# Numbers & Number Systems

You've used numbers your whole life, so it's tempting to assume there's nothing left to understand about
them. But two things almost nobody is taught turn numbers from a blur into a clear map: *why* there are
different **families** of numbers (and what each one was invented to fix), and the fact that the same
number can be **written** in different ways - base 10, base 2, base 16 - which is the entire reason your
computer "thinks in binary."

This guide makes both concrete. You'll see why subtraction forced us to invent negative numbers and
division forced fractions, you'll convert between decimal, binary, and hex until it feels routine, and
you'll meet the "clock arithmetic" (modular arithmetic) that quietly powers hashing, parity checks, and
cryptography. It's the number sense that makes the rest of the Mathematics track - and a lot of code -
suddenly readable.

## How to read this
- **Here for binary and hex?** [Phase 2](02-bases-binary-decimal-hex.md) is the base-conversion phase.
- **Want the whole map?** Read in order - the families (Phase 1) give the rest its footing.

## The phases
1. **[The Families of Numbers](01-the-families-of-numbers.md)** - naturals, integers, rationals,
   irrationals, reals, and why each family had to be invented.
2. **[Bases: Binary, Decimal, Hex](02-bases-binary-decimal-hex.md)** - positional notation, converting
   between bases, and why computers use binary and hex.
3. **[Modular Arithmetic: Clock Math](03-modular-arithmetic-clock-math.md)** - remainders and
   wrap-around, and the everyday tech that runs on them.

> This builds on [Sets, Relations & Functions](/guides/sets-relations-and-functions) (the number
> families are nested sets). Next in the Mathematics track: counting, and probability.