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# Tag Archives: circle

## Book review: Tales of Impossibility

[Disclosure of Material Connection: Princeton Press kindly provided me with a free review copy of this book. I was not required to write a positive review. The opinions expressed are my own.] Tales of Impossibility: The 2000-Year Quest to Solve … Continue reading

Posted in books, review
Tagged algebra, antiquity, circle, constructible, construction, cube, geometry, impossibility, polygon, problems, square, tales, trisect
4 Comments

## The Möbius function proof, part 2 (the subset parity lemma)

Continuing from my previous post, we are in the middle of proving that satisfies the same equation as , that is, and that therefore for all , that is, is the sum of all the th primitive roots of unity. … Continue reading

Posted in arithmetic, combinatorics, complex numbers, primes, proof
Tagged circle, complex, moebius, mu, primitive, proof, roots, sum, unit, unity
3 Comments

## The Möbius function proof, part 1

In my last post, I introduced the Möbius function , which is defined in terms of the prime factorization of : if has any repeated prime factors, that is, if is divisible by a perfect square. Otherwise, if has distinct … Continue reading

## The Möbius function

Time to pull back the curtain a bit! My recent series of posts on complex roots of unity may seem somewhat random and unmotivated so far, but the fact is that I definitely have a destination in mind—we are slowly … Continue reading

## 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

## 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

## 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

## 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

## 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

## Complex multiplication and roots of unity

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