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I've done following steps thus far:

define z:= 1-i; then z=sqrt(2) * e^((-pi/4)i)
(1-i)^n = z^n = (sqrt(2)*e^((-pi/4)i)^n = sqrt(2)^n * e^((-pi/4)in
That gives two conditions:
1) sqrt(2)^n needs to be element N -> n=2k, k element N
2) e^((-pi/4)in needs to be element N -> pi/4 * n = pi * k -> n=4k, k element N

At this point I'm lost, though. I know the solution from straight thinking - n needs to be 8*k, but how do I get to that point from where I am with my steps so far? E.g. From the two conditions I have here, n could also be 4, but (1-i)^n is -4 which is not element N.

Cheers Anton

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  • $\begingroup$ So your two conditions are: (1) $n$ must be a multiple of $2$, and (2) $n$ must be a multiple of $4$. Notice that (2) implies (1). $\endgroup$
    – user169852
    Commented Dec 21, 2020 at 10:26
  • $\begingroup$ $n$ must be a multiple of $2$ and a multiple of $4$. Why do you say that these two conditions contradict each other? $\endgroup$
    – Crostul
    Commented Dec 21, 2020 at 10:32
  • $\begingroup$ Right, I expressed myself wrongly. But how do I write it down? Both conditions - n a multiple of 2 and n a multiple of 4 - are also true for n=4 - but then (1-i)^n is -4 which is not element N... $\endgroup$ Commented Dec 21, 2020 at 10:52
  • $\begingroup$ $(1-i)^4=-4$ therefore $(1-i)^8=16$ $\endgroup$
    – Lozenges
    Commented Dec 21, 2020 at 11:06
  • $\begingroup$ Welcome to Maths SE. Please use MathJax. $\endgroup$
    – Toby Mak
    Commented Dec 21, 2020 at 12:20

1 Answer 1

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Alright, I just found my mistake...

of course coniditon 2) is wrong. It needs to be as followed

2) e^((-pi/4)in needs to be element N -> pi/4 * n = 2*pi * k -> n=8*k, k element N

now 2) implies 1), therefore the only condition that needs to be accounted for is 2). Done.

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