ng. (inside joke, see part one.)
We’re trying to find a reasonable definition for
with real numbers. Continue with the previous strategy – demand the new definition satisfy the normal properties of exponentiation, and see what this leaves you with. The properties I’ll use are:
We can already do , so we’ll just focus on
I did that funny thing with changing the base to just because I want to focus just on complex exponents, and not have to worry about what the base is. If I understand how complex exponents work for one base, I understand them all. I chose as the base because I know where we’re going with this, and it works out well in the end, but you can imagine that is a nice base to work in for exponentiation because, after all, it is the natural logarithm.
Let’s define a new constant so that
It’s true that we expanded the domain of numbers before to allow the complex numbers to extend the real numbers. That let us solve the problem
but if we expand the complex numbers to allow a new type of number beyond that, just so we can answer
where will the process end? (In fact, mathematicians and physicists sometimes expand the complex numbers to things like quaternions and spinors, both of which have some applications in physics, especially mechanics, which I don’t particularly understand.)
Let’s assume that is a complex number of some sort. Then in general
with real-valued functions of . We’ll occasionally denote them by just .
The rule that states
Introduce a new complex exponential and multiply it by the one we already have.
But we should be able to multiply them the FOIL way, too.
Equating the real parts and imaginary parts of the two equations (which is clearly legal given the ordered pair interpretation of complex numbers) gives
Those are just the angle addition formulas for the sine and cosine function. Perhaps we should define
One check on whether this is a good definition is to see if the initial values work out, since we previously identified
That’s all good. The other property we should check is whether
That is just De Moivre’s formula, which can be (and originally was) proven without any calculus. Just use induction or the binomial theorem or something.
I suppose there’s a gap in checking this property, because the binomial theorem can only prove De Moivre’s formula for integer . But we could probably get around this by thinking of as a rational, using repeated multiplication in the numerator and De Moivre’s theorem together, and then taking everything to the power of the denominator to show that it all works out. I’m getting tired and won’t lose sleep over it.
So we have a nice guess, which is that (as it is normally written)
It’s called Euler’s Formula. It’s great because the trig identities are built into it (that might not be the only reason it’s great, but it’s a decent reason for students to like it). For example,
which is the Pythagorean theorem in trigonometry. I exploited that the cosine function is even and the sine function is odd, and simplified the algebra in one step.
We also have
which you can verify by simply plugging in the Euler’s formula. That can make calculus with trig functions much easier.
The question remains whether this is the only viable answer. There might, after all, be other definitions of that also satisfy all the conditions we put on them.
I never realized it before writing this post, but there are other such functions. For example, consider
Those functions will have all the right properties – initial values, adding exponents and multiply exponents-to-exponents. It has some mildly unaesthetic deficiencies. For example,
Which is certainly strange, but not illegal. So I can’t claim to have nailed down Euler’s formula completely without calculus, but we can narrow it down to a class of functions, of which Euler’s formula is obviously the most natural (you can actually prove it must be
for some by considering and , showing with the angle addition formulas that this completely fixes , and finally showing that it corresponds to some choice of in the desired form. I won’t bore you with the details.)
With that in hand, you can take the polar representation of points in a plane to write an arbitrary complex number as
Of course many values of will fit the above prescription, so we’ll restrict to
which shows that the complex numbers are closed under exponentiation (unlike the reals). For example
Finally, be careful with zero.
but is undefined. In fact
is similarly undefined, because it never settles down, but just races faster and faster around the unit circle.