On the muliplication of negative numbers

It might seem strange to consider this plebian topic in a newsgroup
having to do with such an esoteric subject at the theory of
relativity, But, believe it or not, a controversy has arisen between
certain frequent posters in this newsgroup as to the result of
multiplying negative numbers, a topic generally treated in the U.S.
educational system before the 6th year (12-year-olds).

Numbers have interested people for both practical purposes and
amusement for centuries. The negative numbers arose to provide
solutions to questions such as "What number, if any, added to 5 gives
3?" Before the invention of negative numbers, there was no such
number. With negative numbers admitted, there is.

Negative numbers pose no problem in addition and subtraction. And
multiplication of a negative number by a positive integer is quickly
understood as repeated addition. But multiplication of two negative
integers is not so quickly dismissed as a problem. I admit that I
groped for a while to think of a way to make the solution obvious in
terms of more elementary operations. One doesn't want to say: It's a
rule! Obey it!

The way I came up with may not be simplest (if you kinow something
simpler, tell me about it), but it works:

Consider 5 * 5 = 25. If one is silly enough to write 5 as 6+(-1), one
would have
(6+(-1))*(6+(-1)).=25

By the distributive law, one could expand that to

6*6 + 6*(-1) + (-1)*6 + (-1)*(-1) = 25

We all agree that 6*6=36. Now 6*(-1) = (-1)+(-1)+(-1)+(-1)+(-1)+(-1)
which reasonable people will agree amounts to -6. Similarly (-1)*6
= -6 again. So far, we have

25 = 36 +(-6) + (-6) + (-1)*(-1)
or
25 = 24 + (-1)*(-1)

which shows that (-1)*(-1) = 1. Ta-dah!

As a gift, we get sq.rt.(1) = -1, Of course we already knew that
1*1=1, so we have discovered by this simple and obvious means that
numbers can have two different square roots.

sq.rt.(16) = 4 or -4.

Don't laugh,anyone. There is a debate on this matter in one of the
threads in sci.physics.relativity.

I expect that any rational person who did not understand the
multiplication of negative numbers, will now accept the result that
a negative times a negative is a positive. That leaves room for the
rest of us (non-rational) to continue the debate, but maybe now we can
draw the curtain on that debate without any accusation of cruelty.

Uncle Ben

.

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