Back in October, Simon got fascinated with this weird system called p-adic numbers (where p stands for “prime”). It’s a system in which you can have numbers going infinitely far to the left. Simon learned about this system from James Tanton.
Today, while watching Sebastian Lague’s new series about computers work, he discovered that the 2-adic system is used in computers to keep track of negative numbers. We used to think this whole p-adic system was an abstract fantasy.
Computers use a version of this called “2’s compliment”. I was aware of it before but didn’t know it had anything to do with the p-adic system. Computers add with a version of regular long addition but in binary. The only difference is that when the last column carries you don’t write it down. You should just discard it.
There’s a flaw in the system if you use non-prime numbers. There’s a number that’s essentially equal to 5 to the infinite power and a number that’s equal to 2 to the infinite power. If you multiply those two together, you get a number which is equal to 10 to the infinite, which ends up being 0. So you have two non-zero numbers which when multiplied produce zero – that violates the basic laws of arithmetic.