cromulentenough:
jadagul:
jadagul:
Today on the blog I start a new project: where do numbers come from?
By which I mean, mathematicians deal with lots of weird kinds of numbers. Real numbers, complex numbers, p-adic numbers, quaternions, surreal numbers, and more. And if you try to describe the more abstract types of “numbers” you sound completely incomprehensible.
But these numbers all come from somewhere. So I’m going to take you through a fictional history of numbers. Not the real history of the actual people who developed these concepts, but the way they could have developed them, cleaned up and organized. So in the end you can see how you, too, could have developed all these seemingly strange and abstract concepts.
This week in part 1, we cover the most sensible numbers. We start with the basic ability to count, and invent negative numbers, fractions, square roots, and more.
But that will still leave some important questions open—like, what is π? So we’ll have to come back for that in part 2.
stumpyjoepete
One question I had: the definition of algebraic #s is in terms of polynomials with rational coefficients. If you allow coefficients like sqrt(2) or “the smallest solution to {other polynomial}”, does this change anything or is the set closed under this?
Ooh, that’s a good point and I never addressed it explicitly. And you’re right, the answer is it’s closed under this algebra-izing thing; the fancy word is the “algebraic closure”.
And yes, if you just solve all the polynomials with rational coefficients (or integer coefficients, even), you get all the roots of polynomials with algebraic coefficients. And this is basically because if you have a polynomial with algebraic coefficients, you can multiply it by some other polynomials with algebraic coefficients and get a polynomial with rational coefficients.
In general this is, like, “because Galois theory”, but you can see the idea with square roots: If I have sqrt(a)x + sqrt(b), I can multiply by the conjugate sqrt(a)x-sqrt(b) and get ax^2-b. So so anything that’s a root of a polynomial with square-root coefficients is also a root of a polynomial with rational coefficients.
i didn’t realize that the x^5 + x + 3 = 0 situation was a thing. So you there’s algebraic numbers you can’t represent with roots, multiplication and addition? is there no closed form solution at all?
Yeah this is one of the weirder bits. I might write a thing about just this at some point, but it was a tangent from the main goal of this series.
Given any quadratic equation, we can represent the solutions, as seen in the quadratic formula:
This was first developed in India around AD 628, and got its modern form in the Netherlands in 1594.
Given a cubic equation, we can represent all three solutions, as seen in the cubic formula:
The cubic formula was developed in Italy in the 1500s, as part of a running equation-solving competition among roving mathematicians.
The quartic formula looks like this:
It was developed around the same time as the cubic formula. It also looks appalling.
So how bad must the quintic formula be? Well, it doesn’t exist. That was proven in 1824 by Abel, following up an incomplete proof by Ruffini in 1799. In 1832 Galois basically invented group theory to give a really general formulation of these ideas. The fact that the alternating group of order n is not solvable for n>4 is equivalent to the inability to write down solutions with radicals for general polynomials of degree higher than four. (That’s why the term is “solvable”!)
But yeah, we can’t write down a closed-form solution to most higher-degree polynomials, at least not with radicals. I believe you can write down the solution to any quintic if you introduce a symbol for the inverse of f(x) = x^5+x. But you’d need more new symbols for degree six, and more more for degree seven, and… So we mostly just declared bankruptcy.
You can also solve the quintic with functions that don’t look like you obviously just designed them to solve this particular problem, such as elliptic functions and hypergeometric functions. But none that are closed form in the most common sense of the term.