i to the power of i is…an infinite number of real numbers?

Hi all! I’m back after two hectic weeks. I’m tired and super busy, but looking forward to this post. I also have a Cambridge interview on December 4th so pray for me/wish me well! It’ll probably be over by the next time I post…

The Euler formula

So today’s maths hinges on my last post, in which I delved into the Euler formula. The more I’ve thought about it since then, the more I absolutely love it. It goes like this:

image

To understand what’s going on, have a read of The Euler formula: when e starts drawing circles.

So how does this relate to i raised to the power of i?

Well, let’s try to get i out of the equation. To do this, we can start with the right hand side and say that cosx + isinx = i. If x is a real number then cosx is real and isinx is imaginary so cosx must equal 0 and sinx must equal 1. We see that this happens for a number of values of x as shown below:

file

adding 2π each time.

Therefore, we can say:

file2

So then:

file1

and so on for all the values of x.

These are a series of real numbers, with i to the i being a multivalued identity.

Wait…what?

Should we be surprised that i to the i is multivalued? After all, i in itself is clearly defined as one number. How then can it produce an infinite number of values when we step into exponentiation?

Well, it shouldn’t be that surprising. We know that 1 is obviously defined as taking only one value, but its square root, or any even root, is multivalued, producing 1 and -1. This, too, can be explained using the Euler formula:

file-1

where we get 1 and -1 repeating periodically forever, which I think is pretty incredible.

The more I read into the Euler formula, the more impressed I am at just how dynamic and ubiquitous it seems to be. It shows that a seemingly simple piece of maths can have a profound impact, tying together different branches of maths into one elegant conclusion.

Credits

A big thank you to:

Note: I’ve also added a quick edit to my last post concerning how e allows us to define what is meant by an irrational power.

Come back in a fortnight’s time when we’ll talk about the number of possible configurations of a Rubik’s cube (since I tried this myself and got wrong…I’ll have another go and get back to you). See you then;)

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