Clock Arithmetic

Intro to Proofs · Axiom Academy

The familiar 12-hour clock hides a whole arithmetic of its own — where 15 and 3 are the same number. That is modular arithmetic. When 10 o'clock plus 5 hours is 3 o'clock When it's 10 o'clock and you wait 5 hours, the clock reads 3 — not 15. The hand doesn't run off the end; it sweeps past 12 and starts over. That "wraparound" is the entire idea behind modular arithmetic: numbers that would normally keep climbing instead cycle around a fixed loop. Watch the hand start at 10 and sweep 5 hours clockwise. The running count climbs 10, 11, 12 — then keeps going as 1, 2, 3, looping back past the top of the clock. It lands on 3. In symbols, . Fifteen hours of counting, but only twelve marks to land on — so 15 settles onto the same spot as 3. That is what " " means. Add on the clock — and on any clock Set a starting hour and how much to add; the hand lands where the count comes to rest. Then swap the clock for one with a different number of marks. Modular arithmetic isn't special to 12 — every modulus n gives its own loop with values . Adding two clock numbers always gives another clock number — you can never escape the loop. That closure is the first sign of a genuine algebraic structure. One spot on the clock, infinitely many numbers

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