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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?

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    $\begingroup$ Because this is a thing they are already well acquainted with? Familiarity is important in examples. $\endgroup$
    – Thorbjørn Ravn Andersen
    Oct 6, 2019 at 9:49
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    $\begingroup$ 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. $\endgroup$
    – Dave L Renfro
    Oct 6, 2019 at 14:23
  • $\begingroup$ @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. $\endgroup$
    – ELM
    Oct 6, 2019 at 14:34

1 Answer 1

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The answer is simple, practical, and historical. Analog clocks and watch faces were everywhere. We learned to tell time early on, by first or second grade, (age 6 or 7), so the leap to understanding Modulo-12 wasn't too tough. What's 9+6? 3, of course. And for the smart student who resists, insisting it's 15, the teacher offers the fact that military time actually calls this 15, but for regular citizens, whose clocks only show 1-12, we need to answer '3'.

With respect to the rest of your question, offering trig, that was likely a pleasant extension of using the clock face, not the original reason. I recall, once we were comfortable with the clock concept, we moved on to other bases, also illustrated by a circle to maintain that familiarity. And the addition of multiple 'clock' faces to help show how each place represented the next order of magnitude in that base arithmetic.

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    $\begingroup$ This may be a better reason in the US than the UK, where for "regular citizens" almost all timetables, appointments, radio/tv schedules, etc, now use a 24 hour clock rather than am/pm. I wouldn't even think of setting my PC or cellphone to display time modulo 12, for example. $\endgroup$
    – alephzero
    Oct 5, 2019 at 22:20
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    $\begingroup$ @alephzero - when I wrote "historical" I meant it, literally. I was a child in the 1960's. No digital watches then. Were analog wall clocks in the UK not 1 - 12 like in the US? $\endgroup$
    – JTP - Apologise to Monica
    Oct 5, 2019 at 22:40
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    $\begingroup$ @alephzero I can assure you that people in the UK still use am/pm. If you never do, then it's you that's odd. $\endgroup$
    – Jessica B
    Oct 5, 2019 at 23:08
  • $\begingroup$ in most parts of the world it is 15, though, and it is considered "regular" $\endgroup$
    – njzk2
    Oct 6, 2019 at 0:21
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    $\begingroup$ @alephzero I would argue that there is still a well understood high colloquial use, especially when spoken. I don't tell people to meet me for a beer at 20 but at 8. And I don't know any railway station that doesn't have a huge 12 hour analogue clock somewhere near to the 24 hour timetable. And finally, ignoring all that, if you want to insist that today's youth does not understand the 12 hour system, you can simply replace 12 with 24 and get the same educational content. If two hours into the beer, I tell people that I have to go to work in 9 hours, no one is confused by 22+9=7. $\endgroup$
    – mlk
    Oct 6, 2019 at 7:45

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