Why is a negative times a negative a positive?

[Guest]

Hi All,

My assignment is to acquire others' reasonings as to why a postive times a negative is negative and why a negative times a negative is positive.

Then I draw my own conclusions.

My thoughts on this and I would welcome anyone's responses. I do need multiple responses....

The opposite of a positive is a negative. So if you are looking at a number line, and are asked for the opposite of 5, you go to -5. Now if you are at the opposite of 5 and are asked for the opposite of the opposite, you go back to 5. Does that makes sense?

Thanks for the help.

 

[Count Iblis]

Yes, that's basically right. For every number \(\displaystyle x\) there exists a unique negative number \(\displaystyle y\) such that \(\displaystyle x + y=0\). The number \(\displaystyle y\) depends, of course, on \(\displaystyle x\). We denote this by \(\displaystyle y = -x\).

Clearly, if \(\displaystyle y = -x\), then \(\displaystyle x = -y\), because \(\displaystyle x\) satisfies the equation that defines the negative of \(\displaystyle y\).

So, taking the negative twice gives you the original number back. What you now have to do is show that the negative of a number equals minus one multipled by that number. This follows from:

\(\displaystyle x + (-1)*x = (1-1)*x = 0\)

We can thus conclude that minus one times minus one equals one.