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# Help with Complex Number Arithmetic

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Let z be a complex number such that $$z^2=1$$. Compute $$z+\frac{1}{z}+z^2+\frac{1}{z^2}$$

May 8, 2022

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May 8, 2022
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For the last 2 terms, you can just replace z^2 by 1 and calculate its value, so that is done.

For the first 2 terms, you want to rewrite it like this: $$z + \dfrac1z = \dfrac{z^2}z + \dfrac1z = \dfrac{z^2 + 1}z$$

Rewriting and substituting z^2 = 1 gives $$\dfrac{1 + 1}z + 1 + \dfrac11 = 2 + \dfrac2z$$. This indicates the answer still depends on z.

Note that there are exactly 2 complex numbers satisfying z^2 = 1. (Hint: Move the 1 to the left-hand side and try to factorize.)

Depending on which one you consider, the required value will be different.

The answer Guest gave is not entirely correct due to this reason.

May 8, 2022