Built on Facts

We can’t graph here, this is bat country! Complex bat country.

…well ok, let’s stop and take a look anyway. But no graph.

You’ve seen this jewel of mathematics. It’s Euler’s identity.

It comes from the more general expression right below this paragraph, which is today’s Sunday Function. You might wonder where this expression comes from. It’s a long story, but if you want to plow through it, I commend you.

Set θ = π, and Euler’s identity pops right out. But let’s set θ = π/2 instead and see what happens:

In other words, we’ve just figured out a new way to write the imaginary number i:

Now let’s raise that to the ith power.

“Wait, what? What does it even mean to raise a complex number to a complex power? Something like 2³ is easy, it means 2*2*2 = 8. Fractional and negative exponents are a little weirder since they involve roots and division, but conceptually they’re not that much of a stretch. But an imaginary number to an imaginary power? That’s crazy talk!”

Don’t worry. Just follow the rules and be consistent, and even this problem will fall to your skill. Write iⁱ, but for the lower i use that expression we just derived:

Now invoke one or two high school exponent rules:

And as the last step we can rewrite that term on the right:

And that, dear friends, amazes me even more than Euler’s identity. It is almost too numinous for words.

I can tell that you mathematicians are about to fire up your keyboards and note that my exponential expression for i is not unique due to the fact that I have failed to pin down the possibility of a 2πn phase in the exponent. True, but we can worry about that later. For the moment it’s enough just to be awed by the power of writing a complex number as an exponential.

Comments

#1 Pedvi
August 31, 2008

It doesn’t really matter that you have written only the principal value of the exponent, everything will be multiplied by i so all of them are real numbers, what is the amazing thing!
#2 Fergus Gallagher
August 31, 2008

Well here’s pie in your eye.
#3 Carl Brannen
August 31, 2008

The fine structure constant sqrt(alpha), is approximately equal to i raised to the power ln(i), which works out, (I think) to exp(- pi^2 / 4) = sqrt(1/139.04).

In fact, there’s an infinite series that nails it rather well. See the first few posts on this physics forums thread.

For an update to latest experimental data, see post #402 and succeeding posts, which show that the first 4 terms on the infinite series gives the current measurement to within experimental error.

What does this mean? I don’t know. My suspicion is that it is coincidence so I haven’t spent any time on it.
#4 Ron
August 31, 2008

Here’s my visual paean to the complex exponential function: 24 Views of the Complex Exponential Function
#5 IBY
August 31, 2008

Wow, that is really strange! I love these kind of stuffs. Anyways, how do you write the equations in that way, instead of doing it in one line?
#6 CCPhysicist
September 1, 2008

Definitely one of my favorite functions. (The Airy function is another.) BTW, I prefer the series expansion proof when I teach this in Physics 2 because it shows the power of that subject from Calculus 2. Most students don’t see any value to the Taylor series when they take calculus.

And I teach it in Physics 2 because it eliminates the need to ever use a trig identity when working with waves such as you deal with in AC circuits.

But the real test has to be whether your calculator is smart enough to pass that test. It is great fun to tweak students with ln(-1). Is their calculator smarter than they are? Usually the answer, when they see “not real” as the answer and aren’t even smart enough to change mode and see what that means, is YES.
#7 Science
September 2, 2008

Very less explored subject and you have provided valuable information. Thanks a lot.
#8 palmer
September 2, 2008

Careful, “high school exponent rules” are derived in a very specific environment. It still works but you need more justification.
#9 Moliva
December 2, 2011

t doesn’t really matter that you have written only the principal value of the exponent, everything will be multiplied by i so all of them are real numbers, what is the amazing thing!