Show that r e i θ = s e i α ⟹ r = s and θ = α + 2 π k and r e i θ = s e i α ⟹ r = s and θ = α + 2 π k

Question

Show that   r      e          i      θ        =  s      e          i      α          ⟹              r      =      s         and           θ      =      α      +      2      π      k       and   r      e          i      θ        =  s      e          i      α          ⟹              r      =      s         and           θ      =      α      +      2      π      k

Leah Conley · Accepted Answer

HINTClearly we have   r  =  0    ⟺    s  =  0 and if r=s=0 the angles are not important. Assume now   r  ≠  0 and hence   s  ≠  0 and divide the complex numbers. We get  1  =      r    s        e          i      [      θ      −      α      ]        .Taking modulus, we see       |    r      /    s      |    =  1 so r=s since they are both positive. Now we are left with   1  =      e          i      [      θ      −      α      ]      .

Show that r e <mrow class="MJX-TeXAtom-ORD"> i θ

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