Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah

An Example of Logical Construction


Mathematical concepts build on simpler mathematical concepts. It's amazing how quickly one can proceed from simple facts to very complicated ones. I'll try to illustrate this phenomenon on this page by considering powers. Ideally you show know for, example, that two raised to the power three means two times two times two, but not much more. Two is called the base of the power, and three is the exponent (just so we don't have to say "the number on top" or "at the bottom "). So we define for all natural numbers n:


The number a raised to the power n is obtained by writing the base a n times and writing multiplication signs in between the a's. If we wrote plus signs instead we'd get the product n times a. That's exceedingly simple. But big matters flow from little ones.

In the definition (1), the base a may be any real number. But the exponent is restricted to be a natural number. I'm assuming you understand definition (1) and nothing else about powers. (This page would be very effective if that was true.) Will you believe that by the time you are through this page, perhaps half an hour from now, you will understand that


and actually a great deal more?

Eight to the power negative four thirds equals 1 sixteenth!? You mean if you write down eight negative four thirds times and connect those eights with multiplication signs you get one sixteenth? That does not make sense! Still, (2) is true! Just imagine how you can enthrall your date at your next opportunity by casually mentioning cool facts like that!

How do we get to this fact, though? Well, we must extend definition (1) suitably. Negative four thirds is a rational number, but while we are at it we might as well go all the way to real exponents. So how do we "extend definition (1)"? Why not just make (2) a definition and get it over with?

The guiding principle in generalizing a mathematical concept is to take care that everything that was true before the generalization remains true. In other words, we want everything to be consistent.

Let's go in steps and start with the exponent 0. A suitable starting observation is that if we multiply a to the power three and a to the power four then we get a to the power seven, since we wrote a a total of seven times and multiplied all those a's together. In other words, for all natural numbers m and n


Any definition of a to the power zero should preserve the property (3) (and actually a number of others but I'm just giving you the flavor of things). So in particular we must have


In other words, multiplying a to the power n with a to the power 0 does not change a to the power n. How can that be unless


So we make (5) the definition of a to the power zero. We chose that defintion carefully to be consistent with what we did before. If we were developing powers seriosuly we would have to think yet more carefully about whether or not (5) might contradict anything else we thought we understood before. (It actually does if the base is zero. Think about that!)

Now, what about the exponent being a negative integer? Let's invoke that rule (3) again. We should have, for all integers n, that


How could that be unless

? (7)

But we must be careful! What if a=0 (and n=1 , say). Then we'd be dividing by zero which we must not do. (The reason for that is that a division by zero cannot be defined without running into contradictions. There may be a link here some day to a suitable argument.) So to avoid this complication

we require that the base is non-zero.

Now onwards to rational exponents! Let's make another observation. If we take two to the power three, and all that to the power four, then we write down two to the power three four times (do it!). Thus we write down the number two a total of twelve times. Generalizing this observation it is clear that for all natural numbers m and n


Let's first consider the case where the exponent is one half. Rule number 8 should apply in that case, i.e., we should have


So it's clear that a to the power one half must be the square root of a, i.e., that number whose square is a. But again we must be careful. There is no real number whose square is negative one, for example. We could overcome this difficulty by allowing powers whose values are complex numbers. But this would be getting quite technical, and so let us just require that


Now let's generalize the square root idea. For any rational number q we must have (so as to to maintain the validity of rule (8)) that


which clearly requires that we define


To obtain a definition for rational exponents p/q where p and q are integers and q is non-zero, we again employ rule 8 (or more precisely, we make a definition that's consistent with rule (8)):


We can now solve the puzzle that we started out with, using rule (13) and requiring that rule (7) applies to rational exponents:


So we reached our goal. Of course the development given here is not complete. In particular, one can relax the requirement that the base be positive by allowing complex numbers, and by restricting the exponents. However, you saw the basic ideas in action: make definitions that are consistent with what happened before.

One final comment. To define powers (with positive bases) and real exponents one uses techniques developed in Calculus. To give you the flavor, if the exponent is the square root of two one considers a sequence of rational numbers that gets closer and closer to the square root of two, and the corresponding sequence of powers. That sequence gets closer and closer to a real number, which we then define to be the required power:



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