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:
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:
adding 2π each time.
Therefore, we can say:
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.
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:
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.
A big thank you to:
- Matt Samuel, who hinted at the multivalued nature of root1 and e, and Micah, who gave an awesome explanation of multivalued stuff to do with powers that I hope I will be able to fully understand one day… It hints at the really cool nature of maths, though. Both submitted their maths on https://math.stackexchange.com/questions/2161103/high-school-explanation-to-why-is-ii-multi-valued-if-i-is-apparently-a-cons.
- https://www.math.hmc.edu/funfacts/ffiles/20013.3.shtml, which confirmed to me that i to the i is indeed real.
- The people at standup maths for https://www.youtube.com/watch?v=9tlHQOKMHGA, which I watched two weeks ago. It was great, although they didn’t explain about the multivalued stuff, which solidified my resolution that I was going to find an answer, and hence this post was made…
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;)