Tuesday, April 03, 2007

(-3) x (-5) = ?

If you're wondering where I'm getting these random posts from, then please don't worry- all will be revealed soon! Obviously I can't say when, because then I will be putting unwanted pressure on myself, which might result in bad things happening. And I can't say what bad things, because otherwise I will have to actually tell you when soon is! (Use your imagination :D).

It's ok if you didn't understand that mumbo jumbo. BTW you've probably realised that I like that word a lot. It's a handy way of describing things which make no sense whatsoever. Sweet. Oops, getting carried away again.

So my question is what does (-3) x (-5) equal?

My natural response, and I hope that you would agree with me here(!), would be to say that it equals (+)15. But have you ever wondered why a negative number times another negative equals a positive one?

When I was small (read erm... secondary school sometime!) I couldn't subtract and add with negative numbers properly. Actually what I precisely couldn't do was sums like -3 - -5. Obviously now I know that is the same as -3+5 which equals 2, but I really struggled in school. For the other questions like 3 - 8, 8-19 etc. I had to always draw a silly number line and well if we started at 3, I'd count how many steps it'd take for me to get to -8. Very tedious as you can imagine! I think the biggest problem that I had was that I couldn't do these sums in my head. Hence, when we used to have 'mental maths' tests I didn't used to do too well! Maybe that's why I started loving my calculator! Not only did it not require me to draw silly number lines everywhere, but it always
* gave me the correct answer.

I have still kept my primary school books, secondary school and college stuff as well. As I was writing this post I was assuming a lot of things (i.e. those things which I remembered vaguely!). This made me go and find my primary and secondary school maths stuff. It seems that I hadn't done 'negative numbers' in primary school. (
However I'm pretty sure that nowadays they're taught negative numbers in primary school (or is that my imagination?).). But I was/am shocked to read that I could do this in year 5:
I was testing the teacher BTW ;). I can't locate my year 7 book, but it seems we started learning about 'directed numbers' in year 8. However the weird thing is that I can't seem to remember having a problem with them then! I know that in year 9 they caused me 'great' distress but in year 8 it seems that I was happy with them. I have a reason for this which I shall mention in a minute, but looking through my year 9 book you'll find this page.
That makes me think that no wonder I probably struggled- reading it now makes me go 'eh'! OK, I could be deliberately being mean, but you see as I mentioned previously the only time that I've ever hated maths was in year 9 due to the teacher. Maybe that caused me to not understand this topic. You see the teacher that I had in year 8 was the one that I had had again in year 10- the one and the only, who is the reason behind me doing maths. Now I loved my maths lessons in year 8, probably because of the teacher- that is why I seem to not recall having a problem with 'directed numbers'!

Don't worry I'm going somewhere with this! Whenever I used to get stuck in any homework or school work the first person I used to go to was my Dad. So naturally I went to my Dad. Now he taught me how to do addition and subtraction of 'negative' numbers without drawing them number lines. Money was the example he used. Now I can't remember whether he explained to me about the why -3 --5 = -3+5, so I'll assume he didn't. But the example about money did the trick, after this you could say I understood addition and subtraction of negative numbers and was able to do them in my head. (no number lines and calc. needed!).

Now what's addition and subtraction got to do with multiplication, seems the natural question to ask. Well I had to set the scene you see! So onto the initial question. I never had a problem with the fact that + x - = - and - x - =+. All you had to remember was that when dividing or multiplying numbers with different signs, the minus won. No questions asked. 'It's convention' deal with it. I don't think I ever questioned this (I can't remember if I did), but it was what you did. Why I had no idea- well until today that is.

Now this is the bit where I get stuck. Today I read this somewhere (where will be revealed later :D ),

'Think of numbers as representing money in the bank, with the positive numbers being money you possess and negative numbers being debts to the bank. Thus -5 is a debt of £5, so 3 x (-5) is three debts of £5, which clearly amount to a total debt of $15. So 3 x (-5)= -15 and no one seems bothered much about that. But what of (-3) x (-15)? This is what you get when the bank forgives 3 debts of £5. If it does that, you gain £15. So (-3) x (-5)= 15.'

The bit towards the end is what doesn't make sense to me. So because it's -3 the bank is now forgiving you? I don't know. This is not such a big deal, but I tend to use the money example myself when asked about negative numbers, however I've never had to use it to explain multiplication. How else would you phrase that?

After reading this, I finally actually questioned this convention. Why is the answer 15! The question has come a bit late I suppose but it at least came. Funnily or worryingly it has caused me to wonder about why we do other things as well! So if you didn't follow the money explanation here's one which might seem more appealing. Well I like it and I think I'll try to say this in my own words. This calculation has an external and internal reason behind it. The external one was obviously the bank calculation so that leaves the internal one.

'The internal explanation is to work out a sum like (-3) x (5-5)' (oops)

Now obviously this equals zero. But we can also re-write it as (-3) x (5) + (-3) x(-5). From the money example we know that (-3) x (5) = -15 so we have -15 + (-3) x(-5) =0, which implies that (-3) x (-5) = 15! If only I could draw a neat small black square! Ok, that's just me being excited about this. I feel pretty happy upon reading about this, because it makes much more sense about why (-3) x (-5) =15. It's trivial I suppose, but why isn't it taught like this in schools? I'm sure that there is an equally good way of showing that -3 - -5 =2!

It seems that I have a habit of not understanding simple things. Any of you guys still got your old school things? I'm going to see what else I can dig up- hopefully nothing embarrassing, but the stuff brings back good memories. (well some not too good as well, but I can forget about them I suppose).

... and they all lived happily ever after! ;)


KTC said...

You either going to love this or hate this.

We define / let integer be a Group. (There's no better reason than because we want it so. :D)

One property of group is that it has a (unique) inverse element namely (-a) for the element (a) in the case of integer.
a + (-a) = (-a) + a = 0
i.e. the inverse of (a) is (-a), and the inverse of (-a) is (a).

The first '-' in (-(-a)) means we take the inverse of the next bit, and we knows that the inverse of (-a) is (a), so (--a) is (a).

(Okay, I tried and failed miserably, but at least I tried :D)

tdstephens3 said...

thanks guys, this is great!! I am beginning to see some interesting generalities regarding functions expressed as vector valued or scalar valued... ( I call these 'generalities' because they seem to open up an expression into something more profound ) It seems that, along the lines of algebraic factors, scalar objects can be represented by the inner vector product of two vector objects. I noticed this with partial derivatives being expressed as scalar functions and directional derivatives being expressed as the inner product of two vector valued functions - the first simply having the components as the partial derivatives applicable and the other being a vector oriented in the direction the derivative is desired to represent. (Now, I am not sure how the relationship is exactly defined between the domains of the vector functions in question...) I think the above comment concerning Groups points out exactly why this is so and it is beautiful and inspiring and this is why we study math!
BUT of course this all could have been mentioned before hand, explicitly, lest someone miss this along the way and end up a lousy mathematician!!
I am thinking that this suggests some things about the structure of the real number field in relation to the complex number field...
Linear Algebra is supposed to teach us this stuff, what is happening, all I am doing in that class is Gauss-Jordan elimination! (not that those two guys werent amazing, but a few uncooperative numbers in a matrix makes the whole subject seem UGLY!

beans said...

Actually you didn't fail miserably! That made a lot of sense- wow I liked it! :o

I read the words vectors, partial derivates, vectors again and my mind went haywire! Erm, is the inner product the same as the dot-product, because I've got a feeling it isn't!

Understandably we aren't told this is secondary school, but it'd be nice if it was taught in college (I think FP3 had something about groups but I didn't do that module).

haha, I know what you mean about Linear Algebra, although I'd do Gauss-Jordan elimation anyday rather than subspaces, spanning sets etc!! It's interesting but can be tedious I suppose. :/

Once again, thanks KTC, sweet post. :)

KTC said...

'Dot product' \equiv '(Standard) Euclidean inner product' \in 'Inner product space'