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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?

Blue
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Mohd Saad
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  • https://math.stackexchange.com/a/4195982 may hold some wisdom for you. – Arthur Jul 18 '21 at 11:16
  • Also: why is adding a negative number of times the same as subtraction? Then you can go on to complex numbers... – GEdgar Jul 18 '21 at 11:29
  • How is it related to complex numbers? – Mohd Saad Jul 18 '21 at 11:30
  • @Arthur the explanation helped – Mohd Saad Jul 18 '21 at 11:33
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    "Multiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not. Multiplication of natural numbers certainly gives the same result as repeated addition, but that does not make it the same." https://www.maa.org/external_archive/devlin/devlin_06_08.html also https://www.maa.org/external_archive/devlin/devlin_0708_08.html – Gerry Myerson Jul 18 '21 at 13:16

2 Answers2

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

user1337
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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.

Arthur
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    To be philosophical for a minute, I see this as somewhat parallel to Plato's theory of forms. In that the concrete form of a thing can never encapsulate the true nature of that thing. You can recognize a dog basically any time you see one, and just from that have a decent idea of how it will behave, but there is no thing in this world that embodies perfect "dog-ness". In the same way, a concrete interpretation of multiplication cannot contain the true essence of what multiplication is. It is just another manifestation of the more abstract idea of multiplication. – Arthur Jul 18 '21 at 11:43