Express $ z=-32 $ in exponential form.
My reasoning:
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$ z=-32 $ is the same as $ z=-32+0i $
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Exponential form should look like $ z=Re^{\theta i} $
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$ R =\sqrt{(-32)^2 + 0^2} = 32 $
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$ \theta = \tan^{-1}(\frac{0}{-32}) = 0 $
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So answer becomes: $ z = 32e^{0i} = 32 $
But, obviously, $ -32 \ne 32 $
What am I missing?
Best Answer
Since $z=re^{i \theta}$, with $r\geq 0$, you must have $|z| = r$. So $r=32$. Then you need to solve $-32 = 32 e^{i \theta} = 32 (\cos\theta + i \sin \theta)$. Since the imaginary part is zero, you must have $\theta = n \pi$ for some integer $n$. Choosing $n$ to be odd gives all solutions, since $\cos n \pi = -1$ iff $n$ is odd.
Consequently $-32 = 32 e^{i n \pi}$, where $n$ is odd.