The Multiplication Multiverse

Let's say you want to multiply three or more numbers together.

Well, where you put the parentheses doesn't matter.

This property is called associativity.

But what happens if you multiply things that aren't numbers?

And what happens if that multiplication is not associative?

9 00:00:24,710 --> 00:00:25,370 Hi everyone.

My name is Tai Danae, and I'm one of your new co-hosts here on "Infinite Series."

The next two episodes, I'm going to share with you different rules of multiplication and the rich implications in mathematics.

And afterwards, physicist Gabe Perez-Giz will come on board to talk about another exciting topic.

Now, back to this factor of arithmetic.

Multiplication of numbers is associative.

Let's look at an example.

Suppose we want to multiply 2 times 3 times 5.

Well, we can multiply 2 and 3 to get 6, then multiply by 5, or we can first multiply 3 and 5 to get 15, and then multiply by 2.

Either way, the answer is 30.

And so the two ways of multiplying three numbers are the same.

But in mathematics, we can make sense of multiplication between things that aren't numbers.

In that context, we may not have associativity.

For example, if you've heard about vectors, then you might know that the cross-product in three dimensional space is not associative.

In fact, it satisfies something called the Jacobian Identity.

Notice it almost looks associative, except there's that extra term.

And since we're talking about associativity, you might wonder about that other property of real numbers-- you know, when multiplying two numbers, swapping the order doesn't change the answer.

For example, 2 times 4 is name as 4 times 2.

This property is called commutativity.

Now you might be thinking, why would we give a name to such an obvious fact?

But keep in mind, it's a very special property to have.

Not everything in life is commutative.

For example, getting dressed in the morning, because putting on your socks and then your shoes is not the same as first putting on your shoes and then your socks.

Similarly, matrix multiplication is in general not commutative.

If you multiply two matrices-- two arrays of numbers-- the order in which you multiply them matters.

In fact, the cross-product of vectors is also not commutative.

It's what we call anti-commutative.

a cross b is equal to minus b cross a.

Now, here's what's cool.

Because the cross-product is anti-commutative and it satisfies the Jacobi identity, three dimensional vectors with the cross-product form what's called a Lie algebra named after Norwegian mathematician Sophus Lie.

And even better, anytime you have a set of things with a multiplication that satisfies those two properties, it's anti-commutative and it satisfies the Jacobi identity, you have a Lie algebra.

OK, technically that set should be a vector space, but let's not worry about that for now.

So, we've already seen three types of multiplication.

If your product satisfies a certain equation or equations, we say oh, that multiplication's associative or it's commutative or it satisfies the Jacobi identity.

And there is a panoply of other equations your multiplication might satisfy.

And depending on which equations you have, you say ah, my set of things forms an associative algebra or a Lie algebra or Poisson algebra.

And these algebraic structures show up all over math and physics.

Now, I'd like us to take a step back and think about associativity again.

We talked about two examples.

Multiplication of real numbers is associative, the cross-product of vectors is not.

Well, I'd like to tell you about a third example that maybe you haven't heard of.

This example comes from topology, a branch of math that studies shapes.

In particular, topology gives us a way to multiply loops.

Now, a loop is just what you think it is.

It's a loop.

Mathematically, we say it's a continuous function from the interval of length 1 into a topological space X-- some shape, like a donut-- that sends 0 and 1 to the same point.

Here's one way to visualize it.

You can think of this interval as a stopwatch.

And as time goes from 0 seconds to 1 second, you can imagine a car tracing some path in the space.

At 0 seconds, the car starts at some point.

Then it travels around the space somewhere.

And at 1 second, it goes back to where it started.

That's a loop.

Now, the space X for the loop lives isn't really important for us.

It could be a torus, or it could be a sphere or could be a plane or just the blob.

It doesn't matter.

So let's not worry about what it looks like, and let's just focus on the loop itself.

And remember, a loop is just a path where the starting and ending points are the same.

So, this is a perfectly good example of a loop.

And so is this.

Notice, it's totally fine if a car goes to the starting point during its travels, as long as it ends there too.

In particular, if the car goes nowhere, that is a bonafide loop.

The starting and ending points are still the same.

Now, suppose we have two loops.

Let's call them a and b, and let's say they both start and stop at the same point in space.

Is there a way to think of this wedge as a single loop that we can call a times b?

In other words, can we come up with instructions that will tell us how to go around both a and b within-- and here's the key-- a 1-second time interval?

Can we do it?

Absolutely.

Here it is-- just have the red car go around a in the first half second.

Then have the blue car go around b in the next half second.

Now to do this, each car must travel at twice their original speed, but that's fine.

So, this wedge of two loops can really be thought of as a single loop.

Why?

Because one, we have instructions for traversing the whole path within one second.

And two, the starting and ending points are the same.

And this gives us a multiplication.

And what are the multiplication instructions?

Well, you split the time interval in half.

On the first half second, go around the first loop.

And on the next half second, go around the second loop.

And the result is a bonafide third loop.

This multiplication or product of loops is called loop concatenation.

But here's the question-- is it associative?

Well, suppose we have three loops-- a, b, and c. We would like to compare ab times c with a times bc.

Well, let's think about ab times c. According to the multiplication instructions, this is the loop where we go around ab in the first half second, then go around c in the last half second.

But what does going around the product ab in half a second look like?

Well, according to the same instructions, we start by splitting the half second in half, then go around a on the first half of that, and go around b on the second half.

In other words, in the loop ab times c, the red car travels in the first quarter of a second, so at 4 times its original speed.

Then the blue car travels in the next quarter of a second, also at 4 times its original speed.

And then the yellow car travels in the last half second, so only at twice its original speed.

Now, what about a times bc?

Well here, the roles are slightly reversed.

The red car goes a little slower now.

It travels in the first half second, while the blue and yellow cars go around b and c in the last two quarters.

So, are these two loops equal?

Well, the pictures look the same.

But the time in which each car does its traveling is different.

In short, their formulas are not equal.

So no, they're not the same.

Therefore, loop concatenation is an example of a product that is not associative.

Now, not having associativity is not a bad thing.

In fact, because associativity fails, we get some incredibly rich mathematics with implications in topology, algebra, geometry, and mathematical physics.

What exactly?

We'll that all of the different ways of multiplying 100 loops in a topological space can be encoded in a 98-dimensional polyhedron called an associahedron.

And that's what we'll talk about in our next episode.