## My Little 2021 Mathematics A-to-Z: Addition

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: $a + (b + c)$ is the same thing as $(a + b) + c$. So it multiplication: $a \times (b \times c)$ is the same thing as $(a \times b) \times c$.

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:

$a\times(b + c) = a\times b + a\times c$

$(a + b)\times c = a\times c + b\times c$

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. $a + 0$ is always $a$, whatever $a$ 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 $a \times 1$ is always $a$? 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 $a$ you pick, there is some $-a$ so that $-a + a$ 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 $c$, because I need some name. Then of all the things in the world, we know this:

$0 \times c = 1$

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, $(-a + a)$ for some element $a$. So then:

$(-a + a) \times c = 1$

And distribute this away!

$-a\times c + a\times c = 1$

I don’t know what number $ac$ is, nor what its inverse $-ac$ is. But I know its sum is zero. And so

$0 = 1$

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 $b$:

$0 \times c \times b = 1 \times b$

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.

## My Little 2021 Mathematics A-to-Z: Multiplication

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.

# Multiplication

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 $\imath$, the “imaginary unit”. You know it because $\imath^2 = -1$. A bit more multiplying of these and you find a nice tight cycle. This forms a group, with four discernible elements: $1, \imath, -1, \mbox{ and } -\imath$ and regular multiplication. It’s a nice example of a “cyclic group”. We can represent the whole thing as multiplying a single element together: $\imath^0, \imath, \imath^2, \imath^3$. We can think of $\imath^4$ but that’s got the same value as $\imath^0$. Or $\imath^5$, which has the same value as $\imath^1$. With a little ingenuity we can even think of what we might mean by, say, $\imath^{-1}$ and realize it has to be the same quantity as $\imath^3$. Or $\imath{-2}$ which has to equal $\imath^2$. 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 $e$. 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 $e^2$ to mean $e$ times $e$, and $e^3$ to mean $e^2$ times $e$, and so on. Then we have a set that looks like, in part, $\cdots e^{-3}, e^{-2}, e^{-1}, e^0, e^1, e^2, e^3. \cdots$. They multiply together the way we might multiply x raised to powers. $e^2 \times e^3$ is $e^5$, and $e^4 \times e^{-4}$ is $e^0$, and $e^-3 \times e^2$ is $e^{-1}$ and so on.

Those exponents suggest something familiar. In this infinite cyclic group $e^j \times e^k$ is $e^{j + k}$, 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.

## How To Multiply Numbers By Multiplying Other Numbers Instead

I do read other people’s mathematics writing, even if I don’t do it enough. A couple days ago RJ Lipton and KW Regan’s Reductions And Jokes discussed how one can take a problem and rewrite it as a different problem. This is one of the standard mathematician’s tricks. The point to doing this is that you might have a better handle on the new problem.

“Better” is an aesthetic judgement. It reflects whether the new problem is easier to work with. Along the way, they offer an example that surprised and delighted me, and that I wanted to share. It’s about multiplying whole numbers. Multiplication can take a fair while, as anyone who’s tried to do 38 times 23 by hand has found out. But we can speed that up. A multiplication table is a special case of a lookup table, a chunk of stored memory which has computed ahead of time all the multiplications someone is likely to do. Then instead of doing them, you just look them up.

The catch is that a multiplication table takes memory. To do all the multiplications for whole numbers 1 through 10 you need … well, not 100 memory cells. But 55. To have 1 through 20 worked out ahead of time you need 210 memory cells. Can we do better?

If addition and subtraction are easy enough to do? And if dividing by two is easy enough? Then, yes. Instead of working out every pair multiplication, work out the squares of the whole numbers. And then make use of this identity:

$a \times b = \frac{1}{2}\left( \left(a + b\right)^2 - a^2 - b^2\right)$

And that delights me. It’s one of those relationships that’s sitting there, waiting for anyone who’s ever squared a binomial to notice. I don’t know that anyone actually uses this. But it’s fun to see multiplication worked out by a different yet practical way.

## A Leap Day 2016 Mathematics A To Z: Normal Subgroup

The Leap Day Mathematics A to Z term today is another abstract algebra term. This one again comes from from Gaurish, chief author of the Gaurish4Math blog. Part of it is going to be easy. Part of it is going to need a running start.

## Normal Subgroup.

The “subgroup” part of this is easy. Remember that a “group” means a collection of things and some operation that lets us combine them. We usually call that either addition or multiplication. We usually write it out like it’s multiplication. If a and b are things from the collection, we write “ab” to mean adding or multiplying them together. (If we had a ring, we’d have something like addition and something like multiplication, and we’d be able to do “a + b” or “ab” as needed.)

So with that in mind, the first thing you’d imagine a subgroup to be? That’s what it is. It’s a collection of things, all of which are in the original group, and that uses the same operation as the original group. For example, if the original group has a set that’s the whole numbers and the operation of addition, a subgroup would be the even numbers and the same old addition.

Now things will get much clearer if I have names. Let me use G to mean some group. This is a common generic name for a group. Let me use H as the name for a subgroup of G. This is a common generic name for a subgroup of G. You see how deeply we reach to find names for things. And we’ll still want names for elements inside groups. Those are almost always lowercase letters: a and b, for example. If we want to make clear it’s something from G’s set, we might use g. If we want to be make clear it’s something from H’s set, we might use h.

I need to tax your imagination again. Suppose “g” is some element in G’s set. What would you imagine the symbol “gH” means? No, imagine something simpler.

Mathematicians call this “left-multiplying H by g”. What we mean is, take every single element h that’s in the set H, and find out what gh is. Then take all these products together. That’s the set “gH”. This might be a subgroup. It might not. No telling. Not without knowing what G is, what H is, what g is, and what the operation is. And we call it left-multiplying even if the operation is called addition or something else. It’s just easier to have a standard name even if the name doesn’t make perfect sense.

That we named something left-multiplying probably inspires a question. Is there right-multiplying? Yes, there is. We’d write that as “Hg”. And that means take every single element h that’s in the set H, and find out what hg is. Then take all these products together.

You see the subtle difference between left-multiplying and right-multiplying. In the one, you multiply everything in H on the left. In the other, you multiply everything in H on the right.

So. Take anything in G. Let me call that g. If it’s always, necessarily, true that the left-product, gH, is the same set as the right-product, Hg, then H is a normal subgroup of G.

The mistake mathematics majors make in doing this: we need the set gH to be the same as the set Hg. That is, the whole collection of products has to be the same for left-multiplying as right-multiplying. Nobody cares whether for any particular thing, h, inside H whether gh is the same as hg. It doesn’t matter. It’s whether the whole collection of things is the same that counts. I assume every mathematics major makes this mistake. I did, anyway.

The natural thing to wonder here: how can the set gH ever not be the same as Hg? For that matter, how can a single product gh ever not be the same as hg? Do mathematicians just forget how multiplication works?

Technically speaking no, we don’t. We just want to be able to talk about operations where maybe the order does too matter. With ordinary regular-old-number addition and multiplication the order doesn’t matter. gh always equals hg. We say this “commutes”. And if the operation for a group commutes, then every subgroup is a normal subgroup.

But sometimes we’re interested in things that don’t commute. Or that we can’t assume commute. The example every algebra book uses for this is three-dimensional rotations. Set your algebra book down on a table. If you don’t have an algebra book you may use another one instead. I recommend Christopher Miller’s American Cornball: A Laffopedic Guide To The Formerly Funny. It’s a fine guide to all sorts of jokes that used to amuse and what was supposed to be amusing about them. If you don’t have a table then I don’t know what to suggest.

Spin the book clockwise on the table and then stand it up on the edge nearer you. Then try again. Put the book back where it started. Stand it up on the edge nearer you and then spin it clockwise on the table. The book faces a different way this time around. (If it doesn’t, you spun too much. Try again until you get the answer I said.)

Three-dimensional rotations like this form a group. The different ways you can turn something are the elements of its set. The operation between two rotations is just to do one and then the other, in order. But they don’t commute, not most of the time. So they can have a subgroup that isn’t normal.

You may believe me now that such things exist. Now you can move on to wondering why we should care.

Let me start by saying every group has at least two normal subgroups. Whatever your group G is, there’s a subgroup that’s made up just of the identity element and the group’s operation. The identity element is the thing that acts like 1 does for multiplication. You can multiply stuff by it and you get the same thing you started. The identity and the operator make a subgroup. And you’ll convince yourself that it’s a normal subgroup as soon as you write down g1 = 1g.

(Wait, you might ask! What if multiplying on the left has a different identity than multiplying on the right does? Great question. Very good insight. You’ve got a knack for asking good questions. If we have that then we’re working with a more exotic group-like mathematical object, so don’t worry.)

So the identity, ‘1’, makes a normal subgroup. Here’s another normal subgroup. The whole of G qualifies. (It’s OK if you feel uneasy. Think it over.)

So ‘1’ is a normal subgroup of G. G is a normal subgroup of G. They’re boring answers. We know them before we even know anything about G. But they qualify.

Does this sound familiar any? We have a thing. ‘1’ and the original thing subdivide it. It might be possible to subdivide it more, but maybe not.

Is this all … factoring?

Please here pretend I make a bunch of awkward faces while trying not to say either yes or no. But if H is a normal subgroup of G, then we can write something G/H, just like we might write 4/2, and that means something.

That G/H we call a quotient group. It’s a subgroup, sure. As to what it is … well, let me go back to examples.

Let’s say that G is the set of whole numbers and the operation of ordinary old addition. And H is the set of whole numbers that are multiples of 4, again with addition. So the things in H are 0, 4, 8, 12, and so on. Also -4, -8, -12, and so on.

Suppose we pick things in G. And we use the group operation on the set of things in H. How many different sets can we get out of it? So for example we might pick the number 1 out of G. The set 1 + H is … well, list all the things that are in H, and add 1 to them. So that’s 1 + 0, 1 + 4, 1 + 8, 1 + 12, and 1 + -4, 1 + -8, 1 + -12, and so on. All told, it’s a bunch of numbers one more than a whole multiple of 4.

Or we might pick the number 7 out of G. The set 7 + H is 7 + 0, 7 + 4, 7 + 8, 7 + 12, and so on. It’s also got 7 + -4, 7 + -8, 7 + -12, and all that. These are all the numbers that are three more than a whole multiple of 4.

We might pick the number 8 out of G. This happens to be in H, but so what? The set 8 + H is going to be 8 + 0, 8 + 4, 8 + 8 … you know, these are all going to be multiples of 4 again. So 8 + H is just H. Some of these are simple.

How about the number 3? 3 + H is 3 + 0, 3 + 4, 3 + 8, and so on. The thing is, the collection of numbers you get by 3 + H is the same as the collection of numbers you get by 7 + H. Both 3 and 7 do the same thing when we add them to H.

Fiddle around with this and you realize there’s only four possible different sets you get out of this. You can get 0 + H, 1 + H, 2 + H, or 3 + H. Any other numbers in G give you a set that looks exactly like one of those. So we can speak of 0, 1, 2, and 3 as being a new group, the “quotient group” that you get by G/H. (This looks more like remainders to me, too, but that’s the terminology we have.)

But we can do something like this with any group and any normal subgroup of that group. The normal subgroup gives us a way of picking out a representative set of the original group. That set shows off all the different ways we can manipulate the normal subgroup. It tells us things about the way the original group is put together.

Normal subgroups are not just “factors, but for groups”. They do give us a way to see groups as things built up of other groups. We can see structures in sets of things.

## Reading the Comics, March 3, 2016: Let Popeye Do Mathematics Edition

Elzie Segar’s Thimble Theater is a comic strip you maybe vaguely remember hearing about for some reason. The reason is that, ten years into its run, Segar discovered a charismatic sailor named Popeye. People who read my humor blog know I’m a bit Popeye-mad, even still. Comics Kingdom has in its Vintage comics run the strips from the first story where Popeye appeared. This isn’t it. That story resolved, and the comic tried to carry on with the old cast. It didn’t last. After a few dull weeks Segar started making excuses to put Popeye back on-screen. It’s quite like Dickens’s Pickwick Papers and the discovery of Sam Weller, right down to this being the character that made the author famous.

As part of Segar’s excuses to keep Popeye on panel, nominal lead Castor Oyl has hired a tutor. It’s not going well. I blame the tutor, who’s berating Popeye for being wrong and giving no hint what to do right. But in this installment, originally run the 14th of September, 1929, we get around to arithmetic. Popeye is either a natural, has experience we don’t know about, or is quite lucky. It wouldn’t be absurd for Popeye to be good at some kinds of arithmetic. If he’s trained in navigation he’d probably pick up a good bit of practice calculating. I don’t know anything but the most trivial points of how to calculate one’s position at sea. So I can’t say if it’s plausible Popeye would have practiced calculations like “six and a half times 656”. He may just be lucky.

Mark Litzler’s Joe Vanilla for the 26th of February was the anthropomorphized numerals joke for this go-round.

Mark Tatulli’s Lio for the 26th features soap bubbles made into geometry diagrams. I like that; it’s cute. Coincidentally, Guy Gilchrist’s Nancy for the 29th turns the pieces of a geometry puzzle into pizza. I think that’s a lesser version of the joke. It’s less absurd.

Nick Seluk’s The Awkard Yeti for the 2nd of March is a Schrödinger’s Cat reference alongside a butterfly reference. It seems Comic Strip Master Command challenges my “I’ve said all I can say, for now, about Schrödinger’s Cat and Chaos Butterflies” policy.

Missy Meyer’s Holiday Dodles mentions the 2nd of March was World Maths Day. I hadn’t heard about this; had you? Wikipedia indicates it’s a worldwide mathematics competition event sponsored by 3P Learning. Also that the first one was held on “Pi Day”, the 14th of March, which would make sense. I didn’t know it was Dr Seuss’s birthday either until I ran across a third comic strip doing some Dr Seuss jokes. Comic strips sometimes line up by accident. But I’m always impressed when they spontaneously (I assume) line up for some minor event like that.

Charles Schulz’s Peanuts for the 3rd of March originally ran the 6th of March, 1969. It’s part of a storyline in which Linus’s favorite teacher, Miss Othmar, is replaced following a teacher’s strike. This is why he complains to the new teacher about how Miss Othmar never did things that way.

It gets to appear here because Linus suggests that for some problem or other “we could divide instead of subtract”. I’m a little curious what the problem might have been. Division is often presented as a sort of hurried-up subtraction, or at least it was when I was Linus’s age. But they don’t quite address the same sorts of questions. I suppose something like “how many times eight goes into thirty-two”. But I wouldn’t do that by subtraction except to point out how division answers that question so much better. Still, there is a good point in showing how there can be several ways to do a problem. There almost always are. Sometimes a particular approach is faster than another. Sometimes it’s less confusing than another. Sometimes it gives better insight into other problems than another. If all you are interested in is the right answer, then you can use whatever method works, including letting Popeye guess for you. But, except on the frontier of research where we don’t quite know what we’re studying, there are always choices in how to find an answer.

Tom Toles’s Randolph Itch, 2 am for the 3rd I feel confident I’ve shown before. The strip didn’t run long originally and it’s in its third or fourth rerun cycle on Gocomics.com. It’s still an amusing bit of figure drawing, drawn by figures, being figured out. I make it out to 111,193.

Dave Whamond’s Reality Check for the 3rd of March is a little too proud of knowing some common mathematical symbols.

## Ring.

Early on in her undergraduate career a mathematics major will take a class called Algebra. Actually, Introduction to Algebra is more likely, but another Algebra will follow. She will have to explain to her friends and parents that no, it’s not more of that stuff they didn’t understand in high school about expanding binomial terms and finding quadratic equations. The class is the study of constructs that work much like numbers do, but that aren’t necessarily numbers.

The first structure studied is the group. That’s made of two components. One is a set of elements. There might be infinitely many of them — the real numbers, say, or the whole numbers. Or there might be finitely many — the whole numbers from 0 up to 11, or even just the numbers 0 and 1. The other component is an operation that works like addition. What we mean by “works like addition” is that you can take two of the things in the set, “add” them together, and get something else that’s in the set. It has to be associative: something plus the sum of two other things has to equal the sum of the first two things plus the third thing. That is, 1 + (2 + 3) is the same as (1 + 2) + 3.

Also, by the rules of what makes a group, the addition has to commute. First thing plus second thing has to be the same as second thing plus first thing. That is, 1 + 2 has the same value as 2 + 1 does. Furthermore, there has to be something called the additive identity. It works like zero does in ordinary arithmetic. Anything plus the additive identity is that original thing again. And finally, everything in the group has something that’s its additive inverse. The thing plus the additive inverse is the additive identity, our zero.

If you’re lost, that’s all right. A mathematics major spends as much as four weeks in Intro to Algebra feeling lost here. But this is an example. Suppose we have a group made up of the elements 0, 1, 2, and 3. 0 will be the additive identity: 0 plus anything is that original thing. So 1 plus 0 is 1. 1 plus 1 is 2. 1 plus 2 will be 3. 1 plus 3 will be … well, make that 0 again. 2 plus 0 is 2. 2 plus 1 will be 3. 2 plus 2 will be 0. 2 plus 3 will be 1. 3 plus 0 will be 3. 3 plus 1 will be 0. 3 plus 2 will be 1. 3 plus 3 will be 2. Plus will look like a very strange word at this point.

All the elements in this have an additive inverse. Add 3 to 1 and you get 0. Add 2 to 2 and you get 0. Add 1 to 3 and you get 0. And, yes, add 0 to 0 and you get 0. This means you get to do subtraction just as well as you get to do addition.

We’re halfway there. A “ring”, introduced just as the mathematics major has got the hang of groups, is a group with a second operation. Besides being a collection of elements and an addition-like operation, a ring also has a multiplication-like operation. It doesn’t have to do much, as a multiplication. It has to be associative. That is, something times the product of two other things has to be the same as the product of the first two things times the third. You’ve seen that, though. 1 x (2 x 3) is the same as (1 x 2) x 3. And it has to distribute: something times the sum of two other things has to be the same as the sum of the something times the first thing and the something times the second. That is, 2 x (3 + 4) is the same as 2 x 3 plus 2 x 4.

For example, the group we had before, 0 times anything will be 0. 1 times anything will be what we started with: 1 times 0 is 0, 1 times 1 is 1, 1 times 2 is 2, and 1 times 3 is 3. 2 times 0 is 0, 2 times 1 is 2, 2 times 2 will be 0 again, and 2 times 3 will be 2 again. 3 times 0 is 0, 3 times 1 is 3, 3 times 2 is 2, and 3 times 3 is 1. Believe it or not, this all works out. And “times” doesn’t get to look nearly so weird as “plus” does.

And that’s all you need: a collection of things, an operation that looks a bit like addition, and an operation that looks even more vaguely like multiplication.

Now the controversy. How much does something have to look like multiplication? Some people insist that a ring has to have a multiplicative identity, something that works like 1. The ring I described has one, but one could imagine a ring that hasn’t, such as the even numbers and ordinary addition and multiplication. People who want rings to have multiplicative identity sometimes use “rng” to speak — well, write — of rings that haven’t.

Some people want rings to have multiplicative inverses. That is, anything except zero has something you can multiply it by to get 1. The little ring I built there hasn’t got one, because there’s nothing you can multiply 2 by to get 1. Some insist on multiplication commuting, that 2 times 3 equals 3 times 2.

Who’s right? It depends what you want to do. Everybody agrees that a ring has to have elements, and addition, and multiplication, and that the multiplication has to distribute across addition. The rest depends on the author, and the tradition the author works in. Mathematical constructs are things humans find interesting to study. The details of how they’re made will depend on what work we want to do.

If a mathematician wishes to make clear that she expects a ring to have multiplication that commutes and to have a multiplicative identity she can say so. She would write that something is a commutative ring with identity. Or the context may make things clear. If you’re not sure, then you can suppose she uses the definition of “ring” that was in the textbook from her Intro to Algebra class sophomore year.

It may seem strange to think that mathematicians don’t all agree on what a ring is. After all, don’t mathematicians deal in universal, eternal truths? … And they do; things that are proven by rigorous deduction are inarguably true. But the parts of these truths that are interesting are a matter of human judgement. We choose the bunches of ideas that are convenient to work with, and give names to those. That’s much of what makes this glossary an interesting project.

## What Is 13 Times 7?

AbyssBrain, author of the Mathemagical Site blog on WordPress, commented on that 2-plus-2-equals-5 post a couple days ago with a link to an Abbot and Costello Show sketch, in which Lou Costello proves to the landlord that 13 times 7 equals 28. And better than that, he does it three different ways. I didn’t want something fun as that to languish in the comments, so please, enjoy it here on the front page.

I have always liked comedy sketches about complicated chains of mock reasoning so this sort of thing is designed just for me.

## Reading the Comics, January 29, 2015: Returned Motifs Edition

I do occasionally worry that my little blog is going to become nothing but a review of mathematics-themed comic strips, especially when Comic Strip Master Command sends out abundant crops like it has the past few weeks. This week’s offerings bring out the return of a lot of familiar motifs, like fighting with word problems and anthropomorphized numbers; and there’s one strip that suggests a pair of articles I wrote a while back might be useful yet.

Bill Amend’s FoxTrot (January 25, and not a rerun) puts out a little word problem, about what grade one needs to get a B in this class, in the sort of passive-aggressive sniping teachers long to get away with. As Paige notes, it really isn’t a geometry problem, although I wonder if there’s a sensible way to represent it as a geometry problem.

Ruben Bolling’s Super-Fun-Pax Comix superstar Chaos Butterfly appears not just in the January 25th installment but also gets a passing mention in Mark Heath’sSpot the Frog (January 29, rerun). Chaos Butterfly in all its forms seems to be popping up a lot lately; I wonder if it’s something in the air.

## A Forest of 240 Factor Trees

I admit this is a little self-indulgent, but, what the heck. You might also like the factor trees for the extremely factorable number 240 that Ivasallay’s put up at Find The Factors.

• 240 is a composite number.
• Prime factorization: 240 = 2 x 2 x 2 x 2 x 3 x 5, which can be written (2^4) x 3 x 5
• The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 x 2 x 2 = 20. Therefore 240 has 20 factors.
• Factors of 240: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240
• Factor pairs: 240 = 1 x 240, 2 x 120, 3 x 80, 4 x 60, 5 x 48, 6 x 40, 8 x 30, 10 x 24, 12 x 20, or 15 x 16
• Taking the factor pair with the largest square number factor, we get √240 = (√16)(√15) = 4√15 ≈ 15.492.

Because 240 has so many factors, it is possible…

View original post 185 more words

## Combining Matrices And Model Universes

I would like to resume talking about matrices and really old universes and the way nucleosynthesis in these model universes causes atoms to keep settling down to peculiar but unchanging distribution.

I’d already described how a matrix offers a nice way to organize elements, and in ways that encode information about the context of the elements by where they’re placed. That’s useful and saves some writing, certainly, although by itself it’s not that interesting. Matrices start to get really powerful when, first, the elements being stored are things on which you can do something like arithmetic with pairs of them. Here I mostly just mean that you can add together two elements, or multiply them, and get back something meaningful.

This typically means that the matrix is made up of a grid of numbers, although that isn’t actually required, just, really common if we’re trying to do mathematics.

Then you get the ability to add together and multiply together the matrices themselves, turning pairs of matrices into some new matrix, and building something that works a lot like arithmetic on these matrices.

Adding one matrix to another is done in almost the obvious way: add the element in the first row, first column of the first matrix to the element in the first row, first column of the second matrix; that’s the first row, first column of your new matrix. Then add the element in the first row, second column of the first matrix to the element in the first row, second column of the second matrix; that’s the first row, second column of the new matrix. Add the element in the second row, first column of the first matrix to the element in the second row, first column of the second matrix, and put that in the second row, first column of the new matrix. And so on.

This means you can only add together two matrices that are the same size — the same number of rows and of columns — but that doesn’t seem unreasonable.

You can also do something called scalar multiplication of a matrix, in which you multiply every element in the matrix by the same number. A scalar is just a number that isn’t part of a matrix. This multiplication is useful, not least because it lets us talk about how to subtract one matrix from another: to find the difference of the first matrix and the second, scalar-multiply the second matrix by -1, and then add the first to that product. But you can do scalar multiplication by any number, by two or minus pi or by zero if you feel like it.

I should say something about notation. When we want to write out these kinds of operations efficiently, of course, we turn to symbols to represent the matrices. We can, in principle, use any symbols, but by convention a matrix usually gets represented with a capital letter, A or B or M or P or the like. So to add matrix A to matrix B, with the result being matrix C, we can write out the equation “A + B = C”, which is about as simple as we could hope to see. Scalars are normally written in lowercase letters, often Greek letters, if we don’t know what the number is, so that the scalar multiplication of the number r and the matrix A would be the product “rA”, and we could write the difference between matrix A and matrix B as “A + (-1)B” or “A – B”.

Matrix multiplication, now, that is done by a process that sounds like doubletalk, and it takes a while of practice to do it right. But there are good reasons for doing it that way and we’ll get to one of those reasons by the end of this essay.

To multiply matrix A and matrix B together, we do multiply various pairs of elements from both matrix A and matrix B. The surprising thing is that we also add together sets of these products, per this rule.

Take the element in the first row, first column of A, and multiply it by the element in the first row, first column of B. Add to that the product of the element in the first row, second column of A and the second row, first column of B. Add to that total the product of the element in the first row, third column of A and the third row, second column of B, and so on. When you’ve run out of columns of A and rows of B, this total is the first row, first column of the product of the matrices A and B.

Plenty of work. But we have more to do. Take the product of the element in the first row, first column of A and the element in the first row, second column of B. Add to that the product of the element in the first row, second column of A and the element in the second row, second column of B. Add to that the product of the element in the first row, third column of A and the element in the third row, second column of B. And keep adding those up until you’re out of columns of A and rows of B. This total is the first row, second column of the product of matrices A and B.

This does mean that you can multiply matrices of different sizes, provided the first one has as many columns as the second has rows. And the product may be a completely different size from the first or second matrices. It also means it might be possible to multiply matrices in one order but not the other: if matrix A has four rows and three columns, and matrix B has three rows and two columns, then you can multiply A by B, but not B by A.

My recollection on learning this process was that this was crazy, and the workload ridiculous, and I imagine people who get this in Algebra II, and don’t go on to using mathematics later on, remember the process as nothing more than an unpleasant blur of doing a lot of multiplying and addition for some reason or other.

So here is one of the reasons why we do it this way. Let me define two matrices:

$A = \left(\begin{tabular}{c c c} 3/4 & 0 & 2/5 \\ 1/4 & 3/5 & 2/5 \\ 0 & 2/5 & 1/5 \end{tabular}\right)$

$B = \left(\begin{tabular}{c} 100 \\ 0 \\ 0 \end{tabular}\right)$

Then matrix A times B is

$AB = \left(\begin{tabular}{c} 3/4 * 100 + 0 * 0 + 2/5 * 0 \\ 1/4 * 100 + 3/5 * 0 + 2/5 * 0 \\ 0 * 100 + 2/5 * 0 + 1/5 * 0 \end{tabular}\right) = \left(\begin{tabular}{c} 75 \\ 25 \\ 0 \end{tabular}\right)$

You’ve seen those numbers before, of course: the matrix A contains the probabilities I put in my first model universe to describe the chances that over the course of a billion years a hydrogen atom would stay hydrogen, or become iron, or become uranium, and so on. The matrix B contains the original distribution of atoms in the toy universe, 100 percent hydrogen and nothing anything else. And the product of A and B was exactly the distribution after that first billion years: 75 percent hydrogen, 25 percent iron, nothing uranium.

If we multiply the matrix A by that product again — well, you should expect we’re going to get the distribution of elements after two billion years, that is, 56.25 percent hydrogen, 33.75 percent iron, 10 percent uranium, but let me write it out anyway to show:

$\left(\begin{tabular}{c c c} 3/4 & 0 & 2/5 \\ 1/4 & 3/5 & 2/5 \\ 0 & 2/5 & 1/5 \end{tabular}\right)\left(\begin{tabular}{c} 75 \\ 25 \\ 0 \end{tabular}\right) = \left(\begin{tabular}{c} 3/4 * 75 + 0 * 25 + 2/5 * 0 \\ 1/4 * 75 + 3/5 * 25 + 2/5 * 0 \\ 0 * 75 + 2/5 * 25 + 1/5 * 0 \end{tabular}\right) = \left(\begin{tabular}{c} 56.25 \\ 33.75 \\ 10 \end{tabular}\right)$

And if you don’t know just what would happen if we multipled A by that product, you aren’t paying attention.

This also gives a reason why matrix multiplication is defined this way. The operation captures neatly the operation of making a new thing — in the toy universe case, hydrogen or iron or uranium — out of some combination of fractions of an old thing — again, the former distribution of hydrogen and iron and uranium.

Or here’s another reason. Since this matrix A has three rows and three columns, you can multiply it by itself and get a matrix of three rows and three columns out of it. That matrix — which we can write as A2 — then describes how two billion years of nucleosynthesis would change the distribution of elements in the toy universe. A times A times A would give three billion years of nucleosynthesis; A10 ten billion years. The actual calculating of the numbers in these matrices may be tedious, but it describes a complicated operation very efficiently, which we always want to do.

I should mention another bit of notation. We usually use capital letters to represent matrices; but, a matrix that’s just got one column is also called a vector. That’s often written with a lowercase letter, with a little arrow above the letter, as in $\vec{x}$, or in bold typeface, as in x. (The arrows are easier to put in writing, the bold easier when you were typing on typewriters.) But if you’re doing a lot of writing this out, and know that (say) x isn’t being used for anything but vectors, then even that arrow or boldface will be forgotten. Then we’d write the product of matrix A and vector x as just Ax.  (There are also cases where you put a little caret over the letter; that’s to denote that it’s a vector that’s one unit of length long.)

When you start writing vectors without an arrow or boldface you start to run the risk of confusing what symbols mean scalars and what ones mean vectors. That’s one of the reasons that Greek letters are popular for scalars. It’s also common to put scalars to the left and vectors to the right. So if one saw “rMx”, it would be expected that r is a scalar, M a matrix, and x a vector, and if they’re not then this should be explained in text nearby, preferably before the equations. (And of course if it’s work you’re doing, you should know going in what you mean the letters to represent.)

## Factor Finding

I imagine everyone in the world has seen this already, but, over on the Find The factors blog is a string of mathematics puzzles. The one to which I link amounts to writing out a multiplication table, where the rows and columns have been scrambled, and you have to work out which row is which based on the select handful of numbers in the table. That is, the first row might be the multiples of 6, the next row the multiples of 9, the next row the multiples of 4; and the first column the multiples of 4, the second column multiples of 5, the third column multiples of 2, and so on.

I think this is a fun exercise. It’s more challenging than the day’s Jumble (which this year has had a disturbing number of ones that can be solved on sight, without any unscrambling of words), without being so time-consuming as Sudoku, and if you’re trying to learn the times tables (which I admit probably few readers around here are trying to do) there’s a lot of chance to think about what the multiplication tables are to work out the puzzle. There’s a fresh puzzle every week, as well as a good number of tools for people learning multiplication.

## In Defense Of FOIL

I do sometimes read online forums of educators, particularly math educators, since it’s fun to have somewhere to talk shop, and the topics of conversation are constant enough you don’t have to spend much time getting the flavor of a particular group before participating. If you suppose the students are lazy, the administrators meddling, the community unsupportive, and the public irrationally terrified of mathematics you’ve covered most forum threads. I had no luck holding forth my view on one particular topic, though, so I’ll try fighting again here where I can easily squelch the opposition.

The argument, a subset of students-are-lazy (as they don’t wish to understand mathematics), was about a mnemonic technique called FOIL. It’s a tool to help people multiply binomials. Binomials are the sum (or difference) of two quantities, for example, (a + 2) or (b + 5). Here a and b are numbers whose value I don’t care about; I don’t care about the 2 or 5 either, but by picking specific values I avoid having too much abstraction in my paragraph. The product of (a + 2) with (b + 5) is the sum of all the pairs made by multiplying one term in the first binomial by one term in the second. There are four such pairs: a times b, and a times 5, and 2 times b, and 2 times 5. And therefore the product (a + 2) * (b + 5) will be a*b + a*5 + 2*b + 2*5. That would usually be cleaned up by writing 5*a instead of a*5, and by writing 10 instead of 2*5, so the sum would become a*b + 5*a + 2*b + 10.

FOIL is a way of making sure one has covered all the pairs. The letters stand for First, Outer, Inner, Last, and they mean: take the product of the First terms in each binomial, a and b; and those of the Outer terms, a and 5; and those of the Inner terms, 2 and b; and those of the Last terms, 2 and 5.

Here is my distinguished colleague’s objection to FOIL: Nobody needs it. This is true.