Math in… Clock Arithmetic
At 8:00 a.m., what time will it be in 5 hours?
If you’re using a 24-hour clock, the answer is 13:00. But if you’re using a 12-hour clock, it’s 1:00 p.m..
After 12:59, timekeeping “starts over” with 1:00. Although 8 + 5 = 13 and 8 + 5 ≠ 1, you might feel like there should be an arithmetical way to express the fact that 8:00 a.m. + 5 hours = 1:00 p.m..
Mathematicians call this sort of “wrap-around math” modular arithmetic.
We can represent this equation as
8 + 5 ≡ 1 (mod 12)
which translates to
8 plus 5 is congruent to 1 modulo 12.
In other words, 13 and 1 are equivalent if our number system wraps around every 12th integer.
If you can get from one number to another by adding or subtracting copies of 12, those two numbers are congruent modulo 12.
For example, 24 ≡ 0 (mod 12) since 24 - 12 - 12 = 0 and -3 ≡ 9 (mod 12) since -3 + 12 = 9.
For a 24-hour clock, we could similarly work modulo 24.
If today is a Thursday, what day will it be 30 days from now?
Weekday names repeat every 7 days, so we could work modulo 7 and number our days 0-6. We can do some modular arithmetic to discover it’s Saturday:
Since we chose numbering 0 for our first day, subtracting copies of 7 from 34 until we get a number in the range 0-6 is equivalent to finding the remainder when we divide by 34 by 7:
If you’re playing E2 on a piano, how many keys would you need to increase to get to A4?
The notes on a piano similarly repeat (an octave higher) every 7th white key. The note 17 white keys to the right of an E is an A:
What other things have you seen that wrap around in this way? How can you do math with those?
