# Why is it popular to teach modulus via the example of mod 12 and analogue clocks?

Why is it popular to teach modulus via the example of mod 12 and analogue clocks rather than rectangles or tables that have a finite number of columns in each row, and infinitely many rows?

It's natural to identify an origin row that has the number one in the far left-hand column of the origin row, for people who write from left to right. Then it can be indicated that the table extends infinitely far upward by filling in the blanks with zero and negative values, and the table also extends infinitely many rows down, just as for a Cartesian coordinate system. Thus, every integer appears somewhere in the table.

Focus on the clock analogy tends to restrict attention to mod 12, but of course it does also suggest going beyond integer values altogether.

We could define mod $$2\pi$$ as follows:

t is congruent to u (mod $$2\pi$$) if and only if $$\sin(t) = \sin(u)$$ and $$\cos(t) = \cos(u)$$.

Then we can generalize beyond $$2\pi$$ to an arbitrary positive real number $$m$$ by selecting a radius $$r$$ so that $$2\pi r = m$$.

Is that why the clock analogy is used?

• Because this is a thing they are already well acquainted with? Familiarity is important in examples. – Thorbjørn Ravn Andersen Oct 6 '19 at 9:49
• The clock situation is a little different now (so I'd say inertia is now a prime mover of this), but when I was in school everyone was always looking at the wall clocks to see how much time was left before class ended, so it's hard for me to imagine a more immediately apparent example for students than the big clock on the wall above the blackboard or above the classroom door. – Dave L Renfro Oct 6 '19 at 14:23
• @Renfro: is that a compensating feature or "saving grace"? No matter how bad the school experience was from the point of view of the student, at least the student would derive some benefit: motivated focus on an object that can serve an instructive purpose in learning modulus. – ELM Oct 6 '19 at 14:34