Number System Builder

Intro to Proofs · Axiom Academy

Numbers weren't handed down whole. Each system was built — by adding exactly the structure the last one was missing. Every number you know was invented to answer a question Ask the natural numbers to solve x + 5 = 2 and they have no answer — there is no count that works. So mathematicians did something audacious: they invented new numbers until the equation could be solved. Do that four times and you climb from counting all the way to the complex plane. Watch the tower build itself. At each rung, an equation the system below couldn't touch is written in red — then the new numbers switch on and the answer appears in green. The integers, the rationals, the reals, the complex numbers: each one is the smallest world where the previous world's broken equation finally has a solution. Each system contains the one below it whole — — and adds just enough to fix what was broken. Bend the number line into a clock Extending outward isn't the only move. Take the integers and glue n back onto 0 — the line curls into a ring of just n slots, and counting wraps around . This is , modular arithmetic, the same logic a clock uses when 7 hours after 8 o'clock is 3 . Drag the dials and watch a + b land. Add, subtract, multiply — it all still works, but inside a finite circle. That makes a ring : a complete little number system of its own. When can you divide ? The leap to a field

This is the written version of the interactive lesson above. See the full Intro to Proofs course.