# Multiplying Non-Numbers

/In last last week's episode of PBS Infinite Series, we talked about different flavors of multiplication (like associativity and commutativity) to think about when multiplying things that *aren't *numbers. My examples of multiplying non-numbers were vectors and matrices, which come from the land of algebra. Today I'd like to highlight another example:

It's true! To illustrate, here's the multiplication table for a point, a line, a circle, and a square.

**product**$X\times Y$ is the shape whose vertical cross sections look like $Y$ and whose horizontal cross sections look like $X$! Also notice that the point acts like a multiplicative identity: a point times a shape is just the shape.

I'm really stretching my art skills today, but the *idea* is hopefully clear: A 'circle $\times$ line segment' is a vertical cylinder: its horizontal cross sections are circles and its vertical cross sections are lines.

But what about associativity? Let's compare '(point $\times$ line segment) $\times$ circle' with 'point $\times$ (line segment $\times$ circle).' According to the multiplication table:

*is*in general associative. By the way, this multiplication has a name. The product $X\times Y$ of two shapes is called the

**Cartesian product**of $X$ and $Y$. So if you watched last week's video, you've now heard of at least

*four*examples of multiplication* of non-numbers:

*shapes*will make an appearance in our next epsisode on Infinite Series! As we saw last week, loop concatenation is

*not*associative: the two ways of multiplying three loops are not equal. And in the next episode, we'll discover that there are

*infinitely*many ways of multiplying three loops. AND - it gets better -

*all of those ways are encoded in a particular shape*!

What shape, exactly?

You'll have to wait and see!

*binary operation*, which is a way to combine two things (the inputs) to get a third thing (the output). Addition and multiplication of numbers are examples of binary operations. In practice, mathematicians often use the word “addition” to describe a binary operation that is

*commutative*. And although I don’t mention it in the video, loop concatenation is not commutative! So we call it a “multiplication” instead.

*Thanks, John, for suggesting the idea for today's post!*