Let me lead with a provocation. One times two is meaningless.

Yep, this thing: \(1\cdot 2\), is not a thing.

Obviously, if that's true, it's only true in some sense that I'll soon explain. But before doing that, let me tell you where this is going. Consider \(1\cdot 2\) is only a very simple setting to discuss a certain concept, generalization. Eventually everyone struggles with some instance of generalization. Common struggles include:

• 0! = 1 ("zero factorial")
• \(2^1 = 2\)
• \(2^0 = 1\)
• \(x^{-1} = \frac 1 x\)
• \(x^{1/2}=\sqrt x\)
• \(\sqrt{-1}=i\)

In the same way that \(1\cdot 2\) is meaningless, each expression on the left of the equal signs above are meaningless. Again, this can only be in some sense. Because in a much more usual sense, we use these equations every day in mathematics such that their truth is regarded as boring.

The point of this sequence of blog posts is to resolve this paradox.

The paradox

Me: What does multiplication even mean?

You: Repeated addition.

Me: Right, great!
So you agree that \(1\cdot 2\) is meaningless. Because after all, by your own definition, \(1\cdot 2\) would have to mean adding two to itself one time. But addition is intrinsically a binary operation! You cannot have 2+. It has to be 2+(something).

You: Well yeah, but ... like ... \(1\cdot 2\) is just a 2 one time.

Me: So it's not repeated addition, it's just ... repetition?

Well, clearly there's a way to say this more precisely to satisfy everyone. Multiplication is not repeated addition. Rather, it is making repeated copies of a number, and counting up how many things the copies represent. \(4\cdot 2\) is making four copies of 2, so that it means how much is counted up by the list [2,2,2,2]. In cases like this it just means adding all the quantities.

But if you have just one copy, then we think of \(1\cdot 2\) as the length-one list [2] so that counting what this represents requires no addition.

The end

Great, we resolved the paradox, time to go home.

But notice that the resolution to the paradox was to relinquish a very cherished definition. We had to amend "repeated addition" in a gentle way. So gently that we don't mind the modification, because it preserves all the "usual" cases and also generalizes it to further cases which bump up again the limits that the definition seems able to handle.

This is an important case study, and we will try to reproduce this success when we discuss all the other equations listed above.