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# complex numbers

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Find all complex number z such that z^4 = 1.

Jan 10, 2021

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+25658
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Find all complex number z such that $$z^4 = 1$$.

$$\begin{array}{|rcll|} \hline z^4 &=& 1 \\ z^4-1 &=& 0 \\ (z^2-1)(z^2+1) &=& 0 \\ \hline z^2-1 &=& 0 \\ z^2 &=& 1 \quad | \quad \text{sqrt both sides} \\ z &=& \pm\sqrt{1} \\ \mathbf{z} &=& \mathbf{\pm 1} \\ \hline z^2+1 &=& 0 \\ z^2 &=& -1 \quad | \quad \text{sqrt both sides} \\ z &=& \pm \sqrt{-1} \quad | \quad \sqrt{-1}=i \\ \mathbf{z} &=& \mathbf{\pm i}\qquad\text{Complex solutions} \\ \hline \end{array}$$

Jan 10, 2021