Quick question - mathematical claim

[Savage]

I am writing a book and part results in talking about negative exponents. I, for the life of me, could not derive negative exponents without a little BS, so instead I decided to DEFINE them: a^-b = a^-b / 1 = 1/a^b

Used this definition to prove continuity with other theorem's of exponents. Simple right?

My question is, are we allowed to just claim something as a definition and show continuity / would this really be a "definition" / am I breaking any rules someone might call me out on?

Any help would be appreciated.

[andretimpa]

You can define natural exponents quite easily using only multiplication. From there you can extend things to integer exponents by using a^b*a^c = a^(b+c). You can then use ((a^b)^c) = a^(b*c) to define rational exponents (with the help of n-th roots). The hard step is to show that if 2 rational exponents a and b are arbitrarily close, then so are c^a and c^b. With this you'd be justified to assume the existence of a continuous extension of your definition to real exponents.

The big technical difficulties are:
-Proving that this definition is self-consistent (any 2 ways of building the same rational exponent would give the same results).
-The hard step of showing that arbitrarily close rationals give you arbitrarily close powers.


A roundabout way of solving this is to define an exponential function as something with the property
F(x+y) = F(x)*F(y), show that the solution to F'(x) = F(x) is an exponential function exp(x). Show that all exponential functions are essentially exp(a*x). Finally, make the connection with a^x for rational x. Now the fact that exp is automatically a continuous function solves both technical difficulties.

[Savage]

Hi Andretimpa...I think you got off topic with my question.

I was simply asking if for negative exponents: can I (1) define them (as my question stated), then (2) show continuity.

I had no problem proving from the definition that all other theorems of exponents could use negative exponents as well, but I was wondering if I am missing something by using this method of "define and show continuity"

Example:

I proved a^b * a^c = a^(b+c) WHEN b,c are both positive. Then I proved, after DEFINING negative exponents, that a^b * a^c = a^(b+c) even if b and/or c are negative.

I am simply asking if I this is okay.

[andretimpa]

a^(-1) is usualy defined as the thing such that a*(a^(-1)) = 1 and a^(-n) is just a shorthand for (a^(-1))^n.

I'm confused by what you mean by continuity then.

[Savage]

By continuity...I mean showing that once we define something, we can't just assume we can use it with other theorems.

Example:

perhaps I defined a^(-5) as: -a * -a * -a * -a * -a = (-a) ^ 5

Then a^(6) * a^(-5) =/= a^(6 + -5) because -a^11 =/= a^1 for all a.

So if I defined them this way, the addition theorem of exponents could not use negative exponents, only positive.

[Savage]

a^(-1) is usualy defined as the thing such that a*(a^(-1)) = 1 and a^(-n) is just a shorthand for (a^(-1))^n.

I'm confused by what you mean by continuity then.

Expanding off your inverse explanation: I don't want to define "negative" as the inverse of positive. Rather define negative as I have and discover that negative exponents are like inverses later on.

edit: bump

[OldTrees]

a^b = "1 * a * a * a ... b times"
a^-c could be defined as division "1 /a /a /a ... c times" (the easier way)
or as "1 *1/a *1/a *1/a ... c times"

Negative exponents are division so one really can't escape that in one's definition. Any attempts to escape/hide the division are likely to obscure the lesson.

[andretimpa]

If you define exponents only as the number of times that you need to multiply the base to get the result, then no, you can't deduce what a base to a negative power will be. You are basically starting with exponents that equal 1, from them defining what are exponents equal to 2, then 3, 4 and so on. But you can only move in one direction.

To get the negative exponents you will obligatory have to define them by imposing that they are consistent with the rule (a^b)*(a^c) = a^(b+c). This alone is enough to give you the usual definition and you should be able to prove that they correspond to divisions. If you want you can then show that all you needed to do to get to the same definition was defining a^-1 as 1/a.