Tuesday, September 19, 2006

Deep understanding

When learning operations with integers you could just memorize (rote learning?) that a negative number times a negative number is positive or you could go for deep understanding as presented by Dr.Frank Wang of Saxon fame.
This is a brief excerpt of a much longer explanation:

Using the field axioms, we can prove a variety of things that you likely take for granted such as

–(–a) = a

– (a + b) = – a + (– b)

and a x 0 = 0 x a = 0

Now, we finally get to the crux of our explanation (the same explanation as above). In a field, the distributive property must hold. That is, if a, b and c are real numbers, then

a x (b + c) = a x b + a x c


(b + c) x a = b x a + c x a

The distributive property ties together the different operations of addition and multiplication.

Now, we replace a, b and c with -1, 1 and -1 respectively.

That is,

(-1) x (1 + (-1)) = (-1) x (1) + (-1)x(-1)

On the left hand side we see that 1 + (-1) is equal to 0 since any number plus its additive identity is equal to 0. Any number multiplied by 0 is 0 (this can be proven from the axioms for a field). Therefore, replacing the lefthand side with 0, we get

0 = (-1) x (1) + (-1) x (-1)

Since any number times 1, the multiplicative identity, is itself, we can further simplify this equation to get:

0 = –1 + (–1) x (–1)

We now need to figure out what (–1) x (–1) is. Since a number added to its additive inverse is 0, (–1) x (–1) must equal the additive inverse of –1. This is simply 1 so

(–1) x (–1) = 1
Sometimes memorization comes in handy.

1 comment:

Anonymous said...

There's a reason that you are an undergrad in number theory before you attempt that stuff--because it helps to 1) know the truth, and 2) to have several years of mathematical maturity in the form of such truths before you try to prove them.

Asking most 10 or 13 yr olds to do proofs college math majors do is cruel.