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If $z = 1 + i$, calculate the powers $z^j$ for $j = 1,2,3,\ldots,10$ and plot them on an Argand diagram.

I understand how to do this and I'm sure after some tedious work you can do this, however my interest is in whether a certain pattern arises from this plotting and if so, why?

Additionally I wonder what would the smallest positive integer $n$ be such that $z^n$ is a real number?

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    Note that $(1+i)^2=2i$ and so the smallest $n$ for which $z^n=(1+i)^n$ is a real number is what? – 2017-01-14
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    To the last question, does $n=0$ count? :-) – 2017-01-14

2 Answers 2

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There is a pattern:

enter image description here

The general graph of $(1+i)^n$ is given by, for $n\ge0$,

enter image description here

or,

$$(1+i)^n=2^{n/2}(\cos\frac{n\pi}4+i\sin\frac{n\pi}4)$$

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    Wow, nicely done. – 2017-01-14
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    @juniven Thanks! :D (I didn't actually graph that with a complex graphing calculator. Just some polar magic) – 2017-01-14
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    How would you describe this pattern? ;) – 2017-01-14
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    @TripleA see the last line. – 2017-01-14
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    @SimpleArt As in, how would you write in words or describe the pattern? – 2017-01-14
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    @TripleA oh, well, angle and distance from origin. Plot a table of magnitudes, and I hope my first graph helps. – 2017-01-14
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    For searching purposes: that curve is a *logarithmic spiral*. – 2017-01-21
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Hint

Note that $(1+i)^2=2i$.

If $n=2k$ then $(1+i)^{2k}=(2i)^k$;

If $n=2k+1$ then $(1+i)^{2k+1}=(2i)^k(1+i)$.

Also remember that:

  1. $i^{4p}=1$

  2. $i^{4p+1}=i$

  3. $i^{4p+2}=-1$

  4. $i^{4p+3}=-i$

Can you finish?