Negative Numbers

Reading a superb book on the history of mathematics (Mathematics: The Loss of Certainty by Morris Kline) reminded me of the extent to which we take things for granted, especially when we learned them very early in life. One subject that used to plague the ancients was negative numbers. So I got to thinking. If we view mathematics as a language, then the meaning of that language rests on an agreement by all the parties using it what different notations mean. So let us look at one possible explanation for negative numbers:

-9 -8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9

One way to see this series is that negatives belong to one domain, positives to another, and the big divide between them is the zero. When it comes to addition or subtraction, we simply begin at the point indicated by the sign of the first number and then march left or right as indicated by the sign between them and the sign of the second number. Here is 3 + 5:

-9 -8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9

                                     |______________|

Here is 3 + -5. The negativity of the 5 indicates that we need to march to the left.

-9 -8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9

                      |______________|

Now let us take -3 + -5:

-9 -8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9

    |______________|

And its inverse, -3 + 5:

-9 -8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9

                   |______________|

Now I discover what is really a seeming inconsistency in our language—provided, of course, that it is based on equivalent domains separated by a zero. Consider the following. 2 * 2 = 4; this means that we move two positions to the right from 2. And -2 * 2 = -4. That’s also consistent—because, finding ourselves in the negative domain, we move two positions to the left of -2 as shown for both cases here:

-9 -8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9

                |_____|           |_____|

Here the general rule would be that when we multiply, we move in the increasing direction of the domain if the multiplier is positive and in the decreasing direction, still of that domain, if the multiplier is negative. But consider next what happens when we take -2 * -2 = 4 or 2 * -2 = -4. In the negative domain, we should move right if the multiplier is negative; in the positive domain, with a negative multiplier, we should move to the left. As above.  If we applied that rule, both cases would yield zero as shown below.

-9 -8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9

                      |_____|_____|

But they do not. What we do is multiply the two number first and then we assign the sign based on a rule of signs. This is an arbitrary element in our math or, minimally, is no long capable of being tracked in a visual model. The inconsistency continues when we use exponents, which indicate multiplication. For example, both 2² and -2² yield four (one must believe Excel). And -2^-2 yields 0.25. Here we still cling to the explanation that multiplication of values of the same sign yields a positive number.

What these results tell me is that our mathematics uses a different conceptualization for the negative numbers than that of a domain. It uses the notion of gain and loss. Thus -2² is positive because it reduces the losses by 4, as does the equivalent -2 * -2. The last answer, 0.25 as the result of -2^-2, results because the negative exponent signals successive divisions rather than multiplications. Thus that number means:

-2^-2 = (1 / -2) / (-2) = 0.25

But if we chart that result, thus moving to the right, because division means decrease,  we should end up with -0.25, not with a positive value. But the number becomes positive because of our rules. The first result is -0.5 which, divided by -2, turns positive—albeit it is still on the negative side of the zero. We get exactly the same result if we solve for 2^-2. Thus the notion of a domain, as depicted above, corresponding to some physically imaginable plain, has been “suspended” here. 2^-2 should result in 0.25, -2^-2 should yield -0.25.

This field might have been so very different if, instead of the concept of negation, we would have viewed negative numbers as differently colored. Call them red.