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"Minus times a minus is a plus"
By Anonymous on Thursday, October 28, 1999 - 09:57 am:

Does anyone have a good (non-abstract) description suitable for 12 year olds as to why a minus nuber multiplied by a minus number is a positive number?

It's one thing to say "thats the way it is", but if anyone has a real world description that would be very helpful.

Thanks
Tony Ibbotson.

By Arvan Pritchard (T708) on Thursday, October 28, 1999 - 11:13 am:

I think any general description is bound to be a little abstract... Here's my attempt:

I'd start with -1 times (whatever) [a positive number] is -(whatever). Some specific examples perhaps then a general statement [you could derive it formally from the distributive law].

This means you can visualise multiplying by -1 as reversing the number line.

Then split any negative number up as -1 times a positive number.

Shuffle all the -1s to the start and multiply all the positive numbers together.

Finally flip the number line as many times as you have to to get the sign of the answer right.

Negative times negative reverses twice so leaves the answer positive...

By Chris Jefferson (Caj30) on Sunday, October 31, 1999 - 09:22 am:

I don't think there are any real world examples. This kind of thing didn't really come up until people started trying to rigourously define maths, and this is because until then no-one had had reason to think about it....

This is quite abstract I will admit.. but any really inquisitive 12 year olds should take a look at it.

Try writing this out.. it looks nicer on paper...

First of all, see if they will accept that:
1+(-1)=0 by definition of (-1)
So, by taking -1 from both sides,
1+(-1)-(-1)=-(-1)
Now, for any X, X-X=0 by definition, this this condenses to

1=-(-1)
So 1/(-1)=(-1) <- Line 1

Then here is one way of working it out... It requires accepting that what algebra says must be true...

Now, for example why does (-2)×(-3)=6 ? Try writing the following out...

This is the same as why does (-3)=6/(-2)
But 6/(-2)=6/(2×(-1))=(6/2)×(1/-1)

and from line 1, this equals (6/2)×-1

=3×(-1)=-3, so we are done.

By Alastair George (T785) on Sunday, October 31, 1999 - 08:13 pm:

One approach I use -and I admit its not at all mathematical - is to appeal to their "common sense"...
First, they must be completely happy that
5 x -3 = -15
Then, I say, let's think about
-5 x -3
What can this be?
Let them wrestle with the idea that it can't have the same value as the first sum.
"Therefore" it "must" be 15.
(A few holes here I admit...)

By B.A. Lee (T153) on Tuesday, November 2, 1999 - 07:36 am:

Extending Alastair's idea:

5 x -3 = -15
4 x -3 = -12
3 x -3 = -9
2 x -3 = -6
1 x -3 = -3
0 x -3 = 0
-1 x -3 = 3
...

This should be worth a try.

A more complete picture may be to construct a big multiplication table with 0 in the middle, positive integers running up from 0 and to the right of 0, and negative numbers in the opposite directions (just like in a Cartesian plane). The product of the two "coordinates" can be written at the corresponding point in the table.

The first, second and fourth quadrants need little explanation. The third quadrant, which addresses this problem, has a "natural" way of being filled.

By David Sanders on Wednesday, December 22, 1999 - 12:53 pm:

Hi Tony,

There are two real world applications that I can think of where minus numbers become more 'natural'. The first is thermometers, and the second is banks.

Thermometers:
We know that it can get colder and it can get hotter, and we have (by the age of 12, I think?) got used to the idea that we can associate numbers to these changes of temperature: if it gets hotter then the temperature goes up, i.e. increases; if it gets colder, then the temperature goes down. (This is a somewhat arbirary choice! It could have been the other way round.)

So if it gets sufficiently cold, then the temperature is going to get closer and closer to our (again arbitrary) zero value.

But we know that in Siberia, (or Scotland, or wherever!) or even in outer space (in some sense), it's even colder still. We still want to assign a number to the coldness so that we can compare coldnesses ("it's colder today than it was yesterday"). The 'obvious' thing to do is to call it "one below zero" (Erdos thought this was obvious at the age of 3 or 4!!)

This is again an arbitrary name -- we could have called it squiggle or something, but for ease and simplicity, "one below zero" works well.

So we say that if the temperature drops by one "unit" from 1, we reach 0, and if it drops by another unit then we reach "one below zero", which we write as "-1", where the minus sign means "the number which is the following amount below zero".

Now we say that when it gets hotter, the temperature number goes up, so we must ADD to it, and when it gets colder, we must SUBTRACT from it, since the number goes down.

Now we find a problem: unfortunately, the normal mathematical symbol for subtraction is THE SAME as the symbol for "the amount below zero". This is not in fact a surprise, because we will find that
0 -- 1 equals -1. Here I have distinguished the mathematical operation for subtraction by denoting it "--" instead of "-". When I was younger (12ish, I guess), I used a minus sign raised up a bit for "negative" and a normal minus sign in the middle of the lign for subtraction, so that you can see the difference.

So 0 -- 1 is DEFINED to be -1, really: 0 -- 1 means (by definition of subtraction) "the temperature which is one unit colder than 0", which we have CALLED (by definition of -) "-1".

(Here I'm assuming either Celsius or Fahrenheit temperature scales: with the Kelvin scale, things become a bit more complicated, since normally you can't go below 'absolute zero', although 'negative temperatures' do still exist in a sense.)


So we have introduced two things:

a 'binary operator', --, which takes TWO numbers and gives one number as a result, and

a 'unary operator', -, which takes ONE number and gives one number as a result.

Clearly, these are two DIFFERENT things, which is not made clear by the normal notation. They are related by -a = 0 -- a.


The same ideas apply to your supply of money, but it's a bit more complicated, because you have to consider how much money you REALLY have, rather than how much you seem to have:

Suppose you have one pound, which belongs to you. If one of your parents gives you another pound to go and buy a loaf of bread and a pint of milk, but you instead buy one pound's worth of sweets, then you end up with one pound (i.e. 2 -- 1) in your hand (or your pocket), but it doesn't really belong to you -- really you should give it back to your parent.

Similarly, if you now start with ZERO pounds, and you get one pound from your parent and spend it on sweets instead of bread, then you still OWE your parent a pound, but a pound which you don't have! So if you then get a pound by doing a paper round, it won't really be YOUR pound, but it belongs to your parent. So AFTER your paper round, you (your bank account, if you like) have 0 pounds; thus before your paper round, you must have had "one below zero" pounds, or -1 pounds, in our notation from earlier.

So now, we can consider doing things repeatedly:
Say that every day you have a school dinner that costs 1 pound. If your parent gives you 5 pounds at the start of the week, then after the first day, you have 5 -- 1 = 4 pounds, etc. until after the 5th day you have 5 -- (5 times 1) = 0

If one day as well you buy sweets, then by Thursday evening you've got zero pounds, so you have to BORROW some from your parents which you have to repay after your Sunday paper round, so you have -1 pounds.

But you can also think (and this is the hard bit) of your parent giving you a pound because you've spent too much, as them giving you a debt to be repaid, i.e. as them giving you -1 pounds. And if they have to do it twice that week, then 'clearly' you will have 2 * (-1) = -2 pounds --- that is the definition of *, i.e. we repeat the 'operation' that many TIMES.

Now let's suppose that you didn't realise you had to pay the extra money back. Then you thought you were being given one pound (+1), where actually you were being given a debt of one pound (-1). Thus when they told you that you had to pay it back, you had to do the operation '-' (negate -- NOT subtract) to go from 1 to -1.

We then have to convince ourselves, using these interpretations, that -(3*5), or whatever, equals (-3)*5, and then (the very hard bit) that
(-3)*(-5) = -(3*(-5))
= -((-5)*(3)) = (-(-5))*3 = 5 * 3.

I've got to go now, but I have a funny feeling that I've missed out the most important bit of the argument -- how to justify these last steps! But at least I've given an interpretation of negative numbers.

You should also look at the book
"Mathematician's Delight", by Warwick Sawyer, published by Penguin, where he develops these ideas in a similar(ish) way.


David Sanders.