# Category Archives: pictures

## Post without words #16

Posted in pattern, pictures, posts without words | Tagged , , , | 7 Comments

## Post without words #15

Posted in pattern, pictures, posts without words | Tagged , , , , | 8 Comments

## Post without words #14

Posted in pattern, pictures, posts without words | Tagged , , , , | 14 Comments

## Post without words #13

Posted in pattern, pictures, posts without words | Tagged , , | 15 Comments

## Games with factorization diagram cards

Since I published a deck of factorization diagram cards last September, a few teachers have picked up copies of the cards and started using them with their students. I’ve started collecting ideas for games you can play using the cards, … Continue reading

Posted in arithmetic, counting, games, pattern, pictures, primes, teaching | Tagged , , , , , | 4 Comments

## Computing sums of primitive roots

Remember this picture? It, and other pictures like it, express the fact that for a given , if we take the primitive roots for each of the divisors of , together they make up exactly the set of all th … Continue reading

Posted in geometry, pictures | Tagged , , , , , , | 9 Comments

## Sums of primitive roots

In my previous post, we saw that adding up all the complex th roots of unity always yields zero (unless , in which case the sum is ). Intuitively, this is because the roots are symmetrically distributed around the unit … Continue reading

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

## Sums and symmetry

Let’s continue our exploration of roots of unity. Recall that for any positive integer , there are complex numbers, evenly spaced around the unit circle, whose th power is equal to . These are called the th roots of unity. … Continue reading

Posted in geometry, pictures | Tagged , , , , , , | 2 Comments

## Primitive roots of unity

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

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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