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# Prove e^(i*2pi*n) = 1 watch

1. prove e^(i*2pi*n) = 1
where n is just an integer

Using e^(ix) = cos(x) + isin(x)

I get:
e^(i*2pi*n) = cos(2pi*n) + isin(2pi*n)
= cos(2pi)^n + isin(2pi)^n
Im not sure what to do next...

I know the trig identity:
cos^2(x) + sin^2(x) = 1

Is this some way linked to proving it equals 1? and if so where does the i dissappear to?
2. (Original post by Sifr)
prove e^(i*2pi*n) = 1
where n is just an integer

Using e^(ix) = cos(x) + isin(x)

I get:
e^(i*2pi*n) = cos(2pi*n) + isin(2pi*n)
= cos(2pi)^n + isin(2pi)^n
Im not sure what to do next...

I know the trig identity:
cos^2(x) + sin^2(x) = 1

Is this some way linked to proving it equals 1? and if so where does the i dissappear to?
By de Moivre's theorem it's (cos(2pi) + isin(2pi))^n, which I hope you can see is not necessarily the same as what you've written (you'll have various nth order cross terms, unless they evaluate to 0). The required result then follows, from considering what happens to cos and sin at multiples of pi.

As a note, by coincidence what you have written does give you the required result, but only because the cross terms evaluate to 0, which wouldn't necessarily be the case for some general angle. Also you can actually get the result without using de Movie's theorem, by using the periodic properties of cos and sin.
3. cos(2pi*n) = 1
sin(2pi*n) = 0

for any natural n.
4. (Original post by Sifr)
prove e^(i*2pi*n) = 1
where n is just an integer

Using e^(ix) = cos(x) + isin(x)

I get:
e^(i*2pi*n) = cos(2pi*n) + isin(2pi*n)
= cos(2pi)^n + isin(2pi)^n
you've got a couple of nasty misconceptions in there!

As electriic_ink points out, you should be able to write down straightaway what and are from the periodicity of sin and cos.
5. ^^ As long as you know how Euler's formula (famously) simplifies in the case

it's a one-line proof for all integers n.
6. Euler's.

where z is a complex number with modulus a and argument x.

Hence, in the case of then on an Argand diagram (y-axis is "i", and the x-axis is just "x") what you have is a line at an angle of from the positive x-axis.

All integer multiples of 2Pi are going to still be the same as 0 degrees, hence the complex number which results is just a line out that way -> with length 1.

On an Argand diagram, then, it's just 1.

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Updated: March 27, 2013
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