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Chapter 6: Problem 2
Write each expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$ (5+7 i)+(4+6 i) $$
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In Example 6 , we found cube roots of 1 by finding numbers \(\theta\) such that $$ \cos (3 \theta)=1 \text { and } \sin (3 \theta)=0 $$ The three choices \(\theta=0, \theta=\frac{2 \pi}{3},\) and \(\theta=\frac{4 \pi}{3}\) gave us three distinct cube roots of 1 . Other choices of \(\theta\), such as \(\theta=2 \pi, \theta=\frac{8 \pi}{3},\) and \(\theta=\frac{10 \pi}{3},\) also satisfy the equations above. Explain why these choices of \(\theta\) do not give us additional cube roots of \(1 .\)
Suppose \(w\) and \(z\) are complex numbers. Show that $$ |w z|=|w||z| $$.
Find a complex number whose square equals \(21-20 i\).
Describe the subset of the complex plane consisting of the complex numbers \(z\) such that \(z^{3}\) is a real number.
Show that addition of complex numbers is associative, meaning that $$ u+(w+z)=(u+w)+z $$ for all complex numbers \(u, w,\) and \(z\).
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