The p-adic Numbers

On the ordinary number line, two numbers are close when their difference is small. The p-adic numbers keep the same arithmetic but change what "close" means: two numbers are close when their difference is divisible by a high power of a chosen prime.

That one change produces a strange and useful world. Numbers can have infinitely many digits to the left, every point of a disk is its centre, and −1 can have a square root. This video builds the 5-adic numbers from scratch and shows what they are good for.

What it covers

1. A new notion of distance, built on divisibility by powers of 5
2. Numbers as paths down an infinite tree of digits
3. Why …4444 in base 5 equals −1, and how fractions get digits
4. Why the base has to be prime (try 10 and it breaks)
5. Geometry where every point of a disk is its centre
6. A number system with no ordering
7. The product formula, and the "prime at infinity"
8. Monsky's theorem: you cannot cut a square into an odd number of equal-area triangles

Watch on YouTube