The first post of this series, Easy Exponents, defined the fundamental (and familiar) axiom of exponents along with six theorems derived from it. Those seven equivalences are enough for most situations involving exponents. There are some advanced and exotic cases that go beyond the fundamentals. For instance, matrices can both have and be exponents.
This post takes a closer look at the case with negative exponents. The Fourth Theorem from the last post deals with these. It says:
A number with a negative exponent is the inverse of that number if it had a positive exponent. Note that we can rearrange this by multiplying both sides by x-to-the-nth power:
Which brings out the key characteristic of an inverse: multiplying a value by its inverse gives us one (unity). The notion of an inverse is so fundamental that the above applies even for exotic definitions of “multiply” and “one”. For instance, multiplying a matrix by its inverse gives the identity matrix, which is “one” in matrix terms.
Now, consider the following progression:
In particular, notice how the value goes down by a factor of two when the exponent goes down by one. Each result is one-half the previous result. (Two is the base number here, so of course the values change by that factor.)
There’s no reason we need to stop the sequence. We can keep dividing by two and decreasing the exponent by one:
Which are exactly the values given by the Fourth Theorem. For example:
And note what happens when we multiply an exponentiated number by its inverse:
All the above works for any and all bases — it doesn’t have to be two. Here’s an example using a base of four:
Of course, multiplying inverses gives us unity:
Regardless of the base, the decreasing sequence gives us another way to establish the Fourth Theorem. Negative exponents are part of a progression of values, and they form the inverses of the same base number with a positive exponent.
Note also that:
Which is the Third Theorem (so everything ties together).
And that’s all there is for today!