Negative numbers: some whys

Oh, well, it doesn’t have to be spatial concepts. Anything with a notion of direction will do. We could do it temporally, for example.

That having been said, if you really just need to see it demonstrated, then you can illustrate the spatial example for yourself on a piece of paper. Draw an arrow, any arrow you want. Call that arrow A.

Now, anywhere you like, draw another arrow twice as long as A, but pointing in the opposite direction from A. Call that arrow B. By definition, B = -2 * A, since -2 * A just means any arrow twice as long as A but pointing in the opposite direction from A.

Now draw an arrow three times as long as B, but pointing in the opposite direction from B. Call that arrow C. By definition, C = -3 * B = -3 * -2 * A.

Finally, draw an arrow 6 times as long as the original A, pointing in the same direction as A. Call that arrow D. By definition, D = 6 * A.

Do C and D have the same length and point in the same direction? If so, then C = D, and you’ve shown that -3 * -2 * A = 6 * A.

If not, then something very interesting has happened.

I’ve illustrated with -2 and -3 but we can make it much much simpler, trivially so:

Draw an arrow. Call that arrow X.

Draw another arrow, just as long as X, but facing the opposite direction from X. Call this Y = -1 * X.

Draw another arrow, just as long as Y, but facing the opposite direction from Y. Call this Z = -1 * Y.

Finally, draw an arrow 1 times as long as X, facing in the same direction as X. Call this arrow Q = 1 * X.

Do Z and Q have the same length and direction? Then you’ve shown that -1 * -1 * X = 1 * X.

I think the question “Why are they necessary?” could be interpreted in more than one way: What are they used for/applied to? Or, why are they useful within the context of mathematics itself?

I think most of the posts so far have focused on the former. But the more math you study, the more negative numbers (and the rules governing them) come to seem natural, inevitable, and indispensible, from a purely mathematical point of view.

To take just one example: In algebra you learn that an equation of the form y = mx + b, where m and b are any particular numbers, describes the graph of any non-vertical straight line—the m stands for the slope of the line and the b tells you its y-intercept. The m and the b can each be any real number, positive or negative (or zero for that matter). If we didn’t have negative numbers available to us—if the only numbers we could use were positive, we’d have to use several different sorts of equations to describe all the different straight lines that are possible, rather than just one type of equation in which the numbers could be either positive or negative.

As I’ve posted before, consider helium balloons. You’re on a sensitive scale. Some large helium balloons are tied within arm’s reach. You untie three of them and hold them. Your weight has gone down by 6 oz. Obviously each balloon in effect weighs -2 oz. -2 oz. * 3 (number of balloons you took) is -6 oz.

Now you let go of the balloons. Your weight goes up by 6 oz. -2 oz. * -3 (letting the balloons go is the reverse of taking them) = +6 oz.

I don’t think it does if you understand that addition and subtraction can be the same thing. If I add 5 -5 times, I am subtracting 5 from 0 five times. Then, if I add -5 -5 times, I am subtracting -5 from 0 five times, which means I’m adding 5 to 0 five times.