Pi day surprise: i to the i with e and pi by Pinkie Pie · 1:08pm Mar 14th, 2016
Hello dear readers. Me again! I’m here because today is pi day (π=3.14) (or tau/2 day if you define your circles that way). So it’s time for another totally irrational Pinkie Pie special mathematical blog post.
There are lots and lots of fun pi-themed games you can play for pi day, but I’m not going to talk about those, as you’re probably getting bored of that sort of thing and this is a serious post about advanced mathematics for super-smart people and ponies like me. So instead we will continue our discussion of imaginary numbers and other numerical friends. To start you might want to reread the bit about imaginary number in my last post: The Pinkie Pie Guide to Imaginary Numbers and Euler’s Equation.
In this post we are going to calculate the value of i to the power of i, where i is the unitary imaginary number, that is, the square root of minus one. And we want to raise it to the power—that is multiply it by itself—i times!
You might find this taxes your imagination just a little bit. If you find it difficult you might want to practice by first trying to imagine something easy like imagine an ice cream as big as a pony. Then once you can clearly picture a mountain of strawberry/mint/vanilla/chocolate then you can move on to a slightly harder exercise of imagining something abstract like the square root of minus one.
Then once you can hold the concept of 
in your head, you can think about how to calculate it. If you don’t mind spoilers then type i^i into Google and it gives you the answer.
Now how to we do this? If you read my last post closely, you will recall that just as 
(Euler’s Equations), 
. Remember?
So we can write as 
and then we can use what we know about powers (
), and the fact that 
to show that this is 
.
Which is totally real (no imaginary part whatsoever) and we can show it is 0.2078795763507619. So is a little bit more than 1/5.
Happy Pi(e) Day!
See also:
The Pinkie Pie Guide to Irrational and Transcendental Numbers
The Pinkie Pie Guide to Imaginary Numbers and Euler's Equation
Whee! Have a pi pie to go with this post.
upload.wikimedia.org/wikipedia/commons/d/d4/Pi_pie2.jpg
Huh. An imaginary raised to an imaginary power results in something completely real. Go figure. That would certainly explain how Pinkie got crayon directly on the blog.
to much math it hurt brain
Annd, theres me last night going round the long way using logs, exponentials and trigonometrical functions.
Another very close value to another useful one.. the Golden Ratio uses square root 5, so I wonder how much variation in design it would be with this. Then again, 4% variation in 5, so 2% variation in golden ratio? How many different values are percentage close?
Theres a collection of combinations of constants and operators somewhere?
Well would you look at that.
I like how she switched pink crayons since the last blog post
Infinite series are so cool.