Rechnen mit Zahlen

Taschenrechner

((1.23 + 2.25)/3.67)

N[Pi, 500]

Ganzzahlarithmetik

100!

(2^32) - 1

N[%]

((100!)/(2^100))

(∑_(k = 1)^100) (1/k)

%  // N

FactorInteger[100!]

PrimeQ[1234567]

FactorInteger[1234567]

NextPrime[n_] := n + 1 /; PrimeQ[n + 1]
NextPrime[n_] := NextPrime[n + 1]

NextPrime[1234567]

PrimeQ[%]

Algebraische Zahlen

x = (Sqrt[2]) + (Sqrt[3])

(1/x)

(1/x) // Simplify

(1/x) // FullSimplify

y = (Sqrt[3]) - (Sqrt[2])

x*y // Simplify

Sin[(π/5)]

Sin[(π/5)] Cos[(π/5)] // Simplify

Sin[(π/5)] Cos[(π/5)] // FullSimplify

Sin[π]

N[π, 100]

Sin[%]

komplexe Zahlen

((1 + I)/(1 - I))

Re[2  Exp[3  x + I y]]

Clear[x, y]

Re[2  Exp[3  x + I y]]

ComplexExpand[%]

Rechnen mit Symbolen

Polynome und rationale Funktionen

pol = (((x + y))^10) - (((x - y))^10)

Expand[pol]

Factor[pol]

rat = ((1 - (x^10))/(1 - (x^4)))

Together[rat]

Factor[rat]

Factor[(x^6) + (x^2) + 1]

pol = Factor[(x^6) + (x^2) + 1, Modulus -> 13]

Expand[pol - ((x^6) + (x^2) + 1)]

Factor[(x^4) + 1]

Factor[(x^4) + 1, GaussianIntegers -> True]

Factor[(x^4) + 1, Extension -> {(Sqrt[2])}]

Lineare Algebra

HilbertMatrix[n_] := Table[(1/(j + k - 1)), {j, n}, {k, n}]

H = HilbertMatrix[7]

Inverse[H]

Det[H]

Vandermonde[n_] := Table[((Subscript[x, j])^(k - 1)), {j, n}, {k, n}]

V = Vandermonde[5]

Det[V]

Factor[Det[V]]

Gleichungen

s = Solve[(x^2) - 3  x - 1 == 0, x]

x /. s

s = Solve[(x^3) - 3  x - 1 == 0, x]

s // N

NSolve[(x^3) - 3  x - 1 == 0, x]

Plot[(x^3) - 3  x - 1, {x, -3, 3}]

s = Solve[(x^4) - 3  x - 1 == 0, x]

s // N

s = Solve[(x^5) - 3  x - 1 == 0, x]

s // N

(∏_(k = 1)^5)(x /. s[[k]]) // RootReduce

Solve[{(x^2) + (y^2) == 1, -(x^2) + 2  (y^2) + 1 == 0}, {x, 
        y}]

Needs["Graphics`ImplicitPlot`"]

ImplicitPlot[{(x^2) + (y^2) == 1, -(x^2) + 2  (y^2) + 1 == 
          0}, {x, -3, 3}, {y, -3, 3}]

Solve[{((x^2)/4) + (y^2) == 1, -(x^2) + 2  (y^2) + 1 == 0}, {x, 
        y}]

ImplicitPlot[{((x^2)/4) + (y^2) == 1, -(x^2) + 2  (y^2) + 1 == 
          0}, {x, -3, 3}, {y, -3, 3}]

Graphische Darstellungen

Plot[{Sin[x], Cos[x], Tan[x]}, {x, -5, 5}, 
      PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 1, 0], 
          RGBColor[0, 0, 1]}, PlotRange -> {-2, 2}]

Options[Plot]

Plot3D[(x^2) - (y^2), {x, -2, 2}, {y, -2, 2}]

Plot3D[((x y)/((x^2) + (y^2))), {x, -2, 2}, {y, -2, 2}, 
      PlotPoints -> 50]

ContourPlot[Sin[x] Sin[y], {x, -3, 3}, {y, -3, 3}, 
      ColorFunction -> Hue, PlotPoints -> 100]

Analysis

Limit[((Exp[x] - 1)/x), x -> 0]

Series[((Exp[x] - 1)/x), {x, 0, 10}]

ableitung = (∂_x) ((Exp[x] - 1)/x)

∫ableitung \[DifferentialD]x

eingabe = (((x^3) + (x^2) + x - 1)/((x^4) + (x^2) + 1))

Plot[eingabe, {x, -5, 5}]

integral = ∫eingabe \[DifferentialD]x

resultat = (∂_x) integral

Vereinfachung

resultat // FullSimplify