If $e^{i\pi}=-1$, then what does $e^{2i\pi}$ equal?

Question

As the question says. As according to Euler's formula, $e^{i\pi}+1=0$ thus $e^{i\pi}=-1$, what therefore does $e^{2i\pi} $ equal?

$e^{i\theta}=cos\theta+isin\theta$ and the angle is.....$-2\pi$, so... — imranfat, Apr 13 '16 at 16:52
@labbhattacharjee: would that work with $e^{2\pi i}=(e^{i2})^\pi$ ? — , Apr 13 '16 at 16:55
@labbhattacharjee This page is only about $a^b$ for real $a,b$. — Jean-Claude Arbaut, Apr 13 '16 at 16:57
@Jean-ClaudeArbaut, See https://en.wikipedia.org/wiki/Exponentiation#Real_exponents — lab bhattacharjee, Apr 13 '16 at 17:13
@labbhattacharjee You probably mean https://en.wikipedia.org/wiki/Exponentiation#Powers_of_complex_numbers since your second link is also about $a^b$ with real (and even positive) $a$. When $a$ is negative or complex, it's much more difficult to define $a^b$ consistently, but you already knew that I guess. — Jean-Claude Arbaut, Apr 13 '16 at 18:45
Why has @almagest edited my question and post. It should be e^(2ipi) NOT e^-(2ipi) — Pianoman1234, Apr 13 '16 at 21:56
The original title did mention e^-2ipi, with a minus sign. — Did, Apr 13 '16 at 22:13
Not really complex analysis. More like algebra / precalculus. — mathreadler, Apr 13 '16 at 22:14

score 4 · Accepted Answer · edited Apr 13 '17 at 12:21

It equals $(-1)^2 = 1$.

More generally, for a real number $\theta$, we might say that $e^{i \theta}$ is equal to the coordinates of the point on the unit circle at angle $\theta$ with the right horizontal axis, where the coordinates are given as complex numbers. This is sometimes written $$ e^{i\theta} = \cos \theta + i \sin \theta.$$

This might feel funny, but this can be made formal. See for instance this question.

The coordinates of the point at angle $\pi$ are $-1$, and the coordinates of the point at angle $2\pi$ are $1$.

score 2 · Answer 2 · answered Apr 13 '16 at 16:55

2

Observe that: $e^{-2\pi i}$ is equal to $(e^{i\pi})^{-2}$; then

$$\frac{1}{e^{2i\pi}}=1$$

answered Apr 13 '16 at 16:55

InsideOut

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If $e^{i\pi}=-1$, then what does $e^{2i\pi}$ equal?

2 Answers2