## Friday, November 02, 2007

### Multiples and Factors

Hmmmmmm...having sorted out number places I move on to multiples and factors. Obviously I have been multiplying and dividing stuff for years but I had no idea what these things were called.

I do hope that Maths gets a bit harder than this.

Now just in case you, like me, stopped paying attention at school in about year 9, I should point out that, for example the multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96 etc, etc.

Whereas the factors of 8 are only 1, 2, 4 and 8.

So the number of multiples for any given number is infinite whereas the number of factors is finite.

All of which made me wonder (mostly because working out multiples and factors is not very stimulating) what about infinity. Surely since for 'a' to be a factor of 'b' then 'b' has to be a multiple of 'a' (note that 8 is a multiple of 1, 2, 4 and 8) does it mean therefore that infinity is the only number that has an infinite number of factors and a finite number of multiples (ie 1 - infinity). Can you even divide by infinity? If so the answer is probably infinity in which case is infinity some sort of odd (I mean that in the sense of weird) prime number?

Maybe the answer to all these questions is that infinity isn't a proper number because mathematicians are always doing silly stuff like that so that normal people can't understand what they do.

Still these and other questions lay heavy on my mind. Probably because I am a humanities graduate who doesn't really understand mathematics.

Answers on a postcard (or comments box) from anyone with at least an A-level.

#### 6 comments:

Axel Fendersson said...

No, infinity is not prime (even oddly prime).

Dividing a finite number by infinity gives 0. As to multiples of infinity, yes, all multiples of infinity are infinite, but that does not necessarily mean that they are equal. Alas, dealing with infinities that are not equal requires going into fairly high-level number theory, which, alas, was not covered by my A-levels.

For the most part, you're right: it's best to think of infinity as not being a number, or at least not a conventional one.

Adrian said...

Infinity isn't so much a number, as a concept.

You can divide it by whatever you like and you'll still get infinity. So I'm not sure if that counts.

All this talk of infinity has me thinking I might have a book you'd like on this and other semi-related subjects.

{btw. we beat Highfield rifle club today!}

Jon said...

oooh I'm probably qualified to answer this!

I'm sure you know the boring predictable answer to your questions is "Help! Stop him! He's not allowed to do that!" (Turns out, just allowing the possibility of infinity without changing anything else doesn't give you a valid "ring". But you won't learn that until the second year of a maths degree.)

But I guess we can fudge it a little. Trouble is, the definitions you're being taught are simplified to fit the number system you're working in, so to deal with the concept of infinity you need to introduce some more general definitions. For example a prime number is one which if it divides "a times b", then it always divides at least one of "a" and "b".

If infinity divides a product of 2 numbers, then one of those numbers has gotta be infinite so infinity is prime(ish)! Your thing about infinite factors and finite multiples is true too, though as I've mentioned, we're working in a broken number system where some of the normal rules aren't going to work (I can give you examples if you want).

About dividing by infinity. Technically you can't, but almost everyone will let you anyway. If you say to "divide 'a' by infinity", that's maths slang for "divide 'a' by a number, then divide 'a' by a big number, then divide 'a' by a massive number, then make an educated guess about what would happen if you kept on going until you were dividing by something infinitely big". The answer is usually zero.

I'm sure you have more questions than when I started...

Richard Beer said...

BBC 4 did an interesting program on the concept of infinity in the "science you can't see" and I'd recommend watching that. Turns out there are lots of different infinities or different sizes and it all gets quite confusing. I think the mathmatician looking at it also ended up going mad.

Karuna said...

Yes... if you "divide" by infinity, you get zero.

There is a hierarchy of infinities; in fact, there are an infinite number of infinities, each one bigger than the last. Or so I read.

I like infinity.

Thomas said...

I'm going to screw with your mind even more. Because I can.
There is an infinitely large hotel. One day, a bus turns up, full of an infinite number of people. You allocate each person to a room, and your rooms are now full, as you have infinite people occupying infinite rooms. Then, alas, another bus full of an infinite number of people turns up looking for rooms. Can you get them in?
Of course you can! If you move all the people already in the hotel into odd-numbered rooms - so that the guy in room 2 goes into room 3, the guy in room 3 goes into room 5 and so on - you now have all the even numbered rooms free to put the new arrivals in! And there's an infinite number of even-numbered rooms!