Numbers raised to irrational numbers

Question

I realized something the other day that's got my mind in a knot again. Given some constant $ n $, I know that: $$ n^2 = n \cdot n$$ $$ n^3 = n \cdot n \cdot n$$ And so on so forth. I also know that: where, $ n^{1/2} = k $, then: $ k \cdot k = n $ and so on and so forth. Things get a little complicated when $ n^{2/3} = k $, but with some work I understand that $ n \cdot n = k \cdot k \cdot k $. An intuitive explanation might be "the length of one side of that square is equal to the length of one side of that cube." Higher dimensions, of course, break this "intuitiveness" but I can still process mathematically what these rational powers mean.

But what does it mean when you're given $ n^\pi $? How would I visualize this? Is it really as simple as just approximating to the degree of accuracy you need?

$$ n^{31415} = k^{10000} $$

What's a more rigorous definition (or alternative explanation) of $ n^\pi $ than my understanding of powers mentioned above?

I ask this question because I read this thread: Real Numbers to Irrational Powers

Answer 1

The most standard definition uses the idea of a limit: basically, the limit of a function $f(x)$ "as $x$ goes to $c$" is $L$ if you can get $f(x)$ to be as close as you want to $L$ by taking $x$ to be close enough to $c$. For example, the limit of $\frac{1}{x}$ as $x$ goes to $2$ is $\frac{1}{2}$, because no matter how precise I want to measure, I can always choose an $x$ close enough to $2$ that I won't be able to measure the difference between $\frac{1}{2}$ and $\frac{1}{x}$. We write this $\lim_{x \to c}f(x) = L$.

Formally, when $r$ is irrational, we define $a^r$ as $\lim_{x \to a}a^x$. In other words, we say that $a^r$ is the number you get closer and closer to when you raise $a$ to rational powers that are closer and closer to $r$.

Thinking of $n^{\pi}$ as looking at things like $n^{31415} = k^{10000}$ is a pretty good way of thinking about it, but it has the disadvantage of using unreasonably large numbers. I would think of it more as "$n^{\pi}$ is somewhere between $n^{3.1415}$ and $n^{3.1416}$".

Answer 2

Let $\{p_1, p_2, \dots \}$ be a sequence of rational numbers such that $\lim_{i \to \infty} p_i =\pi$. Then one can define $n^\pi = \lim_{i \to \infty} n^{p_i}$