Girlfriend Math: The Exponent Operator

My girlfriend is taking a course on mathematics, and to her delight is finding it to be a pleasant experience. The following post is a re-telling of a discussion we had, so that she can read it again. I believe that the best way to teach anyone anything is to tell them a story, and mathematics is no different. Hence, this will be the story of the exponential function.

One day, there was a mathematician who was poking around with multiplication. He had written down 2*3, 2*4, 7*8, and so on, and so forth, on his piece of paper, and was just sort of idling by. He then, by random chance, decided to focus his idling on multiplication of the same number to itself: 2*2, 3*3, 4*4, and so on.

Mathematicians, not unlike computer programmers, are incredibly lazy individuals. This mathematician didn't like that he had to write down the number twice. Earlier lazy mathematicians had discovered and solved the same problem, only with addition: observe that the expression 6*4 is defined simply as 6+6+6+6. Hence, multiplication was just a shorthand for repeated addition.

Taking a cue from this, the mathematician invented himself a new notation, and wrote 2² next to 2*2, 3² next to 3*3, and 4² next to 4*4, and so on. He called his new notation 'squaring' a number, and he called those numbers that could be expressed using the new notation 'square' numbers. This terminology is rooted in a geometric analogy: if you were to take a square number of beads, you could arrange them in a square such that each row and column had the same number in it. The mathematician referred to this row or column number as the 'square root' of a square number. Right now, he didn't concern himself with what the square root of a non-square number was, because the question doesn't make sense, but we will return to this later.

The mathematician liked his new notation, but he was a good mathematician, which meant that not only was he lazy, but he also liked to generalize things. He looked at his notation, with the little '2' floating up there, and asked himself, "what would happen if I changed that to a '3'?" It didn't take him a lot of thought to answer his question: 2³ was 2*2*2, 3³ was 3*3*3, 4³ was 4*4*4, and so on. He called this new notation 'cubing' a number, and referred to the resulting numbers as 'cubic' numbers that each possessed a 'cube root', using basically the same analogy as that for square numbers.

Once again, he asked himself, "What about if I changed the '3' to a '4'?" He was once again able to define the operation: 2⁴ was 2*2*2*2, 3⁴ was 3*3*3*3, 4⁴ was 4*4*4*4, and so on. It was at this point that the mathematician switched back from generalization mode to lazy mode, and he chose not to try to come up with a new name. Instead, he just called the operation 'raising to the 4th power', and each number that he got by raising to the 4^th power had an associated 4^th root. This let him get away with defining a whole family of operations: fifth powers, sixth powers, and so on and so on.

The mathematician was happy with his work, for a time, as his laziness had been quenched. Eventually, though, his generalist side began to take to the fore once again. His notation for the E^th power had been codified as B^E. It had a definition for any base B he chose to use: whole numbers, integers, rationals, even irrational and complex numbers (even though complex numbers hadn't been invented yet), because if you can multiply a number, you can do so multiple times. However, the exponent E had to be a whole number greater than or equal to 2.

It doesn't immediately make sense to multiply a number only once, or zero times, or a negative number of times, and it certainly didn't make sense to multiply a number two and a half times, or PI times. However, the mathematician wanted to generalize it so that he could do these things, or at least, study something that has a similar behaviour. One of the tricks mathematicians use to generalize something is invent a new something that has the same outputs for the same inputs as the original, but because it has a different definition, it isn't bound by the same rules as the original. They can make up whatever they want, of course: mathematics is at its heart a study of what happens when you make up whatever random set of rules you want, and then start looking at the consequences of those rules. They usually choose to make up something that has similar properties as the original object being studied, though.

The mathematician started simple: "What would the 1^st power of a number be? What about its 0^th power, or its -1^th power?" To solve this, he observed an interesting property of his power notation: 6⁷ was defined as 6*6*6*6*6*6*6, but he could also represent it as (6*6)*(6*6*6*6*6), or (6²) * (6⁵). He could use a nearly identical trick to also write 6⁷ as 6³ * 6⁴. He generalized this property as follows: given two numbers c and d, where both c and d were integers greater than or equal to 2, B^c+d = B^c * B^d. It is an interesting property in its own right, because it hints at a deeper relationship between addition of multiplication: note that multiplying two numbers that are powers of the same base is equivalent to adding the powers together. This fact will turn out to be of central importance to practising mathematicians in years to come.

For now, the mathematician focused on that pesky restriction: 'both c and d are integers greater than or equal to 2'. What would happen if he relaxed this definition? Taking baby steps, he set d = 1, and his property told him that B^c+1 = B^c * B¹. Leaving c in its original domain for now, he looked at the result. It told him that B³ = B² * B¹. He had existing definitions for most of it, and knew that B*B*B = B*B * B¹. Algebra allowed him to divide by B twice on both sides, and this gave him a new definition: B¹ = B.

Note however a critical assumption: that he could divide by B. A major headache in mathematics is the inability to divide by zero, which is why so many theorems have restrictions like B != 0, or other, more complicated expressions that cannot equal zero, because they are at some point used as a divisor. The mathematician was happy with his restriction: his new operation, which he dubbed the 'exponentation' operation, would not allow zero as a base. After all, the original idea for this operator came from repeated multiplication, and multiplying zero together multiple times doesn't provide anything useful.

The mathematician did the same trick again, with d = 0, and found that B^c+0 = B^c * B⁰. He knew he could divide by B, having restricted it already, and thus defined B⁰ as 1. This is interesting, because it does not depend on B. The mathematician was emboldened, and tried to tackle a negative integer for an exponent. To do this, he used d = -c, which his property led him to conclude that B^c-c = B^c * B^-c. He had already defined B⁰, and so he derived the expression B^-c = 1 / B^c. With this, he was pleased. His new exponentiation operation now had a definition for any integer exponent.

Taking a step further, he considered what exponentiation might mean for a rational exponent. As a first step, he went back to the property that had been guiding him to this point, and set a = b = 1/2. His property thus told him that B^{1/2 + 1/2} = B^1/2 * B^1/2, or more simply, B = (B^1/2)². Here we see the square root has returned, and the mathematician took the square root of both sides and concluded that B^1/2 would be defined as the square root of B. By this time, some other mathematician had been poking around with square roots of non-square numbers, and found that although they are not equal to any rational number, they are perfectly usable numbers, and had dubbed them members of the irrational or real numbers.

Note however a second critical assumption: that he could take the square root of B. It is in fact two separate but related assumptions. The lesser assumption is which square root you pick: observe that 3 * 3 is 9, but so is -3 * -3. This didn't bother the mathematician; he simply defined his exponentiation operator to be the positive, or principal, square root, whenever a square root was called for. The greater assumption is that a square root even exists. Negative numbers didn't have square roots defined at the time, and so for now, the mathematician further restricted his exponentiation operation to bases greater than zero. Extending the exponential operation to use negative bases would have to wait for the development of complex numbers, and is beyond the scope of this article, but it can be done, and has very interesting properties.

The mathematician continued along similar lines of reasoning, and eventually defined B^1/d, for whole values of d, to be the d^th root of B. (In particular, for even values of d, it was the positive d^th root of B.) He was then able to apply the ever-more-useful-and-interesting property to define B^c/d as the d^th root of B, raised to the c^th power. If c/d turned out to be a negative value, the earlier definition of B^-x = 1 / B^x would make sense of the pesky negative sign. Going further still, he found that the order of the power and the root didn't matter: he could raise B to the c^th power, and then take the d^th root of that, or he could take the d^th root of B, and then raise that to the c^th power. He was also able to show, based on his expression B = (B^1/2)², that B^cd is equal to B^c*d.

One more step was required to fully flesh out the exponential operator: defining it for irrational exponents. The mathematician wondered, "how can I define B^sqrt(2)?" The mathematician had no answer, and eventually gave up. It can be defined using calculus, but calculus hadn't been invented yet. To define it, a later mathematician used the fact that exponentiation is monotonic: if x < y, then B^x < B^y. He took that one step further: if x < y < z, then B^x < B^y < B^z. Now, the square root of two is between one and two, so B < B^sqrt(2) < B². He then used a better approximation: the square root of two is between 1.4 and 1.5, which are both rational numbers and are cleared for use as exponents, and so B^1.4 < B^sqrt(2) < B^1.5. One more step told him that B^1.41 < B^sqrt(2) < B^1.42, and another that B^1.414 < B^sqrt(2) < B^1.415. The more decimal places he used to approximate the square root of 2, the closer together his upper and lower bounds got. The mathematician thus defined B^sqrt(2) as the limit of Bⁿ for some rational number n that approximates sqrt(2), as the number of decimal places in n tends to infinity.