull9}{ugIntroAssignPageEmpty9} \pastebutton{ugIntroAssignPageFull9}{\hidepaste} \tab{5}\spadcommand{f : Float; f := 2/3\bound{fff }} \indentrel{3}\begin{verbatim} (8) 0.6666666666 6666666667 Type: Float \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroAssignPageEmpty9} \begin{paste}{ugIntroAssignPageEmpty9}{ugIntroAssignPagePatch9} \pastebutton{ugIntroAssignPageEmpty9}{\showpaste} \tab{5}\spadcommand{f : Float; f := 2/3\bound{fff }} \end{paste}\end{patch} \begin{patch}{ugIntroAssignPagePatch10} \begin{paste}{ugIntroAssignPageFull10}{ugIntroAssignPageEmpty10} \pastebutton{ugIntroAssignPageFull10}{\hidepaste} \tab{5}\spadcommand{q} \indentrel{3}\begin{verbatim} (9) q Type: Variable q \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroAssignPageEmpty10} \begin{paste}{ugIntroAssignPageEmpty10}{ugIntroAssignPagePatch10} \pastebutton{ugIntroAssignPageEmpty10}{\showpaste} \tab{5}\spadcommand{q} \end{paste}\end{patch} \begin{patch}{ugIntroAssignPagePatch11} \begin{paste}{ugIntroAssignPageFull11}{ugIntroAssignPageEmpty11} \pastebutton{ugIntroAssignPageFull11}{\hidepaste} \tab{5}\spadcommand{[q, r]} \indentrel{3}\begin{verbatim} (10) [q,r] Type: List OrderedVariableList [q,r] \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroAssignPageEmpty11} \begin{paste}{ugIntroAssignPageEmpty11}{ugIntroAssignPagePatch11} \pastebutton{ugIntroAssignPageEmpty11}{\showpaste} \tab{5}\spadcommand{[q, r]} \end{paste}\end{patch} \begin{patch}{ugIntroAssignPagePatch12} \begin{paste}{ugIntroAssignPageFull12}{ugIntroAssignPageEmpty12} \pastebutton{ugIntroAssignPageFull12}{\hidepaste} \tab{5}\spadcommand{f\free{fff }} \indentrel{3}\begin{verbatim} (11) 0.6666666666 6666666667 Type: Float \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroAssignPageEmpty12} \begin{paste}{ugIntroAssignPageEmpty12}{ugIntroAssignPagePatch12} \pastebutton{ugIntroAssignPageEmpty12}{\showpaste} \tab{5}\spadcommand{f\free{fff }} \end{paste}\end{patch} \begin{patch}{ugIntroAssignPagePatch13} \begin{paste}{ugIntroAssignPageFull13}{ugIntroAssignPageEmpty13} \pastebutton{ugIntroAssignPageFull13}{\hidepaste} \tab{5}\spadcommand{'f} \indentrel{3}\begin{verbatim} (12) f Type: Variable f \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroAssignPageEmpty13} \begin{paste}{ugIntroAssignPageEmpty13}{ugIntroAssignPagePatch13} \pastebutton{ugIntroAssignPageEmpty13}{\showpaste} \tab{5}\spadcommand{'f} \end{paste}\end{patch} \begin{patch}{ugIntroSeriesPagePatch1} \begin{paste}{ugIntroSeriesPageFull1}{ugIntroSeriesPageEmpty1} \pastebutton{ugIntroSeriesPageFull1}{\hidepaste} \tab{5}\spadcommand{series(sin(a*x),x = 0)} \indentrel{3}\begin{verbatim} (1) 3 5 7 9 a 3 a 5 a 7 a 9 a x - �� x + ��� x - ���� x + ������ x 6 120 5040 362880 + 11 a 11 12 - �������� x + O(x ) 39916800 Type: UnivariatePuiseuxSeries(Expression Integer,x,0) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSeriesPageEmpty1} \begin{paste}{ugIntroSeriesPageEmpty1}{ugIntroSeriesPagePatch1} \pastebutton{ugIntroSeriesPageEmpty1}{\showpaste} \tab{5}\spadcommand{series(sin(a*x),x = 0)} \end{paste}\end{patch} \begin{patch}{ugIntroSeriesPagePatch2} \begin{paste}{ugIntroSeriesPageFull2}{ugIntroSeriesPageEmpty2} \pastebutton{ugIntroSeriesPageFull2}{\hidepaste} \tab{5}\spadcommand{series(sin(a*x),x = \%pi/4)} \indentrel{3}\begin{verbatim} (2) a %pi a %pi %pi sin(�����) + a cos(�����)(x - ���) 4 4 4 + 2 a %pi 3 a %pi a sin(�����) a cos(�����) 4 %pi 2 4 %pi 3 - ������������ (x - ���) - ������������ (x - ���) 2 4 6 4 + 4 a %pi 5 a %pi a sin(�����) a cos(�����) 4 %pi 4 4 %pi 5 ������������ (x - ���) + ������������ (x - ���) 24 4 120 4 + 6 a %pi 7 a %pi a sin(�����) a cos(�����) 4 %pi 6 4 %pi 7 - ������������ (x - ���) - ������������ (x - ���) 720 4 5040 4 + 8 a %pi 9 a %pi a sin(�����) a cos(�����) 4 %pi 8 4 %pi 9 ������������ (x - ���) + ������������ (x - ���) 40320 4 362880 4 + 10 a %pi a sin(�����) 4 %pi 10 %pi 11 - ������������� (x - ���) + O((x - ���) ) 3628800 4 4 Type: UnivariatePuiseuxSeries(Expression Integer,x,pi/4) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSeriesPageEmpty2} \begin{paste}{ugIntroSeriesPageEmpty2}{ugIntroSeriesPagePatch2} \pastebutton{ugIntroSeriesPageEmpty2}{\showpaste} \tab{5}\spadcommand{series(sin(a*x),x = \%pi/4)} \end{paste}\end{patch} \begin{patch}{ugIntroSeriesPagePatch3} \begin{paste}{ugIntroSeriesPageFull3}{ugIntroSeriesPageEmpty3} \pastebutton{ugIntroSeriesPageFull3}{\hidepaste} \tab{5}\spadcommand{series(n +-> (-1)**((3*n - 4)/6)/factorial(n - 1/3),x = 0,4/3..,2)} \indentrel{3}\begin{verbatim} 4 10 � �� 3 1 3 5 (3) x - � x + O(x ) 6 Type: UnivariatePuiseuxSeries(Expression Integer,x,0) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSeriesPageEmpty3} \begin{paste}{ugIntroSeriesPageEmpty3}{ugIntroSeriesPagePatch3} \pastebutton{ugIntroSeriesPageEmpty3}{\showpaste} \tab{5}\spadcommand{series(n +-> (-1)**((3*n - 4)/6)/factorial(n - 1/3),x = 0,4/3..,2)} \end{paste}\end{patch} \begin{patch}{ugIntroSeriesPagePatch4} \begin{paste}{ugIntroSeriesPageFull4}{ugIntroSeriesPageEmpty4} \pastebutton{ugIntroSeriesPageFull4}{\hidepaste} \tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f }} \indentrel{3}\begin{verbatim} (4) 2 3 4 5 6 7 8 9 10 1 + x + x + x + x + x + x + x + x + x + x + 11 O(x ) Type: UnivariatePuiseuxSeries(Expression Integer,x,0) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSeriesPageEmpty4} \begin{paste}{ugIntroSeriesPageEmpty4}{ugIntroSeriesPagePatch4} \pastebutton{ugIntroSeriesPageEmpty4}{\showpaste} \tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f }} \end{paste}\end{patch} \begin{patch}{ugIntroSeriesPagePatch5} \begin{paste}{ugIntroSeriesPageFull5}{ugIntroSeriesPageEmpty5} \pastebutton{ugIntroSeriesPageFull5}{\hidepaste} \tab{5}\spadcommand{f ** 2\free{f }} \indentrel{3}\begin{verbatim} (5) 2 3 4 5 6 7 8 1 + 2x + 3x + 4x + 5x + 6x + 7x + 8x + 9x + 9 10 11 10x + 11x + O(x ) Type: UnivariatePuiseuxSeries(Expression Integer,x,0) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSeriesPageEmpty5} \begin{paste}{ugIntroSeriesPageEmpty5}{ugIntroSeriesPagePatch5} \pastebutton{ugIntroSeriesPageEmpty5}{\showpaste} \tab{5}\spadcommand{f ** 2\free{f }} \end{paste}\end{patch} \begin{patch}{ugIntroSeriesPagePatch6} \begin{paste}{ugIntroSeriesPageFull6}{ugIntroSeriesPageEmpty6} \pastebutton{ugIntroSeriesPageFull6}{\hidepaste} \tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f1 }} \indentrel{3}\begin{verbatim} (6) 2 3 4 5 6 7 8 9 10 1 + x + x + x + x + x + x + x + x + x + x + 11 O(x ) Type: UnivariatePuiseuxSeries(Expression Integer,x,0) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSeriesPageEmpty6} \begin{paste}{ugIntroSeriesPageEmpty6}{ugIntroSeriesPagePatch6} \pastebutton{ugIntroSeriesPageEmpty6}{\showpaste} \tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f1 }} \end{paste}\end{patch} \begin{patch}{ugIntroSeriesPagePatch7} \begin{paste}{ugIntroSeriesPageFull7}{ugIntroSeriesPageEmpty7} \pastebutton{ugIntroSeriesPageFull7}{\hidepaste} \tab{5}\spadcommand{g := log(f)\free{f1 }\bound{g }} \indentrel{3}\begin{verbatim} (7) 1 2 1 3 1 4 1 5 1 6 1 7 1 8 x + � x + � x + � x + � x + � x + � x + � x 2 3 4 5 6 7 8 + 1 9 1 10 1 11 12 � x + �� x + �� x + O(x ) 9 10 11 Type: UnivariatePuiseuxSeries(Expression Integer,x,0) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSeriesPageEmpty7} \begin{paste}{ugIntroSeriesPageEmpty7}{ugIntroSeriesPagePatch7} \pastebutton{ugIntroSeriesPageEmpty7}{\showpaste} \tab{5}\spadcommand{g := log(f)\free{f1 }\bound{g }} \end{paste}\end{patch} \begin{patch}{ugIntroSeriesPagePatch8} \begin{paste}{ugIntroSeriesPageFull8}{ugIntroSeriesPageEmpty8} \pastebutton{ugIntroSeriesPageFull8}{\hidepaste} \tab{5}\spadcommand{exp(g)\free{g }} \indentrel{3}\begin{verbatim} (8) 2 3 4 5 6 7 8 9 10 1 + x + x + x + x + x + x + x + x + x + x + 11 O(x ) Type: UnivariatePuiseuxSeries(Expression Integer,x,0) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSeriesPageEmpty8} \begin{paste}{ugIntroSeriesPageEmpty8}{ugIntroSeriesPagePatch8} \pastebutton{ugIntroSeriesPageEmpty8}{\showpaste} \tab{5}\spadcommand{exp(g)\free{g }} \end{paste}\end{patch} \begin{patch}{ugIntroSeriesPagePatch9} \begin{paste}{ugIntroSeriesPageFull9}{ugIntroSeriesPageEmpty9} \pastebutton{ugIntroSeriesPageFull9}{\hidepaste} \tab{5}\spadcommand{f := taylor(exp(x))\bound{f2 }} \indentrel{3}\begin{verbatim} (9) 1 2 1 3 1 4 1 5 1 6 1 + x + � x + � x + �� x + ��� x + ��� x 2 6 24 120 720 + 1 7 1 8 1 9 1 10 11 ���� x + ����� x + ������ x + ������� x + O(x ) 5040 40320 362880 3628800 Type: UnivariateTaylorSeries(Expression Integer,x,0) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSeriesPageEmpty9} \begin{paste}{ugIntroSeriesPageEmpty9}{ugIntroSeriesPagePatch9} \pastebutton{ugIntroSeriesPageEmpty9}{\showpaste} \tab{5}\spadcommand{f := taylor(exp(x))\bound{f2 }} \end{paste}\end{patch} \begin{patch}{ugIntroSeriesPagePatch10} \begin{paste}{ugIntroSeriesPageFull10}{ugIntroSeriesPageEmpty10} \pastebutton{ugIntroSeriesPageFull10}{\hidepaste} \tab{5}\spadcommand{eval(f,1.0)\free{f2 }} \indentrel{3}\begin{verbatim} (10) [1.0, 2.0, 2.5, 2.6666666666 666666667, 2.7083333333 333333333, 2.7166666666 666666667, 2.7180555555 555555556, 2.7182539682 53968254, 2.7182787698 412698413, 2.7182815255 731922399, ...] Type: Stream Expression Float \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSeriesPageEmpty10} \begin{paste}{ugIntroSeriesPageEmpty10}{ugIntroSeriesPagePatch10} \pastebutton{ugIntroSeriesPageEmpty10}{\showpaste} \tab{5}\spadcommand{eval(f,1.0)\free{f2 }} \end{paste}\end{patch} \begin{patch}{ugIntroCallFunPagePatch1} \begin{paste}{ugIntroCallFunPageFull1}{ugIntroCallFunPageEmpty1} \pastebutton{ugIntroCallFunPageFull1}{\hidepaste} \tab{5}\spadcommand{factor(120)} \indentrel{3}\begin{verbatim} 3 (1) 2 3 5 Type: Factored Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCallFunPageEmpty1} \begin{paste}{ugIntroCallFunPageEmpty1}{ugIntroCallFunPagePatch1} \pastebutton{ugIntroCallFunPageEmpty1}{\showpaste} \tab{5}\spadcommand{factor(120)} \end{paste}\end{patch} \begin{patch}{ugIntroCallFunPagePatch2} \begin{paste}{ugIntroCallFunPageFull2}{ugIntroCallFunPageEmpty2} \pastebutton{ugIntroCallFunPageFull2}{\hidepaste} \tab{5}\spadcommand{divide(125,7)} \indentrel{3}\begin{verbatim} (2) [quotient= 17,remainder= 6] Type: Record(quotient: Integer,remainder: Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCallFunPageEmpty2} \begin{paste}{ugIntroCallFunPageEmpty2}{ugIntroCallFunPagePatch2} \pastebutton{ugIntroCallFunPageEmpty2}{\showpaste} \tab{5}\spadcommand{divide(125,7)} \end{paste}\end{patch} \begin{patch}{ugIntroCallFunPagePatch3} \begin{paste}{ugIntroCallFunPageFull3}{ugIntroCallFunPageEmpty3} \pastebutton{ugIntroCallFunPageFull3}{\hidepaste} \tab{5}\spadcommand{quatern(3.4,5.6,2.9,0.1)} \indentrel{3}\begin{verbatim} (3) 3.4 + 5.6 i + 2.9 j + 0.1 k Type: Quaternion Float \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCallFunPageEmpty3} \begin{paste}{ugIntroCallFunPageEmpty3}{ugIntroCallFunPagePatch3} \pastebutton{ugIntroCallFunPageEmpty3}{\showpaste} \tab{5}\spadcommand{quatern(3.4,5.6,2.9,0.1)} \end{paste}\end{patch} \begin{patch}{ugIntroCallFunPagePatch4} \begin{paste}{ugIntroCallFunPageFull4}{ugIntroCallFunPageEmpty4} \pastebutton{ugIntroCallFunPageFull4}{\hidepaste} \tab{5}\spadcommand{factorial 10} \indentrel{3}\begin{verbatim} (4) 3628800 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCallFunPageEmpty4} \begin{paste}{ugIntroCallFunPageEmpty4}{ugIntroCallFunPagePatch4} \pastebutton{ugIntroCallFunPageEmpty4}{\showpaste} \tab{5}\spadcommand{factorial 10} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch1} \begin{paste}{ugIntroCollectPageFull1}{ugIntroCollectPageEmpty1} \pastebutton{ugIntroCollectPageFull1}{\hidepaste} \tab{5}\spadcommand{u := [1,-7,11]\bound{u }} \indentrel{3}\begin{verbatim} (1) [1,- 7,11] Type: List Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty1} \begin{paste}{ugIntroCollectPageEmpty1}{ugIntroCollectPagePatch1} \pastebutton{ugIntroCollectPageEmpty1}{\showpaste} \tab{5}\spadcommand{u := [1,-7,11]\bound{u }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch2} \begin{paste}{ugIntroCollectPageFull2}{ugIntroCollectPageEmpty2} \pastebutton{ugIntroCollectPageFull2}{\hidepaste} \tab{5}\spadcommand{first rest rest u\free{u }} \indentrel{3}\begin{verbatim} (2) 11 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty2} \begin{paste}{ugIntroCollectPageEmpty2}{ugIntroCollectPagePatch2} \pastebutton{ugIntroCollectPageEmpty2}{\showpaste} \tab{5}\spadcommand{first rest rest u\free{u }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch3} \begin{paste}{ugIntroCollectPageFull3}{ugIntroCollectPageEmpty3} \pastebutton{ugIntroCollectPageFull3}{\hidepaste} \tab{5}\spadcommand{concat!(u,[9,1,3,-4]); u\free{u }\bound{u1 }} \indentrel{3}\begin{verbatim} (3) [1,- 7,11,9,1,3,- 4] Type: List Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty3} \begin{paste}{ugIntroCollectPageEmpty3}{ugIntroCollectPagePatch3} \pastebutton{ugIntroCollectPageEmpty3}{\showpaste} \tab{5}\spadcommand{concat!(u,[9,1,3,-4]); u\free{u }\bound{u1 }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch4} \begin{paste}{ugIntroCollectPageFull4}{ugIntroCollectPageEmpty4} \pastebutton{ugIntroCollectPageFull4}{\hidepaste} \tab{5}\spadcommand{lastnode := rest(u,3)\free{u1 }\bound{u2 }} \indentrel{3}\begin{verbatim} (4) [9,1,3,- 4] Type: List Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty4} \begin{paste}{ugIntroCollectPageEmpty4}{ugIntroCollectPagePatch4} \pastebutton{ugIntroCollectPageEmpty4}{\showpaste} \tab{5}\spadcommand{lastnode := rest(u,3)\free{u1 }\bound{u2 }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch5} \begin{paste}{ugIntroCollectPageFull5}{ugIntroCollectPageEmpty5} \pastebutton{ugIntroCollectPageFull5}{\hidepaste} \tab{5}\spadcommand{setrest!(lastnode,rest(u,2)); u\free{u2 }} \indentrel{3}\begin{verbatim} ____ (5) [1,- 7,11,9] Type: List Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty5} \begin{paste}{ugIntroCollectPageEmpty5}{ugIntroCollectPagePatch5} \pastebutton{ugIntroCollectPageEmpty5}{\showpaste} \tab{5}\spadcommand{setrest!(lastnode,rest(u,2)); u\free{u2 }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch6} \begin{paste}{ugIntroCollectPageFull6}{ugIntroCollectPageEmpty6} \pastebutton{ugIntroCollectPageFull6}{\hidepaste} \tab{5}\spadcommand{[factor(i) for i in 2.. by 2]\bound{stream1 }} \indentrel{3}\begin{verbatim} 2 3 2 4 2 2 (6) [2,2 ,2 3,2 ,2 5,2 3,2 7,2 ,2 3 ,2 5,...] Type: Stream Factored Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty6} \begin{paste}{ugIntroCollectPageEmpty6}{ugIntroCollectPagePatch6} \pastebutton{ugIntroCollectPageEmpty6}{\showpaste} \tab{5}\spadcommand{[factor(i) for i in 2.. by 2]\bound{stream1 }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch7} \begin{paste}{ugIntroCollectPageFull7}{ugIntroCollectPageEmpty7} \pastebutton{ugIntroCollectPageFull7}{\hidepaste} \tab{5}\spadcommand{\%.36\free{stream1 }} \indentrel{3}\begin{verbatim} 3 2 (7) 2 3 Type: Factored Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty7} \begin{paste}{ugIntroCollectPageEmpty7}{ugIntroCollectPagePatch7} \pastebutton{ugIntroCollectPageEmpty7}{\showpaste} \tab{5}\spadcommand{\%.36\free{stream1 }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch8} \begin{paste}{ugIntroCollectPageFull8}{ugIntroCollectPageEmpty8} \pastebutton{ugIntroCollectPageFull8}{\hidepaste} \tab{5}\spadcommand{a := oneDimensionalArray [1, -7, 3, 3/2]\bound{a }} \indentrel{3}\begin{verbatim} 3 (8) [1,- 7,3,�] 2 Type: OneDimensionalArray Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty8} \begin{paste}{ugIntroCollectPageEmpty8}{ugIntroCollectPagePatch8} \pastebutton{ugIntroCollectPageEmpty8}{\showpaste} \tab{5}\spadcommand{a := oneDimensionalArray [1, -7, 3, 3/2]\bound{a }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch9} \begin{paste}{ugIntroCollectPageFull9}{ugIntroCollectPageEmpty9} \pastebutton{ugIntroCollectPageFull9}{\hidepaste} \tab{5}\spadcommand{a.3 := 11; a\bound{a1 }\free{a }} \indentrel{3}\begin{verbatim} 3 (9) [1,- 7,11,�] 2 Type: OneDimensionalArray Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty9} \begin{paste}{ugIntroCollectPageEmpty9}{ugIntroCollectPagePatch9} \pastebutton{ugIntroCollectPageEmpty9}{\showpaste} \tab{5}\spadcommand{a.3 := 11; a\bound{a1 }\free{a }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch10} \begin{paste}{ugIntroCollectPageFull10}{ugIntroCollectPageEmpty10} \pastebutton{ugIntroCollectPageFull10}{\hidepaste} \tab{5}\spadcommand{concat!(a,oneDimensionalArray [1,-2])\free{a1 }} \indentrel{3}\begin{verbatim} \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty10} \begin{paste}{ugIntroCollectPageEmpty10}{ugIntroCollectPagePatch10} \pastebutton{ugIntroCollectPageEmpty10}{\showpaste} \tab{5}\spadcommand{concat!(a,oneDimensionalArray [1,-2])\free{a1 }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch11} \begin{paste}{ugIntroCollectPageFull11}{ugIntroCollectPageEmpty11} \pastebutton{ugIntroCollectPageFull11}{\hidepaste} \tab{5}\spadcommand{bits(32,true)} \indentrel{3}\begin{verbatim} (10) "11111111111111111111111111111111" Type: Bits \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty11} \begin{paste}{ugIntroCollectPageEmpty11}{ugIntroCollectPagePatch11} \pastebutton{ugIntroCollectPageEmpty11}{\showpaste} \tab{5}\spadcommand{bits(32,true)} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch12} \begin{paste}{ugIntroCollectPageFull12}{ugIntroCollectPageEmpty12} \pastebutton{ugIntroCollectPageFull12}{\hidepaste} \tab{5}\spadcommand{f := flexibleArray [2, 7, -5]\bound{f }} \indentrel{3}\begin{verbatim} (11) [2,7,- 5] Type: FlexibleArray Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty12} \begin{paste}{ugIntroCollectPageEmpty12}{ugIntroCollectPagePatch12} \pastebutton{ugIntroCollectPageEmpty12}{\showpaste} \tab{5}\spadcommand{f := flexibleArray [2, 7, -5]\bound{f }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch13} \begin{paste}{ugIntroCollectPageFull13}{ugIntroCollectPageEmpty13} \pastebutton{ugIntroCollectPageFull13}{\hidepaste} \tab{5}\spadcommand{insert!(flexibleArray [11, -3],f,2)\free{f }} \indentrel{3}\begin{verbatim} (12) [2,11,- 3,7,- 5] Type: FlexibleArray Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty13} \begin{paste}{ugIntroCollectPageEmpty13}{ugIntroCollectPagePatch13} \pastebutton{ugIntroCollectPageEmpty13}{\showpaste} \tab{5}\spadcommand{insert!(flexibleArray [11, -3],f,2)\free{f }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch14} \begin{paste}{ugIntroCollectPageFull14}{ugIntroCollectPageEmpty14} \pastebutton{ugIntroCollectPageFull14}{\hidepaste} \tab{5}\spadcommand{h := heap [-4,7,11,3,4,-7]\bound{h }} \indentrel{3}\begin{verbatim} (13) [11,4,7,- 4,3,- 7] Type: Heap Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty14} \begin{paste}{ugIntroCollectPageEmpty14}{ugIntroCollectPagePatch14} \pastebutton{ugIntroCollectPageEmpty14}{\showpaste} \tab{5}\spadcommand{h := heap [-4,7,11,3,4,-7]\bound{h }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch15} \begin{paste}{ugIntroCollectPageFull15}{ugIntroCollectPageEmpty15} \pastebutton{ugIntroCollectPageFull15}{\hidepaste} \tab{5}\spadcommand{[extract!(h) while not empty?(h)]\free{h }} \indentrel{3}\begin{verbatim} (14) [11,7,4,3,- 4,- 7] Type: List Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty15} \begin{paste}{ugIntroCollectPageEmpty15}{ugIntroCollectPagePatch15} \pastebutton{ugIntroCollectPageEmpty15}{\showpaste} \tab{5}\spadcommand{[extract!(h) while not empty?(h)]\free{h }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch16} \begin{paste}{ugIntroCollectPageFull16}{ugIntroCollectPageEmpty16} \pastebutton{ugIntroCollectPageFull16}{\hidepaste} \tab{5}\spadcommand{binarySearchTree [5,3,2,9,4,7,11]} \indentrel{3}\begin{verbatim} (15) [[2,3,4],5,[7,9,11]] Type: BinarySearchTree PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty16} \begin{paste}{ugIntroCollectPageEmpty16}{ugIntroCollectPagePatch16} \pastebutton{ugIntroCollectPageEmpty16}{\showpaste} \tab{5}\spadcommand{binarySearchTree [5,3,2,9,4,7,11]} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch17} \begin{paste}{ugIntroCollectPageFull17}{ugIntroCollectPageEmpty17} \pastebutton{ugIntroCollectPageFull17}{\hidepaste} \tab{5}\spadcommand{modTree(8,[2,3,5,7])} \indentrel{3}\begin{verbatim} (16) [0,2,3,1] Type: List Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty17} \begin{paste}{ugIntroCollectPageEmpty17}{ugIntroCollectPagePatch17} \pastebutton{ugIntroCollectPageEmpty17}{\showpaste} \tab{5}\spadcommand{modTree(8,[2,3,5,7])} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch18} \begin{paste}{ugIntroCollectPageFull18}{ugIntroCollectPageEmpty18} \pastebutton{ugIntroCollectPageFull18}{\hidepaste} \tab{5}\spadcommand{fs := set[1/3,4/5,-1/3,4/5]\bound{fs }} \indentrel{3}\begin{verbatim} 1 1 4 (17) {- �,�,�} 3 3 5 Type: Set Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty18} \begin{paste}{ugIntroCollectPageEmpty18}{ugIntroCollectPagePatch18} \pastebutton{ugIntroCollectPageEmpty18}{\showpaste} \tab{5}\spadcommand{fs := set[1/3,4/5,-1/3,4/5]\bound{fs }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch19} \begin{paste}{ugIntroCollectPageFull19}{ugIntroCollectPageEmpty19} \pastebutton{ugIntroCollectPageFull19}{\hidepaste} \tab{5}\spadcommand{multiset [x rem 5 for x in primes(2,1000)]} \indentrel{3}\begin{verbatim} (18) {0,42: 3,40: 1,38: 4,47: 2} Type: Multiset Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty19} \begin{paste}{ugIntroCollectPageEmpty19}{ugIntroCollectPagePatch19} \pastebutton{ugIntroCollectPageEmpty19}{\showpaste} \tab{5}\spadcommand{multiset [x rem 5 for x in primes(2,1000)]} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch20} \begin{paste}{ugIntroCollectPageFull20}{ugIntroCollectPageEmpty20} \pastebutton{ugIntroCollectPageFull20}{\hidepaste} \tab{5}\spadcommand{t : Table(Integer,Integer) := empty()\bound{t }} \indentrel{3}\begin{verbatim} (19) table() Type: Table(Integer,Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty20} \begin{paste}{ugIntroCollectPageEmpty20}{ugIntroCollectPagePatch20} \pastebutton{ugIntroCollectPageEmpty20}{\showpaste} \tab{5}\spadcommand{t : Table(Integer,Integer) := empty()\bound{t }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch21} \begin{paste}{ugIntroCollectPageFull21}{ugIntroCollectPageEmpty21} \pastebutton{ugIntroCollectPageFull21}{\hidepaste} \tab{5}\spadcommand{howMany(k) == (n:=search(k,t); n case "failed" => 1; n+1)\bound{how }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty21} \begin{paste}{ugIntroCollectPageEmpty21}{ugIntroCollectPagePatch21} \pastebutton{ugIntroCollectPageEmpty21}{\showpaste} \tab{5}\spadcommand{howMany(k) == (n:=search(k,t); n case "failed" => 1; n+1)\bound{how }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch22} \begin{paste}{ugIntroCollectPageFull22}{ugIntroCollectPageEmpty22} \pastebutton{ugIntroCollectPageFull22}{\hidepaste} \tab{5}\spadcommand{for p in primes(2,1000) repeat (m:= p rem 5; t.m:= howMany(m)); t\free{how t }} \indentrel{3}\begin{verbatim} (21) table(2= 47,4= 38,1= 40,3= 42,0= 1) Type: Table(Integer,Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty22} \begin{paste}{ugIntroCollectPageEmpty22}{ugIntroCollectPagePatch22} \pastebutton{ugIntroCollectPageEmpty22}{\showpaste} \tab{5}\spadcommand{for p in primes(2,1000) repeat (m:= p rem 5; t.m:= howMany(m)); t\free{how t }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch23} \begin{paste}{ugIntroCollectPageFull23}{ugIntroCollectPageEmpty23} \pastebutton{ugIntroCollectPageFull23}{\hidepaste} \tab{5}\spadcommand{daniel : Record(age : Integer, salary : Float)\bound{danieldec }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty23} \begin{paste}{ugIntroCollectPageEmpty23}{ugIntroCollectPagePatch23} \pastebutton{ugIntroCollectPageEmpty23}{\showpaste} \tab{5}\spadcommand{daniel : Record(age : Integer, salary : Float)\bound{danieldec }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch24} \begin{paste}{ugIntroCollectPageFull24}{ugIntroCollectPageEmpty24} \pastebutton{ugIntroCollectPageFull24}{\hidepaste} \tab{5}\spadcommand{daniel := [28, 32005.12]\free{danieldec }\bound{daniel }} \indentrel{3}\begin{verbatim} (23) [age= 28,salary= 32005.12] Type: Record(age: Integer,salary: Float) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty24} \begin{paste}{ugIntroCollectPageEmpty24}{ugIntroCollectPagePatch24} \pastebutton{ugIntroCollectPageEmpty24}{\showpaste} \tab{5}\spadcommand{daniel := [28, 32005.12]\free{danieldec }\bound{daniel }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch25} \begin{paste}{ugIntroCollectPageFull25}{ugIntroCollectPageEmpty25} \pastebutton{ugIntroCollectPageFull25}{\hidepaste} \tab{5}\spadcommand{daniel.salary := 35000; daniel\free{daniel }} \indentrel{3}\begin{verbatim} (24) [age= 28,salary= 35000.0] Type: Record(age: Integer,salary: Float) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty25} \begin{paste}{ugIntroCollectPageEmpty25}{ugIntroCollectPagePatch25} \pastebutton{ugIntroCollectPageEmpty25}{\showpaste} \tab{5}\spadcommand{daniel.salary := 35000; daniel\free{daniel }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch26} \begin{paste}{ugIntroCollectPageFull26}{ugIntroCollectPageEmpty26} \pastebutton{ugIntroCollectPageFull26}{\hidepaste} \tab{5}\spadcommand{dog: Union(licenseNumber: Integer, name: String)\bound{xint }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty26} \begin{paste}{ugIntroCollectPageEmpty26}{ugIntroCollectPagePatch26} \pastebutton{ugIntroCollectPageEmpty26}{\showpaste} \tab{5}\spadcommand{dog: Union(licenseNumber: Integer, name: String)\bound{xint }} \end{paste}\end{patch} \begin{patch}{ugIntroCollectPagePatch27} \begin{paste}{ugIntroCollectPageFull27}{ugIntroCollectPageEmpty27} \pastebutton{ugIntroCollectPageFull27}{\hidepaste} \tab{5}\spadcommand{dog := "Whisper"\free{xint }} \indentrel{3}\begin{verbatim} (26) "Whisper" Type: Union(name: String,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCollectPageEmpty27} \begin{paste}{ugIntroCollectPageEmpty27}{ugIntroCollectPagePatch27} \pastebutton{ugIntroCollectPageEmpty27}{\showpaste} \tab{5}\spadcommand{dog := "Whisper"\free{xint }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch1} \begin{paste}{ugIntroNumbersPageFull1}{ugIntroNumbersPageEmpty1} \pastebutton{ugIntroNumbersPageFull1}{\hidepaste} \tab{5}\spadcommand{11**13 * 13**11 * 17**7 - 19**5 * 23**3} \indentrel{3}\begin{verbatim} (1) 25387751112538918594666224484237298 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty1} \begin{paste}{ugIntroNumbersPageEmpty1}{ugIntroNumbersPagePatch1} \pastebutton{ugIntroNumbersPageEmpty1}{\showpaste} \tab{5}\spadcommand{11**13 * 13**11 * 17**7 - 19**5 * 23**3} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch2} \begin{paste}{ugIntroNumbersPageFull2}{ugIntroNumbersPageEmpty2} \pastebutton{ugIntroNumbersPageFull2}{\hidepaste} \tab{5}\spadcommand{factor 643238070748569023720594412551704344145570763243\bound{ex1 }} \indentrel{3}\begin{verbatim} 13 11 7 5 3 2 (2) 11 13 17 19 23 29 Type: Factored Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty2} \begin{paste}{ugIntroNumbersPageEmpty2}{ugIntroNumbersPagePatch2} \pastebutton{ugIntroNumbersPageEmpty2}{\showpaste} \tab{5}\spadcommand{factor 643238070748569023720594412551704344145570763243\bound{ex1 }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch3} \begin{paste}{ugIntroNumbersPageFull3}{ugIntroNumbersPageEmpty3} \pastebutton{ugIntroNumbersPageFull3}{\hidepaste} \tab{5}\spadcommand{\% * 12\free{ex1 }} \indentrel{3}\begin{verbatim} 2 13 11 7 5 3 2 (3) 2 3 11 13 17 19 23 29 Type: Factored Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty3} \begin{paste}{ugIntroNumbersPageEmpty3}{ugIntroNumbersPagePatch3} \pastebutton{ugIntroNumbersPageEmpty3}{\showpaste} \tab{5}\spadcommand{\% * 12\free{ex1 }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch4} \begin{paste}{ugIntroNumbersPageFull4}{ugIntroNumbersPageEmpty4} \pastebutton{ugIntroNumbersPageFull4}{\hidepaste} \tab{5}\spadcommand{radix(25937424601,11)} \indentrel{3}\begin{verbatim} (4) 10000000000 Type: RadixExpansion 11 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty4} \begin{paste}{ugIntroNumbersPageEmpty4}{ugIntroNumbersPagePatch4} \pastebutton{ugIntroNumbersPageEmpty4}{\showpaste} \tab{5}\spadcommand{radix(25937424601,11)} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch5} \begin{paste}{ugIntroNumbersPageFull5}{ugIntroNumbersPageEmpty5} \pastebutton{ugIntroNumbersPageFull5}{\hidepaste} \tab{5}\spadcommand{roman(1992)} \indentrel{3}\begin{verbatim} (5) MCMXCII Type: RomanNumeral \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty5} \begin{paste}{ugIntroNumbersPageEmpty5}{ugIntroNumbersPagePatch5} \pastebutton{ugIntroNumbersPageEmpty5}{\showpaste} \tab{5}\spadcommand{roman(1992)} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch6} \begin{paste}{ugIntroNumbersPageFull6}{ugIntroNumbersPageEmpty6} \pastebutton{ugIntroNumbersPageFull6}{\hidepaste} \tab{5}\spadcommand{r := 10 + 9/2 + 8/3 + 7/4 + 6/5 + 5/6 + 4/7 + 3/8 + 2/9\bound{r }} \indentrel{3}\begin{verbatim} 55739 (6) ����� 2520 Type: Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty6} \begin{paste}{ugIntroNumbersPageEmpty6}{ugIntroNumbersPagePatch6} \pastebutton{ugIntroNumbersPageEmpty6}{\showpaste} \tab{5}\spadcommand{r := 10 + 9/2 + 8/3 + 7/4 + 6/5 + 5/6 + 4/7 + 3/8 + 2/9\bound{r }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch7} \begin{paste}{ugIntroNumbersPageFull7}{ugIntroNumbersPageEmpty7} \pastebutton{ugIntroNumbersPageFull7}{\hidepaste} \tab{5}\spadcommand{map(factor,r)\free{r }} \indentrel{3}\begin{verbatim} 139 401 (7) ������� 3 2 2 3 5 7 Type: Fraction Factored Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty7} \begin{paste}{ugIntroNumbersPageEmpty7}{ugIntroNumbersPagePatch7} \pastebutton{ugIntroNumbersPageEmpty7}{\showpaste} \tab{5}\spadcommand{map(factor,r)\free{r }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch8} \begin{paste}{ugIntroNumbersPageFull8}{ugIntroNumbersPageEmpty8} \pastebutton{ugIntroNumbersPageFull8}{\hidepaste} \tab{5}\spadcommand{11@SingleInteger} \indentrel{3}\begin{verbatim} (8) 11 Type: SingleInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty8} \begin{paste}{ugIntroNumbersPageEmpty8}{ugIntroNumbersPagePatch8} \pastebutton{ugIntroNumbersPageEmpty8}{\showpaste} \tab{5}\spadcommand{11@SingleInteger} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch9} \begin{paste}{ugIntroNumbersPageFull9}{ugIntroNumbersPageEmpty9} \pastebutton{ugIntroNumbersPageFull9}{\hidepaste} \tab{5}\spadcommand{123.21@DoubleFloat} \indentrel{3}\begin{verbatim} (9) 123.21000000000001 Type: DoubleFloat \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty9} \begin{paste}{ugIntroNumbersPageEmpty9}{ugIntroNumbersPagePatch9} \pastebutton{ugIntroNumbersPageEmpty9}{\showpaste} \tab{5}\spadcommand{123.21@DoubleFloat} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch10} \begin{paste}{ugIntroNumbersPageFull10}{ugIntroNumbersPageEmpty10} \pastebutton{ugIntroNumbersPageFull10}{\hidepaste} \tab{5}\spadcommand{r :: Float\free{r }} \indentrel{3}\begin{verbatim} (10) 22.1186507936 50793651 Type: Float \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty10} \begin{paste}{ugIntroNumbersPageEmpty10}{ugIntroNumbersPagePatch10} \pastebutton{ugIntroNumbersPageEmpty10}{\showpaste} \tab{5}\spadcommand{r :: Float\free{r }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch11} \begin{paste}{ugIntroNumbersPageFull11}{ugIntroNumbersPageEmpty11} \pastebutton{ugIntroNumbersPageFull11}{\hidepaste} \tab{5}\spadcommand{digits(22)\bound{fewerdigits }} \indentrel{3}\begin{verbatim} (11) 20 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty11} \begin{paste}{ugIntroNumbersPageEmpty11}{ugIntroNumbersPagePatch11} \pastebutton{ugIntroNumbersPageEmpty11}{\showpaste} \tab{5}\spadcommand{digits(22)\bound{fewerdigits }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch12} \begin{paste}{ugIntroNumbersPageFull12}{ugIntroNumbersPageEmpty12} \pastebutton{ugIntroNumbersPageFull12}{\hidepaste} \tab{5}\spadcommand{exp(\%pi * sqrt 163.0)\free{fewerdigits }} \indentrel{3}\begin{verbatim} (12) 26253741 2640768744.0 Type: Float \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty12} \begin{paste}{ugIntroNumbersPageEmpty12}{ugIntroNumbersPagePatch12} \pastebutton{ugIntroNumbersPageEmpty12}{\showpaste} \tab{5}\spadcommand{exp(\%pi * sqrt 163.0)\free{fewerdigits }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch13} \begin{paste}{ugIntroNumbersPageFull13}{ugIntroNumbersPageEmpty13} \pastebutton{ugIntroNumbersPageFull13}{\hidepaste} \tab{5}\spadcommand{digits(40); exp(\%pi * sqrt 163.0)\free{moredigits }} \indentrel{3}\begin{verbatim} (13) 26253741 2640768743.9999999999 9925007259 76 Type: Float \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty13} \begin{paste}{ugIntroNumbersPageEmpty13}{ugIntroNumbersPagePatch13} \pastebutton{ugIntroNumbersPageEmpty13}{\showpaste} \tab{5}\spadcommand{digits(40); exp(\%pi * sqrt 163.0)\free{moredigits }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch14} \begin{paste}{ugIntroNumbersPageFull14}{ugIntroNumbersPageEmpty14} \pastebutton{ugIntroNumbersPageFull14}{\hidepaste} \tab{5}\spadcommand{(2/3 + \%i)**3\bound{gaussint }} \indentrel{3}\begin{verbatim} 46 1 (14) - �� + � %i 27 3 Type: Complex Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty14} \begin{paste}{ugIntroNumbersPageEmpty14}{ugIntroNumbersPagePatch14} \pastebutton{ugIntroNumbersPageEmpty14}{\showpaste} \tab{5}\spadcommand{(2/3 + \%i)**3\bound{gaussint }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch15} \begin{paste}{ugIntroNumbersPageFull15}{ugIntroNumbersPageEmpty15} \pastebutton{ugIntroNumbersPageFull15}{\hidepaste} \tab{5}\spadcommand{conjugate \%\free{gaussint }} \indentrel{3}\begin{verbatim} 46 1 (15) - �� - � %i 27 3 Type: Complex Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty15} \begin{paste}{ugIntroNumbersPageEmpty15}{ugIntroNumbersPagePatch15} \pastebutton{ugIntroNumbersPageEmpty15}{\showpaste} \tab{5}\spadcommand{conjugate \%\free{gaussint }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch16} \begin{paste}{ugIntroNumbersPageFull16}{ugIntroNumbersPageEmpty16} \pastebutton{ugIntroNumbersPageFull16}{\hidepaste} \tab{5}\spadcommand{factor(89 - 23 * \%i)} \indentrel{3}\begin{verbatim} 2 2 (16) - (1 + %i)(2 + %i) (3 + 2%i) Type: Factored Complex Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty16} \begin{paste}{ugIntroNumbersPageEmpty16}{ugIntroNumbersPagePatch16} \pastebutton{ugIntroNumbersPageEmpty16}{\showpaste} \tab{5}\spadcommand{factor(89 - 23 * \%i)} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch17} \begin{paste}{ugIntroNumbersPageFull17}{ugIntroNumbersPageEmpty17} \pastebutton{ugIntroNumbersPageFull17}{\hidepaste} \tab{5}\spadcommand{exp(\%pi/4.0 * \%i)} \indentrel{3}\begin{verbatim} (17) 0.7071067811 8654752440 0844362104 8490392849 + 0.7071067811 8654752440 0844362104 8490392848 %i Type: Complex Float \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty17} \begin{paste}{ugIntroNumbersPageEmpty17}{ugIntroNumbersPagePatch17} \pastebutton{ugIntroNumbersPageEmpty17}{\showpaste} \tab{5}\spadcommand{exp(\%pi/4.0 * \%i)} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch18} \begin{paste}{ugIntroNumbersPageFull18}{ugIntroNumbersPageEmpty18} \pastebutton{ugIntroNumbersPageFull18}{\hidepaste} \tab{5}\spadcommand{decimal(1/352)} \indentrel{3}\begin{verbatim} __ (18) 0.0028409 Type: DecimalExpansion \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty18} \begin{paste}{ugIntroNumbersPageEmpty18}{ugIntroNumbersPagePatch18} \pastebutton{ugIntroNumbersPageEmpty18}{\showpaste} \tab{5}\spadcommand{decimal(1/352)} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch19} \begin{paste}{ugIntroNumbersPageFull19}{ugIntroNumbersPageEmpty19} \pastebutton{ugIntroNumbersPageFull19}{\hidepaste} \tab{5}\spadcommand{continuedFraction(6543/210)} \indentrel{3}\begin{verbatim} 1 � 1 � 1 � 1 � (19) 31 + ����� + ����� + ����� + ����� � 6 � 2 � 1 � 3 Type: ContinuedFraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty19} \begin{paste}{ugIntroNumbersPageEmpty19}{ugIntroNumbersPagePatch19} \pastebutton{ugIntroNumbersPageEmpty19}{\showpaste} \tab{5}\spadcommand{continuedFraction(6543/210)} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch20} \begin{paste}{ugIntroNumbersPageFull20}{ugIntroNumbersPageEmpty20} \pastebutton{ugIntroNumbersPageFull20}{\hidepaste} \tab{5}\spadcommand{partialFraction(1,factorial(10))\bound{partfrac }} \indentrel{3}\begin{verbatim} 159 23 12 1 (20) ��� - �� - �� + � 8 4 2 7 2 3 5 Type: PartialFraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty20} \begin{paste}{ugIntroNumbersPageEmpty20}{ugIntroNumbersPagePatch20} \pastebutton{ugIntroNumbersPageEmpty20}{\showpaste} \tab{5}\spadcommand{partialFraction(1,factorial(10))\bound{partfrac }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch21} \begin{paste}{ugIntroNumbersPageFull21}{ugIntroNumbersPageEmpty21} \pastebutton{ugIntroNumbersPageFull21}{\hidepaste} \tab{5}\spadcommand{padicFraction(\%)\free{partfrac }} \indentrel{3}\begin{verbatim} (21) 1 1 1 1 1 1 2 1 2 2 2 1 � + �� + �� + �� + �� + �� - �� - �� - �� - � - �� + � 2 4 5 6 7 8 2 3 4 5 2 7 2 2 2 2 2 3 3 3 5 Type: PartialFraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty21} \begin{paste}{ugIntroNumbersPageEmpty21}{ugIntroNumbersPagePatch21} \pastebutton{ugIntroNumbersPageEmpty21}{\showpaste} \tab{5}\spadcommand{padicFraction(\%)\free{partfrac }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch22} \begin{paste}{ugIntroNumbersPageFull22}{ugIntroNumbersPageEmpty22} \pastebutton{ugIntroNumbersPageFull22}{\hidepaste} \tab{5}\spadcommand{radix(4/7, 8)\bound{rad }} \indentrel{3}\begin{verbatim} _ (22) 0.4 Type: RadixExpansion 8 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty22} \begin{paste}{ugIntroNumbersPageEmpty22}{ugIntroNumbersPagePatch22} \pastebutton{ugIntroNumbersPageEmpty22}{\showpaste} \tab{5}\spadcommand{radix(4/7, 8)\bound{rad }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch23} \begin{paste}{ugIntroNumbersPageFull23}{ugIntroNumbersPageEmpty23} \pastebutton{ugIntroNumbersPageFull23}{\hidepaste} \tab{5}\spadcommand{\% + 2/3*\%i\free{rad }} \indentrel{3}\begin{verbatim} 4 2 (23) � + � %i 7 3 Type: Complex Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty23} \begin{paste}{ugIntroNumbersPageEmpty23}{ugIntroNumbersPagePatch23} \pastebutton{ugIntroNumbersPageEmpty23}{\showpaste} \tab{5}\spadcommand{\% + 2/3*\%i\free{rad }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch24} \begin{paste}{ugIntroNumbersPageFull24}{ugIntroNumbersPageEmpty24} \pastebutton{ugIntroNumbersPageFull24}{\hidepaste} \tab{5}\spadcommand{(5 + sqrt 63 + sqrt 847)**(1/3)} \indentrel{3}\begin{verbatim} ����������Ŀ 3� �Ŀ (24) \�14\�7 + 5 Type: AlgebraicNumber \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty24} \begin{paste}{ugIntroNumbersPageEmpty24}{ugIntroNumbersPagePatch24} \pastebutton{ugIntroNumbersPageEmpty24}{\showpaste} \tab{5}\spadcommand{(5 + sqrt 63 + sqrt 847)**(1/3)} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch25} \begin{paste}{ugIntroNumbersPageFull25}{ugIntroNumbersPageEmpty25} \pastebutton{ugIntroNumbersPageFull25}{\hidepaste} \tab{5}\spadcommand{x : PrimeField 7 := 5\bound{x }} \indentrel{3}\begin{verbatim} (25) 5 Type: PrimeField 7 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty25} \begin{paste}{ugIntroNumbersPageEmpty25}{ugIntroNumbersPagePatch25} \pastebutton{ugIntroNumbersPageEmpty25}{\showpaste} \tab{5}\spadcommand{x : PrimeField 7 := 5\bound{x }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch26} \begin{paste}{ugIntroNumbersPageFull26}{ugIntroNumbersPageEmpty26} \pastebutton{ugIntroNumbersPageFull26}{\hidepaste} \tab{5}\spadcommand{x**3\free{x }} \indentrel{3}\begin{verbatim} (26) 6 Type: PrimeField 7 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty26} \begin{paste}{ugIntroNumbersPageEmpty26}{ugIntroNumbersPagePatch26} \pastebutton{ugIntroNumbersPageEmpty26}{\showpaste} \tab{5}\spadcommand{x**3\free{x }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch27} \begin{paste}{ugIntroNumbersPageFull27}{ugIntroNumbersPageEmpty27} \pastebutton{ugIntroNumbersPageFull27}{\hidepaste} \tab{5}\spadcommand{1/x\free{x }} \indentrel{3}\begin{verbatim} (27) 3 Type: PrimeField 7 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty27} \begin{paste}{ugIntroNumbersPageEmpty27}{ugIntroNumbersPagePatch27} \pastebutton{ugIntroNumbersPageEmpty27}{\showpaste} \tab{5}\spadcommand{1/x\free{x }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch28} \begin{paste}{ugIntroNumbersPageFull28}{ugIntroNumbersPageEmpty28} \pastebutton{ugIntroNumbersPageFull28}{\hidepaste} \tab{5}\spadcommand{y : IntegerMod 6 := 5\bound{y }} \indentrel{3}\begin{verbatim} (28) 5 Type: IntegerMod 6 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty28} \begin{paste}{ugIntroNumbersPageEmpty28}{ugIntroNumbersPagePatch28} \pastebutton{ugIntroNumbersPageEmpty28}{\showpaste} \tab{5}\spadcommand{y : IntegerMod 6 := 5\bound{y }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch29} \begin{paste}{ugIntroNumbersPageFull29}{ugIntroNumbersPageEmpty29} \pastebutton{ugIntroNumbersPageFull29}{\hidepaste} \tab{5}\spadcommand{y**3\free{y }} \indentrel{3}\begin{verbatim} (29) 5 Type: IntegerMod 6 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty29} \begin{paste}{ugIntroNumbersPageEmpty29}{ugIntroNumbersPagePatch29} \pastebutton{ugIntroNumbersPageEmpty29}{\showpaste} \tab{5}\spadcommand{y**3\free{y }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch30} \begin{paste}{ugIntroNumbersPageFull30}{ugIntroNumbersPageEmpty30} \pastebutton{ugIntroNumbersPageFull30}{\hidepaste} \tab{5}\spadcommand{1/y\free{y }} \indentrel{3}\begin{verbatim} \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty30} \begin{paste}{ugIntroNumbersPageEmpty30}{ugIntroNumbersPagePatch30} \pastebutton{ugIntroNumbersPageEmpty30}{\showpaste} \tab{5}\spadcommand{1/y\free{y }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch31} \begin{paste}{ugIntroNumbersPageFull31}{ugIntroNumbersPageEmpty31} \pastebutton{ugIntroNumbersPageFull31}{\hidepaste} \tab{5}\spadcommand{a := rootOf(a**5 + a**3 + a**2 + 3,a)\bound{a }} \indentrel{3}\begin{verbatim} (30) a Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty31} \begin{paste}{ugIntroNumbersPageEmpty31}{ugIntroNumbersPagePatch31} \pastebutton{ugIntroNumbersPageEmpty31}{\showpaste} \tab{5}\spadcommand{a := rootOf(a**5 + a**3 + a**2 + 3,a)\bound{a }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch32} \begin{paste}{ugIntroNumbersPageFull32}{ugIntroNumbersPageEmpty32} \pastebutton{ugIntroNumbersPageFull32}{\hidepaste} \tab{5}\spadcommand{(a + 1)**10\free{a }} \indentrel{3}\begin{verbatim} 4 3 2 (31) - 85a - 264a - 378a - 458a - 287 Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty32} \begin{paste}{ugIntroNumbersPageEmpty32}{ugIntroNumbersPagePatch32} \pastebutton{ugIntroNumbersPageEmpty32}{\showpaste} \tab{5}\spadcommand{(a + 1)**10\free{a }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch33} \begin{paste}{ugIntroNumbersPageFull33}{ugIntroNumbersPageEmpty33} \pastebutton{ugIntroNumbersPageFull33}{\hidepaste} \tab{5}\spadcommand{b := rootOf(b**4 + a,b)\bound{b }\free{a }} \indentrel{3}\begin{verbatim} (32) b Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty33} \begin{paste}{ugIntroNumbersPageEmpty33}{ugIntroNumbersPagePatch33} \pastebutton{ugIntroNumbersPageEmpty33}{\showpaste} \tab{5}\spadcommand{b := rootOf(b**4 + a,b)\bound{b }\free{a }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch34} \begin{paste}{ugIntroNumbersPageFull34}{ugIntroNumbersPageEmpty34} \pastebutton{ugIntroNumbersPageFull34}{\hidepaste} \tab{5}\spadcommand{2/(b - 1)\free{b }\bound{check }} \indentrel{3}\begin{verbatim} 2 (33) ����� b - 1 Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty34} \begin{paste}{ugIntroNumbersPageEmpty34}{ugIntroNumbersPagePatch34} \pastebutton{ugIntroNumbersPageEmpty34}{\showpaste} \tab{5}\spadcommand{2/(b - 1)\free{b }\bound{check }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch35} \begin{paste}{ugIntroNumbersPageFull35}{ugIntroNumbersPageEmpty35} \pastebutton{ugIntroNumbersPageFull35}{\hidepaste} \tab{5}\spadcommand{ratDenom(\%)\free{check }\bound{check1 }} \indentrel{3}\begin{verbatim} (34) 4 3 2 3 4 3 2 2 (a - a + 2a - a + 1)b + (a - a + 2a - a + 1)b + 4 3 2 4 3 2 (a - a + 2a - a + 1)b + a - a + 2a - a + 1 Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty35} \begin{paste}{ugIntroNumbersPageEmpty35}{ugIntroNumbersPagePatch35} \pastebutton{ugIntroNumbersPageEmpty35}{\showpaste} \tab{5}\spadcommand{ratDenom(\%)\free{check }\bound{check1 }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch36} \begin{paste}{ugIntroNumbersPageFull36}{ugIntroNumbersPageEmpty36} \pastebutton{ugIntroNumbersPageFull36}{\hidepaste} \tab{5}\spadcommand{2/\%+1\free{check1 }\bound{check2 }} \indentrel{3}\begin{verbatim} (35) 4 3 2 3 (a - a + 2a - a + 1)b + 4 3 2 2 (a - a + 2a - a + 1)b + 4 3 2 4 3 2 (a - a + 2a - a + 1)b + a - a + 2a - a + 3 / 4 3 2 3 (a - a + 2a - a + 1)b + 4 3 2 2 (a - a + 2a - a + 1)b + 4 3 2 4 3 2 (a - a + 2a - a + 1)b + a - a + 2a - a + 1 Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty36} \begin{paste}{ugIntroNumbersPageEmpty36}{ugIntroNumbersPagePatch36} \pastebutton{ugIntroNumbersPageEmpty36}{\showpaste} \tab{5}\spadcommand{2/\%+1\free{check1 }\bound{check2 }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch37} \begin{paste}{ugIntroNumbersPageFull37}{ugIntroNumbersPageEmpty37} \pastebutton{ugIntroNumbersPageFull37}{\hidepaste} \tab{5}\spadcommand{ratDenom(\%)\free{check2 }} \indentrel{3}\begin{verbatim} (36) b Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty37} \begin{paste}{ugIntroNumbersPageEmpty37}{ugIntroNumbersPagePatch37} \pastebutton{ugIntroNumbersPageEmpty37}{\showpaste} \tab{5}\spadcommand{ratDenom(\%)\free{check2 }} \end{paste}\end{patch} \begin{patch}{ugIntroNumbersPagePatch38} \begin{paste}{ugIntroNumbersPageFull38}{ugIntroNumbersPageEmpty38} \pastebutton{ugIntroNumbersPageFull38}{\hidepaste} \tab{5}\spadcommand{q:=quatern(1,2,3,4)*quatern(5,6,7,8) - quatern(5,6,7,8)*quatern(1,2,3,4)} \indentrel{3}\begin{verbatim} (37) - 8i + 16j - 8k Type: Quaternion Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroNumbersPageEmpty38} \begin{paste}{ugIntroNumbersPageEmpty38}{ugIntroNumbersPagePatch38} \pastebutton{ugIntroNumbersPageEmpty38}{\showpaste} \tab{5}\spadcommand{q:=quatern(1,2,3,4)*quatern(5,6,7,8) - quatern(5,6,7,8)*quatern(1,2,3,4)} \end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPagePatch1} \begin{paste}{ugIntroDiffEqnsPageFull1}{ugIntroDiffEqnsPageEmpty1} \pastebutton{ugIntroDiffEqnsPageFull1}{\hidepaste} \tab{5}\spadcommand{y := operator 'y\bound{y }} \indentrel{3}\begin{verbatim} (1) y Type: BasicOperator \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPageEmpty1} \begin{paste}{ugIntroDiffEqnsPageEmpty1}{ugIntroDiffEqnsPagePatch1} \pastebutton{ugIntroDiffEqnsPageEmpty1}{\showpaste} \tab{5}\spadcommand{y := operator 'y\bound{y }} \end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPagePatch2} \begin{paste}{ugIntroDiffEqnsPageFull2}{ugIntroDiffEqnsPageEmpty2} \pastebutton{ugIntroDiffEqnsPageFull2}{\hidepaste} \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 }} \indentrel{3}\begin{verbatim} 3 ,,, 2 ,, , 4 (2) x y (x) + x y (x) - 2xy (x) + 2y(x)= 2x Type: Equation Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPageEmpty2} \begin{paste}{ugIntroDiffEqnsPageEmpty2}{ugIntroDiffEqnsPagePatch2} \pastebutton{ugIntroDiffEqnsPageEmpty2}{\showpaste} \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 }} \end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPagePatch3} \begin{paste}{ugIntroDiffEqnsPageFull3}{ugIntroDiffEqnsPageEmpty3} \pastebutton{ugIntroDiffEqnsPageFull3}{\hidepaste} \tab{5}\spadcommand{solve(deq, y, x)\free{e3 }\free{y }} \indentrel{3}\begin{verbatim} (3) 5 3 2 x - 10x + 20x + 4 [particular= ��������������������, 15x 3 2 3 3 2 2x - 3x + 1 x - 1 x - 3x - 1 basis= [�������������,������,������������]] x x x Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPageEmpty3} \begin{paste}{ugIntroDiffEqnsPageEmpty3}{ugIntroDiffEqnsPagePatch3} \pastebutton{ugIntroDiffEqnsPageEmpty3}{\showpaste} \tab{5}\spadcommand{solve(deq, y, x)\free{e3 }\free{y }} \end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPagePatch4} \begin{paste}{ugIntroDiffEqnsPageFull4}{ugIntroDiffEqnsPageEmpty4} \pastebutton{ugIntroDiffEqnsPageFull4}{\hidepaste} \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 }} \indentrel{3}\begin{verbatim} 2 ,, , (4) (x + 1)y (x) + 3xy (x) + y(x)= 0 Type: Equation Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPageEmpty4} \begin{paste}{ugIntroDiffEqnsPageEmpty4}{ugIntroDiffEqnsPagePatch4} \pastebutton{ugIntroDiffEqnsPageEmpty4}{\showpaste} \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 }} \end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPagePatch5} \begin{paste}{ugIntroDiffEqnsPageFull5}{ugIntroDiffEqnsPageEmpty5} \pastebutton{ugIntroDiffEqnsPageFull5}{\hidepaste} \tab{5}\spadcommand{solve(deq, y, x)\free{e5 }\free{y }} \indentrel{3}\begin{verbatim} (5) ������Ŀ � 2 1 log(\�x + 1 - x) [particular= 0,basis= [���������,������������������]] ������Ŀ ������Ŀ � 2 � 2 \�x + 1 \�x + 1 Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPageEmpty5} \begin{paste}{ugIntroDiffEqnsPageEmpty5}{ugIntroDiffEqnsPagePatch5} \pastebutton{ugIntroDiffEqnsPageEmpty5}{\showpaste} \tab{5}\spadcommand{solve(deq, y, x)\free{e5 }\free{y }} \end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPagePatch6} \begin{paste}{ugIntroDiffEqnsPageFull6}{ugIntroDiffEqnsPageEmpty6} \pastebutton{ugIntroDiffEqnsPageFull6}{\hidepaste} \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 }} \indentrel{3}\begin{verbatim} 3 ,, 2 , (6) 2x y (x) + 3x y (x) - 2y(x) Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPageEmpty6} \begin{paste}{ugIntroDiffEqnsPageEmpty6}{ugIntroDiffEqnsPagePatch6} \pastebutton{ugIntroDiffEqnsPageEmpty6}{\showpaste} \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 }} \end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPagePatch7} \begin{paste}{ugIntroDiffEqnsPageFull7}{ugIntroDiffEqnsPageEmpty7} \pastebutton{ugIntroDiffEqnsPageFull7}{\hidepaste} \tab{5}\spadcommand{solve(eq,y,x).basis\free{eq }} \indentrel{3}\begin{verbatim} 2 2 - ���� ���� �Ŀ �Ŀ \�x \�x (7) [%e ,%e ] Type: List Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPageEmpty7} \begin{paste}{ugIntroDiffEqnsPageEmpty7}{ugIntroDiffEqnsPagePatch7} \pastebutton{ugIntroDiffEqnsPageEmpty7}{\showpaste} \tab{5}\spadcommand{solve(eq,y,x).basis\free{eq }} \end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPagePatch8} \begin{paste}{ugIntroDiffEqnsPageFull8}{ugIntroDiffEqnsPageEmpty8} \pastebutton{ugIntroDiffEqnsPageFull8}{\hidepaste} \tab{5}\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)\bound{deqi }\free{y }} \indentrel{3}\begin{verbatim} , y(x) (8) y (x)= ����������������� y(x)log(y(x)) + x Type: Equation Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPageEmpty8} \begin{paste}{ugIntroDiffEqnsPageEmpty8}{ugIntroDiffEqnsPagePatch8} \pastebutton{ugIntroDiffEqnsPageEmpty8}{\showpaste} \tab{5}\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)\bound{deqi }\free{y }} \end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPagePatch9} \begin{paste}{ugIntroDiffEqnsPageFull9}{ugIntroDiffEqnsPageEmpty9} \pastebutton{ugIntroDiffEqnsPageFull9}{\hidepaste} \tab{5}\spadcommand{solve(deq, y, x)\free{deqi y }} \indentrel{3}\begin{verbatim} 2 y(x)log(y(x)) - 2x (9) ������������������� 2y(x) Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPageEmpty9} \begin{paste}{ugIntroDiffEqnsPageEmpty9}{ugIntroDiffEqnsPagePatch9} \pastebutton{ugIntroDiffEqnsPageEmpty9}{\showpaste} \tab{5}\spadcommand{solve(deq, y, x)\free{deqi y }} \end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPagePatch10} \begin{paste}{ugIntroDiffEqnsPageFull10}{ugIntroDiffEqnsPageEmpty10} \pastebutton{ugIntroDiffEqnsPageFull10}{\hidepaste} \tab{5}\spadcommand{x := operator 'x\bound{x }} \indentrel{3}\begin{verbatim} (10) x Type: BasicOperator \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPageEmpty10} \begin{paste}{ugIntroDiffEqnsPageEmpty10}{ugIntroDiffEqnsPagePatch10} \pastebutton{ugIntroDiffEqnsPageEmpty10}{\showpaste} \tab{5}\spadcommand{x := operator 'x\bound{x }} \end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPagePatch11} \begin{paste}{ugIntroDiffEqnsPageFull11}{ugIntroDiffEqnsPageEmpty11} \pastebutton{ugIntroDiffEqnsPageFull11}{\hidepaste} \tab{5}\spadcommand{eq1 := D(x(t), t) = 1 + x(t)**2\free{x }\free{y }\bound{eq1 }} \indentrel{3}\begin{verbatim} , 2 (11) x (t)= x(t) + 1 Type: Equation Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPageEmpty11} \begin{paste}{ugIntroDiffEqnsPageEmpty11}{ugIntroDiffEqnsPagePatch11} \pastebutton{ugIntroDiffEqnsPageEmpty11}{\showpaste} \tab{5}\spadcommand{eq1 := D(x(t), t) = 1 + x(t)**2\free{x }\free{y }\bound{eq1 }} \end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPagePatch12} \begin{paste}{ugIntroDiffEqnsPageFull12}{ugIntroDiffEqnsPageEmpty12} \pastebutton{ugIntroDiffEqnsPageFull12}{\hidepaste} \tab{5}\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)\free{x }\free{y }\bound{eq2 }} \indentrel{3}\begin{verbatim} , (12) y (t)= x(t)y(t) Type: Equation Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPageEmpty12} \begin{paste}{ugIntroDiffEqnsPageEmpty12}{ugIntroDiffEqnsPagePatch12} \pastebutton{ugIntroDiffEqnsPageEmpty12}{\showpaste} \tab{5}\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)\free{x }\free{y }\bound{eq2 }} \end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPagePatch13} \begin{paste}{ugIntroDiffEqnsPageFull13}{ugIntroDiffEqnsPageEmpty13} \pastebutton{ugIntroDiffEqnsPageFull13}{\hidepaste} \tab{5}\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])\free{x }\free{y }\free{eq1 }\free{eq2 }} \indentrel{3}\begin{verbatim} (13) 1 3 2 5 17 7 62 9 11 [t + � t + �� t + ��� t + ���� t + O(t ), 3 15 315 2835 1 2 5 4 61 6 277 8 50521 10 1 + � t + �� t + ��� t + ���� t + ������� t 2 24 720 8064 3628800 + 11 O(t ) ] Type: List UnivariateTaylorSeries(Expression Integer,t,0) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroDiffEqnsPageEmpty13} \begin{paste}{ugIntroDiffEqnsPageEmpty13}{ugIntroDiffEqnsPagePatch13} \pastebutton{ugIntroDiffEqnsPageEmpty13}{\showpaste} \tab{5}\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])\free{x }\free{y }\free{eq1 }\free{eq2 }} \end{paste}\end{patch} \begin{patch}{ugIntroGraphicsPagePatch1} \begin{paste}{ugIntroGraphicsPageFull1}{ugIntroGraphicsPageEmpty1} \pastebutton{ugIntroGraphicsPageFull1}{\hidepaste} \tab{5}\spadgraph{draw(cos(5*t/8), t=0..16*\%pi, coordinates==polar)} \center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugIntroGraphicsPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugIntroGraphicsPage1}} \end{paste}\end{patch} \begin{patch}{ugIntroGraphicsPageEmpty1} \begin{paste}{ugIntroGraphicsPageEmpty1}{ugIntroGraphicsPagePatch1} \pastebutton{ugIntroGraphicsPageEmpty1}{\showpaste} \tab{5}\spadgraph{draw(cos(5*t/8), t=0..16*\%pi, coordinates==polar)} \end{paste}\end{patch} \begin{patch}{ugIntroGraphicsPagePatch2} \begin{paste}{ugIntroGraphicsPageFull2}{ugIntroGraphicsPageEmpty2} \pastebutton{ugIntroGraphicsPageFull2}{\hidepaste} \tab{5}\spadgraph{draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction == (x,y) +-> argument atan complex(x,y))} \center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugIntroGraphicsPage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugIntroGraphicsPage2}} \end{paste}\end{patch} \begin{patch}{ugIntroGraphicsPageEmpty2} \begin{paste}{ugIntroGraphicsPageEmpty2}{ugIntroGraphicsPagePatch2} \pastebutton{ugIntroGraphicsPageEmpty2}{\showpaste} \tab{5}\spadgraph{draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction == (x,y) +-> argument atan complex(x,y))} \end{paste}\end{patch} \begin{patch}{ugIntroCommentsPagePatch1} \begin{paste}{ugIntroCommentsPageFull1}{ugIntroCommentsPageEmpty1} \pastebutton{ugIntroCommentsPageFull1}{\hidepaste} \tab{5}\spadcommand{2 + 3 -- this is rather simple, no?} \indentrel{3}\begin{verbatim} (1) 5 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroCommentsPageEmpty1} \begin{paste}{ugIntroCommentsPageEmpty1}{ugIntroCommentsPagePatch1} \pastebutton{ugIntroCommentsPageEmpty1}{\showpaste} \tab{5}\spadcommand{2 + 3 -- this is rather simple, no?} \end{paste}\end{patch} \begin{patch}{ugIntroSolutionPagePatch1} \begin{paste}{ugIntroSolutionPageFull1}{ugIntroSolutionPageEmpty1} \pastebutton{ugIntroSolutionPageFull1}{\hidepaste} \tab{5}\spadcommand{S(t) == [x**2-2*y**2 - t,x*y-y-5*x + 5]\bound{S1 }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSolutionPageEmpty1} \begin{paste}{ugIntroSolutionPageEmpty1}{ugIntroSolutionPagePatch1} \pastebutton{ugIntroSolutionPageEmpty1}{\showpaste} \tab{5}\spadcommand{S(t) == [x**2-2*y**2 - t,x*y-y-5*x + 5]\bound{S1 }} \end{paste}\end{patch} \begin{patch}{ugIntroSolutionPagePatch2} \begin{paste}{ugIntroSolutionPageFull2}{ugIntroSolutionPageEmpty2} \pastebutton{ugIntroSolutionPageFull2}{\hidepaste} \tab{5}\spadcommand{solve(S(19),1/10**20)\free{S1 }} \indentrel{3}\begin{verbatim} (2) 2451682632253093442511 [[y= 5,x= - ����������������������], 295147905179352825856 2451682632253093442511 [y= 5,x= ����������������������]] 295147905179352825856 Type: List List Equation Polynomial Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSolutionPageEmpty2} \begin{paste}{ugIntroSolutionPageEmpty2}{ugIntroSolutionPagePatch2} \pastebutton{ugIntroSolutionPageEmpty2}{\showpaste} \tab{5}\spadcommand{solve(S(19),1/10**20)\free{S1 }} \end{paste}\end{patch} \begin{patch}{ugIntroSolutionPagePatch3} \begin{paste}{ugIntroSolutionPageFull3}{ugIntroSolutionPageEmpty3} \pastebutton{ugIntroSolutionPageFull3}{\hidepaste} \tab{5}\spadcommand{complexSolve(S(19),10.e-20)\free{S1 }} \indentrel{3}\begin{verbatim} (3) [ [y= 5.0, x= 8.3066238629 1807485256 1669055295 290320373] , [y= 5.0, x= - 8.3066238629 1807485256 1669055295 290320373] , [y= - 3.0 %i,x= 1.0], [y= 3.0 %i,x= 1.0]] Type: List List Equation Polynomial Complex Float \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSolutionPageEmpty3} \begin{paste}{ugIntroSolutionPageEmpty3}{ugIntroSolutionPagePatch3} \pastebutton{ugIntroSolutionPageEmpty3}{\showpaste} \tab{5}\spadcommand{complexSolve(S(19),10.e-20)\free{S1 }} \end{paste}\end{patch} \begin{patch}{ugIntroSolutionPagePatch4} \begin{paste}{ugIntroSolutionPageFull4}{ugIntroSolutionPageEmpty4} \pastebutton{ugIntroSolutionPageFull4}{\hidepaste} \tab{5}\spadcommand{radicalSolve(S(a),[x,y])\free{S1 }} \indentrel{3}\begin{verbatim} (4) ������Ŀ ������Ŀ [[x= - \�a + 50 ,y= 5], [x= \�a + 50 ,y= 5], �������Ŀ �������Ŀ �- a + 1 �- a + 1 [x= 1,y= �������� ], [x= 1,y= - �������� ]] \� 2 \� 2 Type: List List Equation Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSolutionPageEmpty4} \begin{paste}{ugIntroSolutionPageEmpty4}{ugIntroSolutionPagePatch4} \pastebutton{ugIntroSolutionPageEmpty4}{\showpaste} \tab{5}\spadcommand{radicalSolve(S(a),[x,y])\free{S1 }} \end{paste}\end{patch} \begin{patch}{ugIntroSolutionPagePatch5} \begin{paste}{ugIntroSolutionPageFull5}{ugIntroSolutionPageEmpty5} \pastebutton{ugIntroSolutionPageFull5}{\hidepaste} \tab{5}\spadcommand{eqns := [x**2 - y + z,x**2*z + x**4 - b*y, y**2 *z - a - b*x]\bound{e }} \indentrel{3}\begin{verbatim} 2 2 4 2 (5) [z - y + x ,x z - b y + x ,y z - b x - a] Type: List Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSolutionPageEmpty5} \begin{paste}{ugIntroSolutionPageEmpty5}{ugIntroSolutionPagePatch5} \pastebutton{ugIntroSolutionPageEmpty5}{\showpaste} \tab{5}\spadcommand{eqns := [x**2 - y + z,x**2*z + x**4 - b*y, y**2 *z - a - b*x]\bound{e }} \end{paste}\end{patch} \begin{patch}{ugIntroSolutionPagePatch6} \begin{paste}{ugIntroSolutionPageFull6}{ugIntroSolutionPageEmpty6} \pastebutton{ugIntroSolutionPageFull6}{\hidepaste} \tab{5}\spadcommand{solve(eqns,[x,y,z])\free{e }} \indentrel{3}\begin{verbatim} (6) 2 a a [[x= - �,y= 0,z= - ��], b 2 b 3 2 2 z + 2b z + b z - a [x= ��������������������, y= z + b, b 6 5 2 4 3 3 4 2 z + 4b z + 6b z + (4b - 2a)z + (b - 4a b)z + 2 3 2 - 2a b z - b + a = 0 ] ] Type: List List Equation Fraction Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroSolutionPageEmpty6} \begin{paste}{ugIntroSolutionPageEmpty6}{ugIntroSolutionPagePatch6} \pastebutton{ugIntroSolutionPageEmpty6}{\showpaste} \tab{5}\spadcommand{solve(eqns,[x,y,z])\free{e }} \end{paste}\end{patch} \begin{patch}{ugIntroPreviousPagePatch1} \begin{paste}{ugIntroPreviousPageFull1}{ugIntroPreviousPageEmpty1} \pastebutton{ugIntroPreviousPageFull1}{\hidepaste} \tab{5}\spadcommand{10 ** 10\bound{prev }} \indentrel{3}\begin{verbatim} (1) 10000000000 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroPreviousPageEmpty1} \begin{paste}{ugIntroPreviousPageEmpty1}{ugIntroPreviousPagePatch1} \pastebutton{ugIntroPreviousPageEmpty1}{\showpaste} \tab{5}\spadcommand{10 ** 10\bound{prev }} \end{paste}\end{patch} \begin{patch}{ugIntroPreviousPagePatch2} \begin{paste}{ugIntroPreviousPageFull2}{ugIntroPreviousPageEmpty2} \pastebutton{ugIntroPreviousPageFull2}{\hidepaste} \tab{5}\spadcommand{\% - 1\free{prev }\bound{prev1 }} \indentrel{3}\begin{verbatim} (2) 9999999999 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroPreviousPageEmpty2} \begin{paste}{ugIntroPreviousPageEmpty2}{ugIntroPreviousPagePatch2} \pastebutton{ugIntroPreviousPageEmpty2}{\showpaste} \tab{5}\spadcommand{\% - 1\free{prev }\bound{prev1 }} \end{paste}\end{patch} \begin{patch}{ugIntroPreviousPagePatch3} \begin{paste}{ugIntroPreviousPageFull3}{ugIntroPreviousPageEmpty3} \pastebutton{ugIntroPreviousPageFull3}{\hidepaste} \tab{5}\spadcommand{\%\%(-1)\free{prev1 }\bound{prev2 }} \indentrel{3}\begin{verbatim} (3) 9999999999 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroPreviousPageEmpty3} \begin{paste}{ugIntroPreviousPageEmpty3}{ugIntroPreviousPagePatch3} \pastebutton{ugIntroPreviousPageEmpty3}{\showpaste} \tab{5}\spadcommand{\%\%(-1)\free{prev1 }\bound{prev2 }} \end{paste}\end{patch} \begin{patch}{ugIntroPreviousPagePatch4} \begin{paste}{ugIntroPreviousPageFull4}{ugIntroPreviousPageEmpty4} \pastebutton{ugIntroPreviousPageFull4}{\hidepaste} \tab{5}\spadcommand{\%\%(1)\free{prev2 }} \indentrel{3}\begin{verbatim} (4) 10000000000 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ugIntroPreviousPageEmpty4} \begin{paste}{ugIntroPreviousPageEmpty4}{ugIntroPreviousPagePatch4} \pastebutton{ugIntroPreviousPageEmpty4}{\showpaste} \tab{5}\spadcommand{\%\%(1)\free{prev2 }} \end{paste}\end{patch}