multiplication on a circle
Clock Arithmetic
We learn pretty early about clock arithmetic. Instead of counting evenly spaced points on a line, we count evenly spaced points on a circle. It’s convenient, since a circle can fit on a piece of paper, but lines are just sooo long. But arithmetic on a circle can also be much more interesting.
Consider counting. If you are counting on a line, and you start at dot 5, and then you add 8 by counting 8 dots to the right, you are at 13. But on a clock, you are back at 1. That probably took a bit more thinking. If you start at 5 on a line, and add 1000, you know immediately you are at 1005. But where would you be on the clock?
A zero-based clock
Our familiar 12-hour clock is thousands of years old, though the circular format is much more recent. And the clock arithmetic we are introduced to when we are young is confined to the 12-hour clock with numbers 1 through 12.
It will turn out to be convenient to have our clock start at zero. And we will allow for clocks to vary the number of equally spaced points, as long as the number of points is greater than one.
Modular arithmetic
…is what mathematicians call zero-based clock arithmetic. Eight hours after 5 o’clock is 1 o’clock. In modular arithmetic, we would say 8 plus 5 is 1 modulo 12. We can write that like this:
(8+5) mod 12 ≡ 1
The modulus, 12, is the number of equally spaced points on our clock. Righting 12 every time is verbose, but we will soon look at clocks that have a different modulus, i.e. a different number of equally spaced points. But we will make it clear what the modulus is, and can drop it from our notation. So the above equation can be written like this:
8+5 ≡ 1
Modular arithmetic is remarkably important in our lives, as it is key to many concepts in computer science and therefore in the software around us.
Modular multiplication
What does it mean to multiply in modular arithmetic? Well, multiplication is just repeated adding. So that’s what we do. 7 o’clock plus 7 hours is 2 o’clock. So 7 · 2 ≡ 2. What is 7 · 3 modulo 12?
Modular exponentiation
You probably thought I was going to talk about division. I might do that later,but let’s first talk about raising a number to a power. Exponentiation is repeated multiplication. So 22 ≡ 4, 23 ≡ 8, and 24 ≡ 4. What is 21000?
What is 102, 103, 104? Investigate powers of some other numbers.
From clock to line art
Dave Richeson gave a presentation at Bridges 2023 about line art on a circle, where the dots are connected via a common multiplier.