Monthly Archives: October 2016

Post without words #12

Posted on October 28, 2016 by Brent
Posted in fractals, geometry, pattern, posts without words | Tagged , , , , | 1 Comment

Primitive roots of unity

Posted on October 20, 2016 by Brent

So we have now seen that there are always different complex th roots of unity, that is, complex numbers whose th power is equal to , equally spaced around the circumference of the unit circle. Consider the first th root … Continue reading →

Posted in geometry, pictures | Tagged , , , , , | 4 Comments

Complex multiplication: proof

Posted on October 18, 2016 by Brent

In my previous post, I claimed that when multiplying two complex numbers, their lengths multiply and their angles add, like this: In particular, this means that there are always different complex numbers whose th power is equal to : they … Continue reading →

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Complex multiplication and roots of unity

Posted on October 14, 2016 by Brent

If played around with the question from my previous post, you probably found something like the following: That is, as the powers of we get , , and with all possible sign combinations. Of course, since , if we continue … Continue reading →

Posted in geometry, pictures | Tagged , , , , , | 4 Comments

Complexifying our dots

Posted on October 11, 2016 by Brent

It’s time to up our game a bit. Previously we have considered some cool pictures with dots and bespoked circles, looking for patterns, without really considering what sort of mathematical objects these circles might represent. In fact, they turn out … Continue reading →

Posted in geometry, pictures | Tagged , | 3 Comments

Totient sums

Posted on October 7, 2016 by Brent

I took a bit of a break to travel to Japan for a conference, but I’m back now to continue the series I started with Post Without Words #10, a follow-up post, and Post Without Words #11. Recall that we … Continue reading →

Posted in geometry, pattern, pictures, posts without words, proof | Tagged , , , , , | 4 Comments