## Friday, May 06, 2011

### Proof of -x * -y = x * y

An inquisitive person posted a question on pakgrid: Why a negative number multiplied by a negative number gave a positive result? An example given by the poster was

While (-6) + (-6) + (-6) + (-6) = -24
why (-6) * (-4) = 24?

To help him, many people came forward with their examples and proofs. Here is one interesting proof by an assistant professor at NUST. Produced verbatim.

The proof of
-x * -y = x * y

is as follows:
-x * - y = -(x * -y) = --(x * y) = x * y

I have my sympathies with the professor's students. For one, what's this -- symbol? What on earth does that mean? Secondly, what property of arithmetic operators are you using when you convert -x * -y to -(x * -y), and finally isn't the whole argument cyclic? Isn't mathematical rigidity of your proofs the first thing they teach to a PhD candidate? Or have things changed these days?

On a side note, other interesting "logical" posts can be found here:

Jokes aside, the example given of -6 and -4 is plain wrong but the question "why a negative multiplied by a negative number gives a positive number" is a very genuine one. It's very sad that as school children we are not encouraged to ask such questions when a new concept is introduced. But sooner or later one realizes that all of the rules in Mathematics must have a rationale.

1. I can somehow relate this with division operator when the denominator is a number between 0 and 1. E.g. 1/0.9 is greater than 1 even though we're dividing 1.

In the meanwhile, I am trying to get my matheurons to start pumping.

2. Yes, it seems counter-intuitive. How could "dividing" 1 result in something larger than 1?

The problem is with how we understand Arithmetic (and Mathematics) based on everyday life experiences as a child, and how some concepts beyond natural numbers do not fit with every day experience.

Consider subtraction, for example. How could you subtract 6 sheep from 4 sheep to get -2 sheep? The only remedy is to "explain" the concept of "-2" by equating positive integers with "possession" and negative integers with "owing" something.

But then what's multiplication by a negative number?