Comments on: Humpty Dumpty’s Math Puzzle

By: Mike H

Mike H — Tue, 10 Apr 2012 00:40:34 +0000

“Sacrificing 0^0=1 would be as unsettling to me as sacrificing 2^2 = 4″

f(x,y) = x^y, x,y\in R^+ doesn’t have a well defined limit as (x,y) -> (0,0). If you take the limit along paths approaching (0,0), you might get 1 if you choose the path along y=x, but you might get 0 if you chose a different path, or in fact any positive number, or infinity, or no well-defined limit.

Choosing 0^0=1 makes sense when discussing the function f(x)=x^x, but it makes g(x)=0^(x^2) look rather strange.

By: Steve Landsburg

Steve Landsburg — Sat, 07 Apr 2012 17:52:33 +0000

KS: It’s still a function, I think. To make it a true function, you’ve got to pick one value for each x. As you point out, there’s an obvious way to do that when x is positive, but there’s not when x is negative.

By: KS

KS — Sat, 07 Apr 2012 01:19:23 +0000

Btw it’s been a while since I’ve taken complex analysis, but I think (1/2)^(1/2) is defined as the positive solution only. For instance, there are two distinct solutions to x^2=4, but 4^(1/2) is defined as 2 (and not -2). Does that make sense?

In addition, for -x, the function f(x) = x^x yields complex values sometimes, so the curve is actually shooting up into the z-axis (not depicted in your image) and then coming back into the real-number-only xy plane at times. It’s still a function, I think.

By: KS

KS — Sat, 07 Apr 2012 01:15:05 +0000

Check out the plot of f(x) = sin(e^x).

If you saw that curve alone first, would you guess it had a simple equation associated with it?

By: v

Fri, 06 Apr 2012 22:00:06 +0000

Wolfram’s Mathematica favors a slightly different solution (or gets a B- for the real-valued plot):
http://www.wolframalpha.com/input/?i=x^x

By: Ken B

Ken B — Fri, 06 Apr 2012 16:01:00 +0000

@Ken Aromdee:
There’s a natural hierarchy. Start with Peano Axioms, define N, define Z, define R, define C – all unique in a very strong sense. Define a^b on N and extend it.

By: Ken Arromdee

Ken Arromdee — Fri, 06 Apr 2012 14:37:29 +0000

@Steve: What is a definition?

There are mathematical concepts which are related in certain ways, such that if you pick enough of them as given you can describe the others in terms of the ones you picked. But which ones you pick are arbitrary. I could understand * but not understand / , and explain / in terms of *. Or I could understand / but not *, and explain * in terms of /. Neither one is “the definition” as if some were more elementary than others and there is some order in which you must understand them.

By: Ken

Ken — Fri, 06 Apr 2012 06:49:04 +0000

Keshav,

The number b=a^(1/n) is defined as any number such that b^n = a.

Thus, (-1/2)^(-1/2) doesn’t really make sense if you don’t use the definition e^(xLog(x)). After all there is no rational or real a such that 1/(a^2) = -1/2.

For for all positive rational x, x^x makes sense and then x^x = e^(xlog(x)) through the basic properties of rational exponentiation (LR Theorem 1.21), the exponential, and the logarithm (LR pages 178-182). Extending to negative numbers, you have to resort to Log (the complex logarithm) and x^x can really only be reasonably defined as e^(xLog(x)).

Lastly, I’m not sure your definition makes sense. 1/2^1/2 is not rational. Just because xn is rational and you can construct an irrational x from rational Cauchy sequence doesn’t mean you can do that same with xn^xn. Even when xn is rational, xn^xn is not guaranteed to be rational (in fact it almost never will be and if any xn is negative, then it’s not even real), so you can’t define x^x, for real x, as the limit of a rational Cauchy sequence.

*LR = Principles of Mathematical Analysis by Walter Rudin

By: Ken B

Ken B — Fri, 06 Apr 2012 05:09:55 +0000

@Keshav: I’m not sure that works well when you toss i into the mix.

By: Keshav Srinivasan

Keshav Srinivasan — Fri, 06 Apr 2012 04:43:43 +0000

Steve, you say “any plausible definition of x^x is going to look a lot like e^(x log(x))”. But I can think of one definition that doesn’t look like that: define the exponentiation of real numbers a^b in terms of exponentiation of rational numbers and Cauchy sequences of rational numbers.