Lösungen / Solutions

Remove["Global`*"]

1

a

x=7.EW

x = 1*E^(I 2 Pi/7)

x^0 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6

x^0 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 // Simplify

x^0 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 // N // Chop

(x^7 - 1)/(x - 1)

Sum[x^k, {k, 0, 6}]

b

Sum[x^k, {k, 0, 999}]

Sum[x^k, {k, 0, 999}] // Simplify

Sum[x^k, {k, 0, 999}] // N // Chop

Sum[x^k, {k, 0, 994}] // N // Chop

Sum[x^k, {k, 0, 999}] == 
      Sum[x^k, {k, 0, 994}] + Sum[x^k, {k, 995, 999}]

Sum[x^k, {k, 995, 999}]

Sum[x^k, {k, 995, 999}] // N

2

a = 2 + I;  b = 1 - 2  I;

a

(a^21  - 1)/(a - 1)

b

Conjugate[a] Conjugate[b] Conjugate[a + b]

Conjugate[a] Conjugate[b] Conjugate[a + b]^(-1)

Remove[x]

3

a

Remove[a1, a2, a3, x]

a1 = 3 - I; a2 = 3 + I; a3 = 4 + I;

s = Solve[x^5 == a1/a2 a3, {x}]

N[%]

s1 = Table[x[k] = x /. s[[k]], {k, 1, 5}] // Flatten // 
        N

s2 = Table[x[k] = x^3 /. s[[k]], {k, 1, 5}] // Flatten // 
        N

b

so1 = 
        Show[Graphics[
            Prepend[Table[
                Point[{Re[s1[[t]]], Im[s1[[t]]]}], {t, 1, 5}], 
              PointSize[0.03]]], AspectRatio -> Automatic];

so2 = 
        Show[Graphics[{Line[{{-1, 0}, {1, 0}}], 
              Line[{{0, -1}, {0, 1}}]}]];

so3 = 
        Show[Graphics[
            Prepend[Table[
                Point[{Re[s2[[t]]], Im[s2[[t]]]}], {t, 1, 5}], 
              PointSize[0.02]]], AspectRatio -> Automatic];

Show[so1, so2, so3];

Abs[a1/Conjugate[a2] Conjugate[a3]]

Abs[a1/a2 a3]

4

Abs[x]/Abs[a1] == Abs[x/a1]

Solve[Abs[x]/Abs[a1] == Abs[x/a1], {x}]

{Abs[x]/Abs[a1], Abs[x/a1]}

Jedes x ist Lösung

5

a = {12, 4}; b = {7, -4}; c = {18, -5};  
    o = {0, 0};

c == λ a  +  μ b

s = 
      Solve[c == λ a  +  μ b, {λ, μ}] // 
        Flatten

{l1, l2} = {λ, μ} /. s

N[%]

<< Graphics`Arrow`

Show[
        Graphics[{Arrow[o, a],  
            Arrow[o, l1 a], 
Arrow[o, b],  
            Arrow[o, l2 b], 
Arrow[o, 
              c], 
Line[{{-1, 0}, {1, 0}}], 
            Line[{{-0, -1}, {0, 1}}] , Line[{c, l1 a}], 
            Line[{c, l2 b}]}, AspectRatio -> Automatic]];

6

a

?Mod

t1 = Table[Table[Mod[x + y, 6], {x, 0, 5}], {y, 0, 5}]

t1 // MatrixForm

t2 = Table[Table[Mod[x*y, 6], {x, 0, 5}], {y, 0, 5}]

t2 // MatrixForm

b

Remove[a]

Mod[(x + 2)*4, 6] == Mod[a, 6] // ExpandAll

Mod[4  x, 6] == Mod[a - 8, 6] // ExpandAll

Table[Mod[4  x, 6] == Mod[a - 8, 6], {a, 0, 3}]

Table[
        Table[Mod[4  x, 6] == Mod[a - 8, 6], {a, 0, 3}], {x, 0, 5}] // 
      MatrixForm

Mod[4*0, 6] == Mod[2 - 8, 6]

Mod[4*1, 6] == Mod[0 - 8, 6]

Mod[4*3, 6] == Mod[2 - 8, 6]

Mod[4*4, 6] == Mod[0 - 8, 6]