Tag Archives: circle

Book review: Tales of Impossibility

Posted on January 1, 2020 by Brent

[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)

Posted on December 3, 2016 by Brent

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

Posted on November 29, 2016 by Brent

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 →

Posted in Uncategorized | Tagged circle, complex, moebius, mu, primitive, proof, roots, sum, unit, unity | 9 Comments

The Möbius function

Posted on November 22, 2016 by Brent

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 →

Posted in Uncategorized | Tagged circle, complex, moebius, mu, primitive, roots, sum, unit, unity | 5 Comments

Computing sums of primitive roots

Posted on November 15, 2016 by Brent

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 circle, complex, primitive, roots, sum, unit, unity | 9 Comments

Sums of primitive roots

Posted on November 9, 2016 by Brent

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 circle, complex, primitive, roots, sum, unit, unity | 4 Comments

Sums and symmetry

Posted on November 4, 2016 by Brent

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 circle, complex, roots, sum, symmetry, unit, unity | 2 Comments

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 circle, complex, primitive, roots, unit, unity | 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 →

Posted in geometry, pictures | Tagged circle, complex, multiplication, proof, rotation, unit | Comments Off

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 circle, complex, multiplication, roots, unit, unity | 4 Comments