Re: On the muliplication of negative numbers


"Dirk Van de moortel" <dirkvandemoortel@xxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote
in message news:g6k7qu$r2l$1@xxxxxxxxxxxxxxxxxxxxxx
Uncle Ben <ben@xxxxxxxxxxx> wrote in message
5277af6b-2456-4669-bec1-24e62a6f14a0@xxxxxxxxxxxxxxxxxxxxxxxxxxxx
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,

If by sqrt(...) you mean the square root function (as the notation
strongly suggests), or the standard root sign, then you write
sqrt(1) = 1
and
-sqrt(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.

Yes, in a sloppy way of speakoing, but *the* square root of a
positive real number is defined as a positive number and it is unique.


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

No, we write
sqrt(16) = 4
and
-sqrt(16) = -4

Dirk Vdm

It's storm in a wine (or whine) glass:
http://en.wikipedia.org/wiki/Square_root
Harald


.

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