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Algebra Level 2

e 2 π i = ? \Large e^{2πi } =\, ?

Clarification : i = − 1 i = \sqrt{-1} .


The answer is 1.

Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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2 solutions

Joshua Chin
Nov 19, 2015

We know that

e i π = − 1 { e }^{ i\pi }=-1

e 2 i π = ( e i π ) 2 = − 1 2 = 1 { e }^{ 2i\pi }={ ({ e }^{ i\pi }) }^{ 2 }\\ \qquad ={ -1 }^{ 2 }\\ \qquad =1

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Moderator note:

Simple standard approach.

Complex Number
Dec 15, 2015

e i x = cos ⁡ ( x ) + i × sin ⁡ ( x ) e^{ix} = \cos(x) + i \times \sin(x) Therefore by replacing x with two times pi, we can see that it indeed equates to 1

5 Helpful 0 Interesting 0 Brilliant 0 Confused

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