Mathematics

The numbers e, i and pi are related. Is this natural or a consequence of the way we do our mathematics? Iain Nicholson

Comments

I'm not sure what's meant by "natural" here. But the numbers e, i, and π are the numbers they are, and are related as they are, quite independently of how we choose to do mathematics, just as the stars are hot balls of fiery gasses whether or not anyone regards them as such. Whether Euler's Equation (see question 393) plays any important role in our mathematical theories is, on the other hand, a consequence of how we choose to formulate them, and so one might ask why we should choose to formulate mathematics as we do instead of some other way. But again, this question is no different, in principle, from the question why we should accept the astronomical theories we do. Mathematicians have, and can give, good reasons for formulating analysis in the way they do, and there are sometimes disagreements about how best to proceed. These disagreements get resolved (or not) on broadly mathematical grounds. One or another formulation leads to a fruitful way of conceiving the problem space; others do not, or do so less well.

I assume the relationship you have in mind is eiπ= -1. I wouldn't go quite as far as Richard in saying that this is"independent of how we choose to do mathematics," because it doesdepend on how we choose to define exponentiation for complex numbers.Mathematicians were faced with a choice when they were deciding how to defineexponentiation with complex numbers, and there was no one definitionthat was the unique "right" definition. However, the definition thatmathematicians chose is a very natural one, and once you choose thatdefinition the relation eiπ = -1 follows.

Perhaps it would be of interest for me to give an explanation, forthose who are familiar with power series, of why the definition ofexponentiation with complex numbers is natural. If you studied powerseries in a calculus class, then you are probably familiar with thefollowing formulas:

ex = 1 + x + x2/2 + x3/3! + x4/4! + ...
cos(x) = 1 - x2/2 + x4/4! - ...
sin(x) = x - x3/3! + x5/5! - ...

Now, mathematicians could have defined ez, where z is a complex number, however they pleased, but as was suggested in question 324, mathematicians often try to make their definitions in a way that preserves general rules. If we want the formula for exgiven above to be preserved when we go from real numbers to complexnumbers, then we will want to define exponentiation with complexnumbers so that:

eix = 1 + ix + (ix)2/2 + (ix)3/3! + (ix)4/4! + ...
= 1 + ix - x2/2 - i(x3)/3! + x4/4! + ...
= (1 - x2/2 + x4/4! - ...) + i (x - x3/3! + x5/5! - ...) = cos(x) + i sin(x).

Dan says that he wouldn't say that whether eiπ = -1 isn't "'independent of how we choose to do mathematics' because it does depend on how we choose to define exponentiation for complex numbers". But to what does the emphasized "it" refer? Presumably, to the claim that eiπ = -1. But we really should not say that whether eiπ = -1 depends upon how we have chosen to define exponentiation. Whether "eiπ = -1" expresses a truth depends upon how we define exponentiation, but whether eiπ = -1 does not, just as whether "3+4=7" expresses a truth depends upon what "3", "4", "7", "+", and "=" mean, but whether 3+4=7 does not.

Richard makes a good point, but I still think that I had a good point also, although I may not have expressed it as well as I might have. It is often said that Euler proved that eix = cos(x) + i sin(x), but it seems to me this is somewhat misleading. Many (most?) modern complex analysis books present Euler's equation as a definition, not a theorem--it is the definition of eix. On this view, what Euler did is not to prove that eix is equal to cos(x) + i sin(x), but rather to show that it would be a good idea to define it that way. If you want to regard Euler's equation as a theorem, then you have to take something else to be the definition of eix. You could do that--you could, for example, take the power series for ex, which was derived for x a real number, and extend it into the complex numbers to define ez when z is a complex number, and then Euler's equation would be a theorem. But this requires developing much more of the theory of complex analysis (one has to work out the theory of power series with complex numbers before one can understand ez), which is most likely why many textbook authors don't choose to follow this approach. What approach did Euler take? I'm not a historian, but I suspect Euler didn't worry too much over these details. He probably just did his calculations without worrying too much about what justified them (but of course with an uncanny intuition for when it would be fruitful to calculate in a particular way).

It is not always so easy to say what's a definition and what's a theorem. All mathematicians agree that Euler's equation and the power series representation of ez are correct, but authors of complex analysis textbooks have a choice about which is the definition and which is the theorem. But they can't both be theorems--unless, of course, some third fact about ez is taken to be the definition.

I suspect that some readers of this web site may have the following view: We all learned what exponentiation means (for real numbers) when we were in high school. When we then go on to define the complex numbers, and we define addition and multiplication of complex numbers, then there is a unique correct answer to the question of what exponentiation with complex numbers means, even before we have made a definition, and Euler discovered what that correct answer is. My point is that that view is mistaken.

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