1

Here's a question that's bothered me ever since highschool, and I've never heard a good answer.

I know that mathematicians can define operators to mean whatever they want, as long as their system of mathematics is self-consistent. But some definitions are less useful than others when one wishes to use math to model our physical world.

And so my question regards the soundness of using non-integer exponents when modeling physical systems. I accept the validity of positive-integer exponent rules, such as $a^n \times a^m = a^{n+m}, n \in I, m \in I$, because this degenerates into straight-forward multiplication. And that, in turn, maps pretty well to grids of apples on a picnic blanket.

But why do people so readily (or at least without public discussion) accept that $a^x \times a^y = a^{x+y}$ even when $x$ and $y$ are fractions, irrationals, or even complex numbers? More precisely, why do people who are using equations to model physical systems not demand a justification for the validity of such a definition of exponentiation, before being willing to apply it in the manipulation of their mathematical physical models?

Christian Convey
  • 131
  • 1
    The rational case is pretty simple; you can just raise both sides to an appropriate power and see that the results are equal. If you take a course in analysis, you will likely prove the equality holds for real numbers. Most people are introduced to exponentials well before they're ready for analysis; similarly, we introduce multiplication without any formal justification that laws like distributivity hold, because elementary schoolers aren't ready for proofs. – user2357112 Apr 22 '14 at 17:30
  • @user2357112 It sounds like you're making a good case for how the generally accepted rules of exponentiation are consistent with the definitions of equality of reals, etc.

    But the part I'm missing is why someone should accept the soundness of using such a definition when modeling physical systems. That is, I don't see how we get from (a) the internal consistency of defining non-positive-integer exponentiation in this manner, to (b) believing that it's sound in manipulating the usual physical models.

    I suspect I'm missing something fundamental about mathematical modeling.

     – Christian Convey Apr 22 '14 at 17:32
  • 1
    This might help. http://math.stackexchange.com/questions/435751/proving-the-product-rule-for-exponents-with-the-same-base – RandomUser Apr 22 '14 at 17:35
  • @randomuser Thanks for that link, but I think it might miss my main point. I don't see why a physicist would ever consider it valid for any of his model input variables, which measure things such as position for example, to ever by raised to a non-positive-integer exponent in the first place. It seems to me that if/when the model did that, even as an intermediate step, there might be a serious question about the mathematical model continuing to be a valid model of the physical system in question. – Christian Convey Apr 22 '14 at 17:38
  • Do you have an example? I'm having trouble understanding what you mean. Is it that fractional exponents in nature don't make intuitive sense? When it comes to seeing if an equation fits a physical model, it's likely done case by case. – RandomUser Apr 22 '14 at 17:43
  • That's more of a dimensional analysis concern. It's entirely reasonable to raise units to negative powers (velocity is in dimensions of distance * time^-1) or fractional powers (area is in dimensions of distance^2, so distance is in dimensions of area^(1/2)), but if you ever add together distance and distance^2, you have a problem. – user2357112 Apr 22 '14 at 17:44
  • I don't understand what you mean by "validity." Certainly there are mathematical models you can write down under this assumption, certainly those mathematical models are internally consistent, and certainly it's an empirical fact that they model many interesting phenomena (bacteria growth being perhaps the least interesting, but at least a typical example). – Qiaochu Yuan Apr 23 '14 at 04:16
  • @ChristianConvey I'm confused -- as mentioned above, it's simply a fact that $a^x\times a^y = a^{x+y}$ holds over the real numbers. Is your suggestion that physical systems might not resemble the real numbers as much as we typically assume and that systems that we usually model with exponents should not be modelled in such a way until we verify that they do in fact behave like the real numbers? -- I think we usually do empirically verify that things resemble exponentiation on the real numbers before we slap an exponential model on them, but I can't speak for the physicists. – Jack Crawford May 23 '19 at 12:15

0 Answers0