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If we have $ x^y$, where $ y $ is irrational, would the following expression capture the way raising a real number to an irrational exponent is usually defined?

$$x^y= \lim_{n\rightarrow \infty }{{x^{\frac {\left \lfloor 10^ny\right \rfloor}{10^n}}}}=\lim_{n\rightarrow \infty }{\sqrt[10^n]{x^{\left \lfloor 10^ny\right \rfloor}}}$$

user132181
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  • I might be wrong, but I think this is all right. It would work for anything apart from $10$ too, for instance $3$ maybe. Whatever you want. Side note: This isn't a very useful expression. Why not $e^{y\log(x)}$ where $$e^t=\lim_{n\to\infty} \left(1+\frac{t}{n}\right)^n$$ and $\log$ is just its inverse? – Guy Mar 30 '14 at 13:14
  • This definition seems more natural to me :) – user132181 Mar 30 '14 at 13:18
  • Well yes it does seem natural. I agree. And I guess you want $10$ because decimal. Weirdly I don't feel binded to the decimal system. I agree with the general notion of converting $y$ to an arbitrarily close rational though. – Guy Mar 30 '14 at 13:20
  • Of course there should be '$ b $' instead of '$10 $', but base ten is the most polular one, so :) – user132181 Mar 30 '14 at 13:28
  • @Sabyasachi You don't want to introduce exponential and logarithmic function before having the power! This is a stage in sequential generalization of operations to reals by closing the $\mathbb{Q}$ with limits of Cauchy sequences (a standard formal procedure, regularly taught in school). $\exp$ and $\log$ are transcendental functions that still have to be defined, so you are building the roof before the foundations. – orion Mar 30 '14 at 13:28
  • @orion $\exp$ can be defined by the limit. $\log$ as its inverse. – Guy Mar 30 '14 at 13:38

1 Answers1

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Yes this converges. It's a very specific flavour of the general idea that involves constructing a rational sequence

$$(a_n)=\frac{p_n}{q_n}$$ that converges to $$\lim_{n\to \infty} a_n=y$$

And define

$$x^y=\lim_{n\to \infty} \sqrt[q_n]{x^{p_n}}$$

Of course for real $x>0$.

You can use any convergent sequence. You used the sequence of increasing number of decimal digits, but you could just as well use convergents of the continued fraction expansion, or virtually any rational sequence.

If you want to capture behaviour, it's nice to have a monotonous sequence, but it doesn't have to be.

orion
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