I wanted to start the Little 2021 Mathematics A-to-Z with more ceremony. These glossary projects are fun and work in about equal measure. But an already hard year got much harder about a month and a half back, and it hasn’t been getting much better. I’m even considering cutting down the reduced A-to-Z project I am doing. But I also feel I need to get some structured work under way. And sometimes only ambition will overcome a diminished world. So I begin, and with luck, will keep posting weekly essays about mathematical terms.
Today’s was a term suggested by Iva Sallay, longtime blog friend and creator of the Find The Factors recreational mathematics puzzle. Also a frequent host of the Playful Math Education Blog Carnival, a project quite worth reading and a great hosting challenge too. And as often makes for a delightful A-to-Z topic, it’s about something so commonplace one forgets it can hold surprises.
A friend pondering mathematics said they know you learn addition first, but that multiplication somehow felt more fundamental. I supported their insight. We learn two plus two first. It’s two times two where we start seeing strange things.
Suppose for the moment we’re interested only in the integers. Zero multiplied by anything is zero. There’s nothing like that in addition. Consider even numbers. An even number times anything gives you an even number again. There’s no duplicating that in addition. But this trait isn’t even unique to even numbers. Multiples of three, or four, or 237 assimilate the integers by multiplication the same way. You can find an integer to add to 2 to get 5; you can’t find an integer to multiply by 2 to get 5. Or consider prime numbers. There’s no integer you can make by only one, or only finitely many, different sums. New possibilities, and restrictions, happen in multiplication.
Whether this makes multiplication the foundation of mathematics, or at least arithmetic, is a judgement. It depends how basic your concepts must be, and what you decide is important. Mathematicians do have a field which studies “things that look like arithmetic”, though. We call this algebra. Or call it abstract algebra to clarify it’s not that stuff with the quadratic formula. And that starts with group theory. A group is made of two things. One is a collection of elements. The other is a thing to do with pairs of elements. Generically, we call that multiplication.
A possible multiplication has to follow a couple rules. It has to be a binary operation on your group’s set. That is, it matches two things in the set to something in the set. There has to be an identity, something that works like 1 does for multiplying numbers. It has to be associative. If you want to multiply three things together, you can start with whatever pair looks easier. Every element has to have an inverse, something you can multiply it by to get 1 as the product.
That’s all, and that’s not much. This description covers a lot of things. For example, there’s regular old multiplication, for the set of rational numbers (other than zero and I intend to talk about that later). For another, there’s rotations of a ball. Each axis you could turn the ball around on, and angle you could rotate it, is an element of the set of three-dimensional rotations. Multiplication we interpret as doing those rotations one after the other. There’s the multiplication of square matrices, ones that have the same number of rows and columns.
If you’re reading a pop mathematics blog, you know of , the “imaginary unit”. You know it because . A bit more multiplying of these and you find a nice tight cycle. This forms a group, with four discernible elements: and regular multiplication. It’s a nice example of a “cyclic group”. We can represent the whole thing as multiplying a single element together: . We can think of but that’s got the same value as . Or , which has the same value as . With a little ingenuity we can even think of what we might mean by, say, and realize it has to be the same quantity as . Or which has to equal . You see the cycle.
A cyclic group doesn’t have to have four elements. It needs to be generated by doing the multiplication over and over on one element, that’s all. It can have a single element, or two, or two hundred. Or infinitely many elements. Suppose we have a set built on the powers of an element that we’ll call . This is a common name for “an element and we don’t care what it is”. It has nothing to do with the number called e, or any number. At least it doesn’t have to.
Please let me use the shorthand of to mean times , and to mean times , and so on. Then we have a set that looks like, in part, . They multiply together the way we might multiply x raised to powers. is , and is , and is and so on.
Those exponents suggest something familiar. In this infinite cyclic group is , where j and k are integers. Do we even need to write the e? Why not just write the j and k in a normal-size typeface? Is there a difference between cyclic-group multiplication and regular old addition of integers?
Not an important one. There’s differences in how we write the symbols, and what we think they mean. There’s not a difference in the way they interact. Regular old addition, in this light, we can see as a multiplication.
Calling addition “multiplication” can be confusing. So we deal with that a few ways. One is to say that rather than multiplication what a group has is a group operation. This lets us avoid fooling people into thinking we mean to take this times that. It lacks a good shorthand word, the way we might say “a times b” or “a plus b”. But we can call it “the group operation”, and say “times” or “plus” as fits our sentence and our sentiment.
I’ve left unanswered that mention of multiplication on the rational-numbers-except-zero making a group. If you include zero in the set, though, you don’t have multiplication as a group operation. There’s no inverse to zero. There seems to be an oversight in multiplication not being a multiplication. I hope to address that in the next A-to-Z essay, on Addition.
This, and my other essays for the Little 2021 Mathematics A-to-Z, should be at this link. And all my A-to-Z essays from every year should be at this link. Thanks for reading.