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Why is Negative * Negative = Positive ??? ( Trivia # 13 )

on Friday, May 1, 2009

One way to think of numbers is in terms of money. Let's say you and I are playing poker. To make life convenient, we use chips instead of real money. A green chip is worth Rs.5. A red chip means that you owe Rs.5. So if you lose Rs.5, you can represent that by giving up a green chip, or (if you're out) by picking up a red chip. Of course, you are always allowed to pick up a green chip and a red chip at the same time, because that doesn't change your total sum. (At the end, presumably, we'll cash in all our chips and see who gains or loses what money.)

I hope you see the mathematical analogy I'm drawing here: A green chip represents +Rs.5, and a red chip represents -Rs.5. Make sense? If so, here comes multiplication in terms of chips.

If you gain three green chips, what happens? Intuitively, you know that you gain Rs.15. Mathematically, we take the +Rs.5 that one chip is worth, and multiply it by +3, to indicate that you are gaining three chips. 3 x Rs.5 = Rs.15; positive times positive is positive.

If you gain three red chips, what happens? If you think about it, I think you'll agree that you just lost Rs.15. Mathematically, this looks the same as the previous example, except that the +Rs.5 that represents a green chip, is replaced by -Rs.5 to represent a red chip. 3 x -Rs.5 = -Rs.15; positive times negative is negative.

Now, what if you lose three green chips? Once again, you have lost Rs.15. We represent this "loss" mathematically by changing the 3 into a -3, so our equation is: -3 x Rs.5 = -Rs.15. Negative times positive is negative.

And finally, what if you lose three red chips? Hooray! This is happy news, it means you have actually gained money. Mathematically, this is -3 (since you lost) times -Rs.5 (since they were red chips). -3 x -Rs.5 = +Rs.15. Negative times negative is positive.

Finally, for people who like algebraic sorts of proofs, consider this. A and B are both positive numbers.

A + (-A) = 0
(A)(B) + (-A)(B) = 0

Since the term on the left is positive, the term on the right must be negative. (I'm assuming here that if two non-zero numbers add up to zero, then one of them is negative and one is positive—I think that is too obvious to require a proof.) This proves that negative times positive is negative. Now:

A + (-A) = 0
(A)(-B) + (-A)(-B) = 0

Since the term on the left is negative (as we just proved!), the term on the right must be positive.


Mukesh said...

Brilliantly explained!

Anonymous said...


eureka123 said...

almost same explaination as mine...
here it goes...

It is kind of like two wrongs make a right. Let us say that x and

y are both positive. Then the meaning of -(x) + (y) is just y - x and (y) - -

(x) = y + x, subtracting a negative number is the same as adding the positive

number. Now, let us say n is a positive integer. Then n * x can be thought of

as adding x to itself n times which explains why a positive number times a

negative number is always negative. So, what does -(n) * x? Well,

multiplication is commutative, i.e., x * y = y * x. So -(n) * x = x * -(n)

or -(n) * x = (-1 * n ) * x = ( n * -1 ) * x = n * ( -1 * x ) because

multiplication is also associative = n * -(x). All of which is to say that -n

* x is the same as adding -x to itself n times. Therefore, -n * -x is the same

as adding -(-x) to itself n times. And, I think we all agree that -(-x)

better be +x. Hope this is not too confusing!

Ankit Mahato said...

yup .. urs is another good explaination !!

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