John Golden, whom so far as I know doesn’t have an active blog, suggested this week’s topic. It pairs nicely with last week’s. I link to that in text, but if you would like to read all of this year’s Little Mathematics A to Z it should be at this link. And if you’d like to see all of my A-to-Z projects, pleas try this link. Thank you.
When I wrote about multiplication I came to the peculiar conclusion that it was the same as addition. This is true only in certain lights. When we study [abstract] algebra we look at things that look like arithmetic. The simplest useful thing that looks like arithmetic is a group. It has a set of elements, and a pairwise “group operation”. That group operation we call multiplication, if we don’t have a better name. We give it two elements and it gives us one. Under certain circumstances, this multiplication looks just like addition does.
But we have reason to think addition and multiplication aren’t the same. Where do we get addition?
We can make a meaningful addition by giving it something to interact with. By adding another operation. This turns the group into a ring. As it has two operations, it’s hard to resist calling one of them addition and the other multiplication. The new multiplication follows many of the rules the addition did. Adding two elements together gives you an element in the ring. So does multiplying. Addition is associative: is the same thing as . So it multiplication: is the same thing as .
And then the addition and the multiplication have to interact. If they didn’t, we’d just have a group with two operations. I don’t know anyone who’s found a good use for that. The way addition and multiplication interact we call distribution. This is represented by two rules, both of them depending on elements a, b, and c:
This is where we get something we have to call addition. It’s in having the two interacting group operations.
A problem which would have worried me at age eight: do we know we’re calling the correct operation “addition”? Yes, yes, names are arbitrary. But are we matching the thing we think we’re doing when we calculate 2 + 2 to addition and the thing for 2 x 2 to multiplication? How do we tell these two apart?
For all that they start the same, and resemble one another, there are differences. Addition has an identity, something that works like zero. is always , whatever is. Multiplication … the multiplication we use every day has an identity, that is, 1. Are we required to have a multiplicative identity, something so that is always ? That depends on what it said in the Introduction to Algebra textbook you learned on. If you want to be clear your rings do have a multiplicative identity you call it a “unit ring”. If you want to be clear you don’t care, I don’t know what to say. I’m told some people write that as “rng”, to hint that this identity is missing.
Addition always has an inverse. Whatever element you pick, there is some so that is the additive identity. Multiplication? Even if we have a unit ring, there’s not always a reciprocal. The integers are a unit ring. But there are only two integers that have an integer multiplicative inverse, something you can multiply them by to get 1. If your unit ring does have a multiplicative inverse, this is called a division algebra. Rational numbers, for example, are a division algebra.
So for some rings, like the integers, there’s an obvious difference between addition and multiplication. But for the rational numbers? Can we tell the operations apart?
We can, through the additive identity, which please let me call 0. And the multiplicative identity, which please let me call 1. Is there a multiplicative inverse of 0? Suppose there is one; let me call it , because I need some name. Then of all the things in the world, we know this:
I can replace anything I like with something equal to it. So, for example, I can replace 0 with the sum of an element and its additive inverse. Like, for some element . So then:
And distribute this away!
I don’t know what number is, nor what its inverse is. But I know its sum is zero. And so
This looks like trouble. But, all right, why not have the additive and the multiplicative identities be the same number? Mathematicians like to play with all kinds of weird things; why not this weirdness?
The why not is that you work out pretty fast that every element has to be equal to every other element. If you’re not sure how, consider the starting line of that little proof, but with an element :
So there, finally, is a crack between addition and multiplication. Addition’s identity element, its zero, can’t have a multiplicative inverse. Multiplication’s identity element, its one, must have an additive inverse. We get addition from the thing we can’t un-multiply.
It may have struck you that if all we want is a ring with the lone element of 0 (or 1), then we can have addition and multiplication be indistinguishable again. And have the additive and multiplicative identities be the same thing. There’s nothing else for them to be. This is true, and we can. Unfortunately this ring doesn’t do much that’s interesting, except maybe prove some theorem we were working on isn’t always true. So we usually draw a box around it, acknowledge it once, and then exclude it from division algebras and fields and other things of interest. It’s much the same way we normally rule out 1 as a prime number. It’s an example that is too much bother to include given how unenlightening it is.
You can have groups and attach to them a multiplication and an addition and another binary operation. Those aren’t of such general interest that you study them much as an undergraduate.
And this is what we know of addition. It looks almost like a second multiplication. But it interacts just enough with multiplication to force the two to be distinguishable. From that we can create mathematics structures as interesting as arithmetic is.
16 thoughts on “My Little 2021 Mathematics A-to-Z: Addition”
This is fascinating and I’m not enough of a mathematician to discuss the rings, except to say that when I studied college level number properties to take my credential test I thought I was in a Tolkien novel.
However, I explicitly tell my kids that multiplication is NOT repeated addition and that it’s a mistake to limit it in this form. Three cases:
Multiplying fractions: 1/4 * 1/2 can not be represented with addition. Nor can 1/2 * 2, for that matter.
Multiplying negatives (either by positives or other negatives) can’t be represented by addition.
While 1 is the identity value for multiplication and division (and I don’t quite understand your discussion of why you rejected this), 0 either should not be included in multiplication’s “ring” or it’s a special case. Zero is Siva the Destroyer in multiplication, and of course you can’t divide by 0–and these are related. As I tell my kids, you can’t ctrl-z from total nuclear destruction, and you can’t undo multiplying by 0. I usually mention at this point that I don’t know enough math to discuss how you include this in the definition of multiplication.
I tell them that the best way I have to understand multiplication is that it’s scalar (it was when I first learned of vectors that I saw it’s the best way to understand multiplication) , in the sense of it “scales” the number. It’s only a coincidence that some of its operations look like arithmetic (or if not a coincidence, then incomplete). That is, when I multiply 2×3 I can say that I’m adding 2 3 times, but I can also say that I’m tripling the scale of 2, and the second one is more consistent.
I also say that I think, but am not sure, that division by zero is also a way of describing infinity that addition can’t match.
As you are a real mathematician, please tell me what part of this I have wrong or incomplete.
I thank you so for your thoughts. And also for your patience in my reply; I’ve not been at my best lately.
I’m enough out of the loop in thinking about arithmetic-education that I’m reluctant to say anything about how multiplication and addition should be taught. I can see the use, if you’re interested in helping someone understand whether they’re multiplying whole numbers correctly, in presenting multiplication as a sort of accelerated addition. And I can see the argument that, at least for people who aren’t going to specialize in mathematics, it’s enough if they can do a calculation efficiently rather than that they have a ‘better’ mental model for it.
I think that the multiplication of fractions can be explained in repeated-addition terms, if someone’s already comfortable with fractions and so doesn’t mind (say) turning 3/4 times 2/5 into
3 times 2, and then take a quarter of that, and then take a fifth of that''. This depends on them already having an idea ofa quarter of 6” or so, but if we’re just designing the curriculum from scratch we can put that in. This is rubbish in setting people up to understand multiplication by an irrational number. But there’s some handwaving that has to be done in going from whole numbers to integers to rationals to real numbers anyway, not without getting into some really abstract territory.
A negative times a negative, yeah, that’s a disaster to try explaining as repeated-addition. I think the only way to make it make sense is first to try justifying positive-times-negative as repeated-subtraction, and then put in the idea that you have two parties and adding to one person’s total is subtracting from the other’s.
My gut is to say the vividness of describing multiplication by zero as destruction, and of a kind of destruction that can’t be undone, is compelling. But it also suggests that zero is a ruin left over where a number used to be, and that’s a bit of trouble too. People already suspect zero of being too weird to be a number, and I’d be afraid of feeding that.
I like the notion of talking about multiplication as scaling. Particularly since a reliable mathematical trick is in rescaling a quantity, as a change of variable or something similar. But I’m not sure how to describe what scaling is without describing it as a kind of multiplication. One trouble with teaching the basic concepts is that you have few tools to build an understanding on, which I guess is why elementary school education is so hard.
I’m always wary of infinities as a way of explaining something strange. This may be just my idiosyncratic bias. I feel like people treat infinity as having this godlike power to create or abolish any property that’s useful, and it kind of is, but you need to reason carefully through why it can. So I’m not sure that it’s a satisfying explanation for why it would cause trouble to allow division by zero. On the other hand my explanation wasn’t very satisfying either, looks like. Hm.
Hey, I somehow missed that you’d responded. Thanks so much!
“But I’m not sure how to describe what scaling is without describing it as a kind of multiplication. One trouble with teaching the basic concepts is that you have few tools to build an understanding on, which I guess is why elementary school education is so hard.”
Yes, in spades. NAEP scores have dropped in the last decade, and I think part of it could be due to the attempts of elementary teachers to explain math conceptually when there really aren’t good explanations. In high school, I try hard not to confuse kids, but I do want to get them thinking.
“But I’m not sure how to describe what scaling is without describing it as a kind of multiplication. ”
Good lord. I never thought of that. I try to describe it with vectors or distance. Doubling distance, half the distance, and so on. But you’re right.
John Golden’s blog is Math Hombre: https://mathhombre.blogspot.com. It’s not super-active, but he still posts interesting things from time to time.
Oh thank you so. I’m embarrassed not to have realized this since I have his blog in my bookmarks and he even had a post just a couple weeks ago. Shows how scattered I’ve been lately, I suppose.