# Thread: Why a plus when multiplying minuses?

1. ## Why a plus when multiplying minuses?

Hi

I can understand (and see from the elevator anology) why two minus numbers are added together and still have a minus sign.
I know that to multiply two minus numbers gives a plus answer but I cannot understand or see why. Could someone point me in the right direction for an 'easy to understand' explanation to this please?

2. Originally Posted by Meggomumsie
Hi

I can understand (and see from the elevator anology) why two minus numbers are added together and still have a minus sign.
I know that to multiply two minus numbers gives a plus answer but I cannot understand or see why. Could someone point me in the right direction for an 'easy to understand' explanation to this please?

Think of the numbers along a number line - neg to the left, pos to the right.

Multiplying by -1 can be considered as being the operation that "reverses the direction" of the line, a bit like reflecting it in the zero point.

Let's work towards -3 x -2.

To start with, we'll consider 3 x 2. Easy enough, that's 6. You can, if you like, think of the number line as a piece of elastic, with a dot at the 3 point. Multiplying by 2 is like stretching the elastic to twice its length. The 3 ends up being where 6 was. Thus we illustrate 2x 3 = 6.

Now take 2 x -3. Same piece of elastic, this time the dot is at the -3 point. Stretch it to twice its length, but this time imagine it being pulled from the opposite end as well, so the negative numbers get stretched as well. The -3 ends up where the -6 was.

Now take -2 x 3. You start with the dot at the 3 point, but this time not only do you stretch the elastic to twice its length, you pull the right hand end with your left hand and the left hand end with your right hand. So the entire piece of elastic "swaps ends" before being stretched, and the 3, having been swapped with -3, then gets stretched so it now lies on the -6.

Now take -2 x -3. Same thing, except the dot is at the -3 point, and when it swaps ends, it ends up at 3 before being stretched to 6.

The idea of "reversing direction" is a useful one for imagining what "multiplication by a negative number" is, it's like "multiply by its positive value, then change its sign."

When you get onto some applications in which negative number multiplication is actually used for something real-world, it will start to make proper sense, believe me - till then it's all this picturesque imaginings about elastic.

Let me know if it doesn't make sense. Point out the first line where I've lost you if not, and I'll try and make it clearer. (Might not be tonight, it's almost 11 where I am and it's a work day tomorrow and I'm *tired*!)

All the best - keep the faith!

3. Thanks for the answer Simon. I made a number line and got some elastic, still not quite there. Are you saying that once the minus sign is encountered, it reverses the sign? Why does it do that? I can see that 2 * -3 is 2 lots of -3, and -3 * 2 ???? Oh!!! That's where I'm at, sorry, using my elastic didn't help yet.

Marilyn

4. Originally Posted by Meggomumsie
Thanks for the answer Simon. I made a number line and got some elastic, still not quite there. Are you saying that once the minus sign is encountered, it reverses the sign? Why does it do that? I can see that 2 * -3 is 2 lots of -3, and -3 * 2 ???? Oh!!! That's where I'm at, sorry, using my elastic didn't help yet.

Marilyn
Exactly it, when you have a minus sign in a multiplication (or division, for that matter), it reverses the sign. If you have two minus signs, it reverses it twice. (So minus times minus is plus).

At this stage, the best answer to "Why?" is "It just does, dunnit?" It's just how numbers behave. "Multiplying by -1" is the same as "changing the sign".

Take -2 x -3 = 6 again.

As -2 = -1 x 2, and -3 = -1 x 3, it follows that -2 x -3 = -1 x -1 x 2 x 3.

And so you have two -1's in the calculation: the first -1 changes the sign, and the second -1 changes the sign again, and so they cancel out and you get left with the 2 x 3 = 6.

BTW how did you know my middle name was Simon?

5. Perhaps it would help to think in terms of inequalities. 2< 3 but, multiplying both sides by -1 we get -2 and -3 and -3< -2. The direction has been reversed. Multiplying by -4, we go from 2< 3 to -12< -8. The "directed" segment from 2 to 3 has been stretched (by the "4") and reversed by the negative. Of course, reversing a second time puts us right back where we were before. Multiplying 2< 3 by -1 gives -3< -2 but doing it again gives 2< 3 again.

6. Sorry Matt, I wondered why your name was Simon when your user name is Matt. I must have had a psychic moment. Back to the maths - soooo,

+3*-2=-6 because I am looking at the the minus which is not changed by the plus

-3*+2=-6 because I am looking at the + which is changed by the minus

-3*-2=+6 because I am looking the minus (2) which is changed to a plus by the minus

Sooooooooo - when I am working on this type of question at this level, I think to myself - +x so the answer is whatever the next sign is plus or minus - OR - -x so the answer will be the opposite sign; plus to minus or minus to plus.

If this is correct, then I have something to 'see' when I approach the question.

7. Hello Marilyn
Originally Posted by Meggomumsie
Hi

I can understand (and see from the elevator anology) why two minus numbers are added together and still have a minus sign.
I know that to multiply two minus numbers gives a plus answer but I cannot understand or see why. Could someone point me in the right direction for an 'easy to understand' explanation to this please?

Think of (+3) as a cheque for £3 (or \$3 or whatever!), and (-3) as a bill (an invoice - a debt to be paid) for £3.

One morning the postman gives you 5 envelopes each containing a cheque for £3. So you're £15 better off than you were before the postman came. That's like saying (+5) x (+3) = (+15).

On another day, the postman brings 5 envelopes each containing a bill for £3. So you're £15 worse off than before he came. That's like saying (+5) x (-3) = (-15).

The postman comes a third time, saying there's been a mistake, and takes away 5 cheques for £3 each. You're £15 worse off now: that's (-5) x (+3) = (-15).

Finally a friendly postman comes, once again saying a mistake has been made, but this time takes away 5 envelopes each containing a bill for £3. So you're £15 better off than you were before he came: that's (-5) x (-3) = (+15).

Simple, really.