Being a mathematician who spends much of his time talking to nonmathematicians, I find I often have to censor myself, because the way I want to say something uses mathematical jargon. Times when I forget to censor myself are often mistaken for trying to make myself seem more confusing or complicated. Actually, it’s just the opposite — mathematicians have lots of words and constructions to deal very precisely with complex ideas, and it’s not unusual for them to be useful in “real life” too.
Hence the first in a series of short and potentially useful posts. Whether you want to understand a mathematician better or to try to pass yourself off as one at a cocktail party, I’m here to help.
Today’s word: modulo.
Examples of usage.
- Eleven plus two is one, modulo twelve.
Translation 1: The difference between eleven plus two and one is evenly divisible by twelve.
Translation 2: If you don’t distinguish between twelve and zero, you also don’t distinguish between eleven plus two and one.
- These two sudoku puzzles are the same modulo rotations and reflections.
Translation 1: You can rotate and/or reflect this sudoku puzzle to make it identical to that one.
Translation 2: If you consider two things the same if they are mirror images of each other or if they are rotations of the same image, then these two sudoku puzzles are identical as far as you are concerned
- I proved Theorem A modulo Theorem B.
Translation 1: I wrote down a proof of Theorem A, but it assumes the truth Theorem B (which I may or may not know how to prove, and which may or may not even be true).
Translation 2: If I knew how to prove Theorem B, I would instantly be able to write a proof of Theorem A.
- The original usage is as in the first example. You probably learned about (and forgot about) modular arithmetic in some math class along the way. In this context, twelve is called the modulus. The weird o ending and peculiar syntax come from Latin. Latin buffs already realized that modulo is just the ablative case of modulus. So “modulo twelve”, usually written (and sometimes pronounced) just “mod 12” would expand to something like “using twelve as a modulus”. This is unwieldy, and we are obliged to those crazy Latin grammar rules for the use of their ablative case.
- The common thread here is that working “modulo [stuff]” means that we are treating [stuff] as trivial. We don’t distinguish between [stuff] and nothing. Nothing might mean zero, or it might mean something else as dictated by context.
- Note that “modulo” doesn’t mean merely “except”. One would never say “I finished getting groceries modulo buying eggs.”