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Multiplication of Two Negative Numbers

Date: 06/28/2001 at 13:36:35
From: Daniel Alexandre
Subject: Multiplication of two negative numbers

I was wondering about the history of numbers and possible arguments 
against the birth of a new concept of number. There is always 
rejection at first; that was the case for irrationals, complexes, and 
so on. So I was thinking also, to what extent do the negative numbers 
we use deserve indeed to be called numbers?

I thought of this problem:

Multiplication of A by B is: A + .... + A , with B parcels of A.
Now if A = -1 and B = -1 :
    A * B = A + .... + A , with B parcels of A
<=> (-1) * (-1) = (-1) + .... + (-1) , with (-1) parcels of (-1).

But what does this last expression (on the right) mean? To be 
meaningful, the expression (-1) * (-1) = (+1), as we know it is.
So... is (-1) really a number?
Does this multiplication have anything to do with i * i = -1 ?


Date: 06/28/2001 at 15:20:07
From: Doctor Ian
Subject: Re: Multiplication of two negative numbers

Hi Daniel,

These are excellent questions that you're asking, and they all have
basically the same answer, which is that new types of numbers are 
created by asking questions that seem to have the right form, but no 
clear answer... until someone makes one up. 

The constraints on making sense out of something that used to make no 
sense are pretty tight: everything that used to work has to keep 
working. For example, when you introduce negative numbers, all the 
rules for positive numbers have to keep working; when you introduce 
rationals, all the rules for integers have to keep working; when you 
introduce irrationals... well, you get the idea. 

But how do we come up with something like negative numbers? At first, 
it would seem that while something like

  7 - 5 = 2

makes perfect sense, something like

  5 - 7 = ?

makes no sense at all. After all, if you have only 5 things, how can 
you take 7 things away from them?  

The answer is to create the mathematical equivalent of credit. How can  
you spend $200 when you have only $100 in the bank? By keeping track 
of what you've spent, so you can pay it back later. In a sense, that's 
what we're doing when we do this:

  5 - 7 = -2

Consider the square root function. It makes perfect sense to say that

  3^2 = 9, so sqrt(9) = 3

But what do we do with this?

  sqrt(7) = ?

There is no integer or fraction that we can put on the right side to 
make this true. What we do is use the square root sign (or, more 
generally, the non-integer exponent) as another kind of credit - a 
promise that the number on the right side does exist, even though we 
can't represent it in any of the ways that we're used to. 

Suppose I ask you to point to the inside of a brick. You can't do it.  
If you break the brick in half, you're just showing me the outside of 
some smaller bricks. But we can both agree that there _is_ an inside, 
even if we can't ever look directly at it. Square roots work in sort 
of the same way.

So, eventually we get to the point where we're comfortable writing 
stuff like 

  blah blah blah = sqrt(2) 

But we're not comfortable with 

  blah blah blah = sqrt(-2)

because we can't imagine any number that we can multiply by itself to 
get a negative number... at least, not at first. The answer, as you 
know, is to add a second 'dimension' to the number system.

Now let's get back to multiplying negative numbers. We can get 
comfortable with multiplying two positive numbers. Once we understand 
that negative numbers represent a kind of 'credit', we can even get 
comfortable multiplying a negative number by a positive number. (If I 
put two items on my charge card, I owe twice as much as if I'd bought 
only one of them.)

But what's the deal with a negative number times a negative number?  
That can't possibly make sense, can it? 

Well, if we can agree that a negative number is just a positive number
multiplied by -1, then we can always write the product of two negative
numbers this way:

  (-a)(-b) = (-1)(a)(-1)(b) = (-1)(-1)ab
For example,

  -2 * -3 = (-1)(2)(-1)(3)
          = (-1)(-1)(2)(3)

          = (-1)(-1) * 6
So the real question is,

  (-1)(-1) = ?
and - as you know - the answer is that the following convention has 
been adopted:

  (-1)(-1) = +1               

This convention has been adopted for the simple reason that any other
convention would cause something to break. 

For example, if we adopted the convention that (-1)(-1) = -1, the
distributive property of multiplication wouldn't work for negative 

  (-1)(1 + -1) = (-1)(1) + (-1)(-1)
       (-1)(0) = -1 + -1

             0 = -2
As Sherlock Holmes observed, "When you have excluded the impossible,
whatever remains, however improbable, must be the truth."

Since everything except +1 can be excluded as impossible, it follows 
that, however improbable it seems, (-1)(-1) = +1.

So now we have to consider the question:  Given that this definition 
works, should we throw it out just because we can't think of a nice 
picture to illustrate it?  

Does this help? 

- Doctor Ian, The Math Forum   

Date: 06/28/2001 at 15:32:38
From: Doctor Rob
Subject: Re: Multiplication of two negative numbers

Hi Daniel,

Thanks for writing to Dr. Math.  

I think the following section of the Dr. Math FAQ will tell you
what you want to know:   

-1 is definitely a number. There is no obvious connection with
i*i = -1.

I hope this helps. Let me know if you'd like to talk about this some
more, or if you have any other questions.

- Doctor Rob, The Math Forum   
Associated Topics:
High School Number Theory

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