What does it mean to add something half times?

Question

I want to ask what does to mean to add something "half times". Because, if we multiply fractions like $16 \times \frac12$, we are essentially adding $16$ one-half times, but it doesn't make any sense.

So is there any way to explain the multiplication do fractions like this?

Answer 1

Here is some wisdom that you might not think answers your question. But I can assure you it does, in its own way:

It is unhelpful to try to understand what multiplication is. Instead, try to focus on how multiplication behaves.

What do I mean? Well, repeated addition isn't the only way to think about multiplication. Here is another: take the number line, hold $0$ in place, and stretch the line until $1$ moves to where $2$ was. Where did $3$ go? That's $2\cdot3$.

There are dozens of other ways to think of multiplication. Who is to say which one is the "true" multiplication? No one. What unifies them? The laws that they obey. Laws like $xy=yx$, and $1x=x$, and $(x+y)z=xz+yz$, and $x(yz)=(xy)z$, and so on.

These laws are incredibly useful. They make it so that every single time you encounter something that obeys these laws, you know that you have encountered another incarnation of multiplication, and you can apply everything you already know about multiplication to this new thing.

So the way we define multiplication on fractions and on negative numbers is as follows:

1. Start with multiplication on the natural numbers
2. Define the product of two numbers from this new domain to be whatever it has to be to agree with multiplication on the natural numbers, and at the same time obey the laws

That's it. No need for interpretations. No fuss about what multiplication is. Because it is so many different things. But they all obey the laws.

This process is one example of what mathematicians call generalisations. And some times it is not possible to do. Some times you have to let go of a law or two to make things work. Deciding what laws need to be cut isn't always an easy task. And you certainly have to let go of interpretations (although I personally like to play with the interpretations some times and see whether they can be forced to make sense in some absurd way). Some times there are multiple ways to do it, and you have to add new laws that this new domain must obey to ensure that you get a unique result.

Coincidentally, the number line scaling above is a multiplication interpretation that works without modification for multiplication by any positive real number. And for negative numbers, you only need to include the ability to "flip" the number line around.

Answer 2

The operation "adding half-times" can be defined implicitly, by requiring that repeating it twice is the usual addition.