I want to give you a sense of what it’s like doing math “in the wild”. Doing mathematics is not just about learning what other people have already done: it’s about exploring and playing around with a system to figure out what’s going on. Let’s give it a go!
We are all familiar with the 12 hour system for counting time on a clock. The numbers count each digit from 1 to 12 and then circle back to 1. Since 12 is the number just before 1, we can think of 12 just like 0. We will ignore AM/PM for all of this.
We don’t usually talk about it this way, but we could say that adding 2 and 3 o’clock gives 5, in that 5 o’clock is 2 hours after 3, or 3 hours after 2.
We can ask what 7 + 7 would be. The answer is 14, but this is the same as 2, since 12 is a whole circle around the clock. That is, since 12 is the same as 0.
The next few examples are a little more complicated. Math is a much more interesting subject if you interact with it. It’s rare to find a mathematician reading without something to write on. Find a pencil and paper if you don’t have them around already.
We can do this with larger sums as well. Say, 5 + 5 + 5 + 5 = 20 = 8 + 12 = 8 + 0 = 8. (We can think “20 is 8 past 12.” or “Since 12 = 0, I can add or subtract 12 and still write equals.”) Go ahead and try 10 + 5, 10 + 10, 10 + 15, and 10 + 20. Find each sum and then subtract 12 until the result is less than 12. (I’ll wait.)
The mathematical way to say “in clock math” is just “working modulo 12”. This is our way of saying to consider 12 and 0 to be the same number.
This method, doing the operation as usual and then subtracting or adding 12 until we have a number from 0 to 11, also works with multiplication. We can take 5*4 = 20 = 8, or 5*5 = 25 = 1, or 5*6 = 30 = 6. (This gives the same answer as repeated addition!)
Something funny happens with multiplication modulo 12. With normal numbers, the only time a product is 0 is if one of the factors is 0: two nonzero numbers have a nonzero product. But on a clock, there are numbers which do multiply together to give 0, after you adjust by subtracting 12. Can you find them? There is actually a number which is 0 when multiplied by itself. Can you find this one?
The answers are at the bottom of this post, but it’s good to look around for yourself!
Two nonzero numbers whose product is zero are called zero divisors. There are no normal numbers like this, but modulo 12 this property appears.
Finally, what if we look at clock math on a clock with only 6 numbers? This would be working “modulo 6”, and 6 would be considered equal to 0. We would count 0, 1, 2, 3, 4, 5, 6, and 6 would be equal to 0. Since we can add, subtract, and multiply modulo 6, we can look for zero divisors again, although they may be different than zero divisors modulo 12. What are they? Do we find any? We could also work in modulo 2 and look for them, or look in modulo 3, or modulo 4, and so on.
This process we’re going through–taking a structure and experimenting with it, noticing strange objects and occurrences–is a fundamental part of the mathematical research process. It’s the beginning. We experiment to become familiar with the operations and objects, and then as patterns emerge, we try to find some way to verify that the pattern will always hold. This is where proofs come in.
In science, hypotheses are verified by various experiments, and hypotheses that hold up to this scrutiny gradually become theories. In mathematics, our hypotheses–which we call conjectures–become theorems not through experiments, but through proofs: a sequence of logical inferences linking our assumptions to our conclusions.
I hope you enjoyed playing around with something new, and I hope this gives you a little better idea what’s going on behind the scenes in Rawles Hall!
Just for fun: There is actually a pattern that allows you determine which modular systems contain zero divisors. In other words, when working modulo n, we can figure out whether or not the system has zero divisors based entirely on properties of the number n. I’ll leave the pattern for you to find. Try working up: Start with modulo 2, then modulo 3, and so on, marking whether each modular system has zero divisors.
Zero divisors modulo 12: 2, 3, 4, 6
Modulo 12, if we take 6*6 we get 12 = 0. So 6 times itself is 0, modulo 12.
Zero divisors modulo 6: 2, 3
Zero divisors modulo 2: None!
Zero divisors modulo 3: None!
Zero divisors modulo 4: 2