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:

(1)

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

(2)

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*

(3)

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

(4)

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

(5)

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

(6)

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

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*

(8)

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

(9)

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

(10)

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

(11)

which clearly requires that we define

(12)

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)):

(13)

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

(14)

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:

(15)

[26-Nov-1996]

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[08-Apr-1996]