My youngest son is obsessed with $\pi$.  “How much is $\pi^\pi$,” he asked me one day recently.  “I don’t know, a little more than 27,” was my unsatisfactory reply (which, it turns out, is only accurate for large values of “a little”).  Google Calculator tells me that it’s (approximately) 36.4621596 but what does it really mean to raise an irrational number to an irrational power?

Despite being inspired by my elementary-school-aged son, this post is not about elementary or  middle school math, in my usual style.  (For elementary Pi Day ideas, check out last year’s Pi Day post.)  But stick with me, if you will, and maybe you will learn something fascinating about irrational numbers.

Let’s step back a moment.  What does it mean for a number to be irrational?  My son knows that the decimal expansion of pi goes on forever and never repeats.  An irrational number is defined to be a number that cannot be expressed as a ratio or fraction (and a rational number is a number that can be expressed as a fraction).  All rational numbers have decimal expansions that are either terminating (such as 1/2) or repeating (such as 1/3).    Maria over at Homeschool Math has an excellent introduction to irrational numbers and proof that $\sqrt{2}$ is irrational — the former should be accessible to middle schoolers, the latter high schoolers and beyond.   (Pi, as it turns out, is not only irrational but also transcendental, which means that it is not a solution to any polynomial with rational coefficients.)

Alexander Bogomolny at Cut the Knot offers an easy-to-understand proof that it is possible to raise an irrational number to an irrational power and obtain a rational result!  Many are familiar with Euler’s famous equation $e^{i\pi} = -1$ or equivalently $e^{i\pi} +1 = 0$.  Somehow one can raise an irrational number to an imaginary irrational power (!!!) and get an integer!  But what does it mean to raise a number to an irrational power??

Most of us have an intuition about what it means to raise a whole number to a whole number power.  $3^7$ means you multiply 3 by itself 7 times: $3\times3\times3\times3\times3\times3\times3$.

What does it mean to take a number to a rational power? $3^{\frac{1}{2}} = \sqrt{3}$, but why is this the case?  One intuition is to think that because it’s taken to the 1/2 power, you need 2 of them multiplied together just to get up to the original base, in this case, 3.  Or if you’re comfortable with the rules of exponents, you’ll know that $3^{\frac{1}{2}}\cdot 3^{\frac{1}{2}} = 3^{\frac{1}{2}+\frac{1}{2}} = 3^1 = 3$.  So if $3^{\frac{1}{2}}$ times itself equals 3, it must be equal to $\sqrt{3}$  Similarly $5^{\frac{1}{3}} = \sqrt[3]{5}$

But is there any intuition that can help us undertand what it means to take a number to an irrational power?  What could it mean to multiply $\pi$ by itself $\pi$ times?  The best explanation I can find is over at Ask Dr. Math, where Doctor Rob explains that we find the value of a number to an irrational exponent by raising the number to successively more precise rational approximations of the irrational exponent, and then using calculus to find where that sequence converges. (Although, that does not explicitly cover the case where the base is also irrational, the same general approach applies.)

Finally, Thomas Christie offers a nice explanation of Euler’s Equation.  (This link has been broken for the past few days, but I hope it will be back!)  In case Christie’s page doesn’t come back, you might wish to look here or here.

I hope you had fun exploring irrational numbers and powers with me today!  Now, go eat some pie. ;-) Happy Pi Day!

Edited to add: The Math Less Traveled has a nice series of posts up going through the proof of the irrationality of pi, starting here.

1. March 14, 2009 6:01 pm

I like your post! I suspect there must be some more intuitive explanation of an irrational power than having to invoke calculus (calc 2, no less), but nothing occurs to me. I was trying to think of a pattern that would result in an irrational exponent, in the same way we make sense of negative exponents…hm…

2. March 14, 2009 6:18 pm

well… you can skip the calculus if you just want an approximation, of course ;-)

I thought about but decided against including negative powers in the post as well.

3. April 26, 2009 6:50 am

Isn’t there as way of expressing a^b using exponetials and logs something like exp b lna, though i cant remember it exactly.

Anyway interesting post

david

4. May 19, 2009 2:07 pm

David: Yes! Since the natural logarithm and natural exponential functions are inverses of one another, any positive quantity A can be expressed as e^ln(A). In particular, when A = a^b, you have a^b = e^(b ln(a) ).

This fact gets used (a fair amount) in the calculus curriculum to analyze exponential functions, their derivatives, and antiderivatives; the analysis is straightforward once you know everything you need to know about e^t and ln(t), as well as the chain rule and substitution integrals.

For calculation, though, reducing pi^pi to instead being able to compute ln(pi), then multiply by pi, then raise e to the resulting power, is hardly an improvement. I suppose you could replace e^ln(A) with 2^log(A), where the log(A) is computed base 2, and then have irrational powers of an integer. But that’s still rather horrid.

In a prior-to-calculus (but not just precalculus!) environment, I’d be tempted to approach pi^pi using the successive estimates 3^3, 3.1^3.1, 3.14^3.14, etc…. Each of those is a rational power of a rational number, so in theory they’re simple objects. (But ugly: 3.14^3.14 would be the 100th root of (3.14)^314, which is not something for the faint of heart.)

And technically we should probably worry about a^b being a double limit rather than a single limit — what if a -> pi at a different rate than b -> pi?

Sigh. Irrationals: crazy stuff.

5. July 29, 2009 11:31 pm

Good article.

6. December 18, 2009 9:02 pm

Hello Math Mom-

Not sure how to leave a general comment.. Thought you might like this post on math education-

http://blogontheuniverse.org/2009/12/18/lets-ban-english-in-school-except-in-english-class/

If you do can you help spread the word? You in fact might like Blog on the Universe given the blend of math, science and education.

Jeff Goldstein, Center Director
National Center for Earth and Space Science Education

7. June 5, 2010 9:45 am

Very Interesting Post. Thanx :)