Prove conjugate of imaginary number

Question

This is statement is used in one of the proofs in my book and I am struggling to figure out how to deduce it or proof it:

$, \overline{i^{n - k}} = i^{n - k}(-1)^{n - k})$

What is the intuition behind the statement? Why is the $(-1)$ raised to a power?

Wouldn't the conjugate of $i$ be $-i$ and thus the conjugate of $i^n$ be $-i^n$?

$\begingroup$ $(-i)^n$, not $-i^n$. $\endgroup$ – Gerry Myerson Sep 28 '15 at 13:45 — Gerry Myerson, Sep 28 '15 at 13:45

Nizar · Accepted Answer · 2015-09-28 14:06:13Z

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Simply, we have $\overline{zz'}= \bar{z}\bar{z'}$. So $$ \overline{i^{n-k}}=\bar{i}\bar{i}...\bar{i}= (-i)(-i)....(-i)= (-1)^{n-k} \; \;i^{n-k}$$

answered Sep 28 '15 at 14:06

Nizar

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$\begingroup$ Thank you! I was being potato $\endgroup$ – Luke Taylor Sep 28 '15 at 14:13

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Surb · Accepted Answer · 2015-09-28 13:51:51Z

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Big Hint :

$$i=e^{i\frac{\pi}{2}},$$

and discuss the cases where $n-k\in \{4m, 4m+1,4m +2,4m+3\}.$

answered Sep 28 '15 at 13:46

Surb

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