Numbers & Number Systems

Numbers come in expanding families — counting numbers, integers, fractions, the reals — each invented to fix something the last couldn't do. And the same number can be written in different bases, which is the whole reason computers think in binary and hex.

1. The Families of Numbers Naturals, integers, rationals, irrationals, reals — each family was invented to make an operation always work: subtraction needed negatives, division needed fractions, geometry needed irrationals. They nest like Russian dolls.
2. Bases: Binary, Decimal, Hex A base is how many digits you count with before carrying. Decimal uses ten, binary two, hex sixteen. Positional notation explains all three — and why computers store binary and humans read it as hex.
3. Modular Arithmetic: Clock Math Modular arithmetic is the math of remainders and wrap-around — the reason 15:00 is 3 o'clock. It quietly runs hashing, parity checks, cyclic buffers, and cryptography.