A few days ago, I mentioned the series of math-related posts on the “Opinionator” blog at the New York Times Web site. Prof. Steven Strogatz, professor of applied mathematics at Cornell and the author of the series, has a new post up, “The Enemy of My Enemy”. Although the title sounds like something from a conventional OpEd article, Prof. Strogatz in this installment focuses on subtraction, and on the negative numbers that sometimes result.
He does some pretty wide-ranging exploration of how the idea of negative numbers, and of negativity in general, relates to different areas of life — and you will have to read his column to discover how its title is relevant. Here I want to talk a bit about one of the ideas he presents, one which I think is relevant to getting a real understanding of math in general, and numbers in particular.
The idea of counting things with numbers is pretty natural. Anthropologists have discovered that essentially all human cultures, no matter how primitive from our perspective, have names (words) for the number one, two, and three, and have the concept of counting objects. That is, three rocks is represented by the same number as three birds. Addition, similarly, is quite natural. When we get to subtraction, however, things get a bit more complicated. It is easy enough to visualize having three rocks, taking away one, and having two left. But what does it mean if we start with three rocks, and take away five?
Subtraction forces us to expand our conception of what numbers are. Negative numbers are a lot more abstract than positive numbers — you can’t see negative 4 cookies and certainly can’t eat them — but you can think about them, and you have to, in all aspects of daily life, from debts and overdrafts to contending with freezing temperatures and parking garages.
In a certain sense, negative numbers are an abstraction; but they are not just any old abstraction, because we define them in a particular way, one that allows the normal rules of arithmetic, which apply to positive numbers, to continue to work. For example, just as
4 + 2 = 2 + 4
it’s also true that
4 + (-2) = (-2) + 4 and
(-2) + (-3) = (-3) + (-2)
In some sense, negative numbers represent a generalization of the idea of positive numbers, a generalization done in such a way that all the rules of arithmetic continue to work.
Many of the other classes of numbers that mathematicians talk about, and that occasionally bedevil math students, arise from similar considerations. Take rational numbers, for example. A rational number is one that can be written as a fraction, or ratio, with integers in the numerator and the denominator. The idea of fractions probably arose naturally in the context of measuring things. We might measure length in feet, for example; but some lengths will be longer than one foot but shorter than two feet. A natural impulse might be to divide the foot into sub-intervals, as of course was done by introducing inches. But the same problem of “gaps” will arise on the smaller scale, and eventually the conceptual leap was made to the idea of the fraction as a number, not tied to a particular measurement framework, so that one could speak of half a foot, half an inch, or half a potato. And, once again, these numbers were defined in such a way that the rules of arithmetic continue to hold.
The possibility and necessity of a further step was realized by the Greeks, back around the time of Pythagoras. They were, of course, very interested in geometry, and discovered that, if they constructed a square one unit on a side, the length of the square’s diagonal could not be written as a rational number of the same units. To see this, consider the square bisected by its diagonal as two right triangles, and apply the Pythagorean Theorem:
a² + b² = c²
where ‘a’ and ‘b’ are two sides of the triangle, and ‘c’ is the hypotenuse. Then the length of the diagonal is √2, and that number cannot be written as a fraction; it is an irrational number. It’s easy to come up with others, such as √3 or √7. Once again, though, we define the properties of these numbers so that they obey the same rules as the integers and rational numbers.
There are, of course, additional members of the number zoo. Transcendental numbers are irrational numbers which, unlike √2, are not the roots of any algebraic equation. Probably the best known example is π (pi), the ratio of the circumference of a circle to its diameter. The union of the sets of rational and irrational numbers (which includes transcendentals) is the set of real numbers. The idea of imaginary numbers arose from asking the question: What is the value of √-1? The answer is an imaginary number, whose name is i. We can then form a complex number, which has the form:
a + b i
In all these cases, what we have done is to generalize the idea of numbers that we first got from counting, defining the generalizations so that arithmetic (and other properties of numbers) still works.