Amazing Numbers in a Complex World

(This is for Jordan because he appreciates really cool, weird looking math writing. I do too!)

ε^iπ = -1

What the heck does that mean? What is ‘ε’ ? Why are ‘i’ and ‘π’ written above it? This is so strange. Did Martians write this? When were they here? Oh, let’s worry about that another day shall we?

ε is a letter in the Greek alphabet. It’s name is pronounced “epsilon.” Sounds like a good name to give your dog.

π is another Greek letter. It’s name is pronounced “Pi”, sort of like “pie” but pie are round and

πr² . (That’s a dorky math joke.)

‘i’ is just the 9th letter of the good old American (actually Arabic) alphabet.

Each of these letters is used to represent some very peculiar numbers. We use letters because the numbers we want to talk about are hard to write. For example, take π. We all learn this number in grade school. It’s value is 3.14. Well actually, it’s more like 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 …. PHEW! Are you kidding me? Hey I have an idea. Let’s call it π.

But what’s this ε ? Another whopper. 2.71828 18284 59045 23536 02874 71352 66249 …
JEEZ LOUISE! You’re killing me already!

But the real puzzle, the real ENIGMA" is ‘i’. (I like that word "enigma". It's kind of sexy and it makes me sound smart because lots of people don't use it much.) Its value is simple to write. The value of the number it represents is √(-1).

What’s so puzzling about that?

Don’t you know anything? Don’t you pay attention in school? Everyone KNOWS that you just can’t take the square root of a negative number. Why? Because any number multiplied by itself will always have a positive number for a result. A negative number times the same negative number will be a positive number. So, if there is no number that you can “square” and get a negative number, how can you have a negative number to take a square root from? Hey, go get a glass of juice or take a bike ride or something and stop hyper-ventilating. Your mother always said it was not healthy to be so "negative", (Yuk! Yuk!) Ahem! Sorry.

You're back? GOOD!

Oh! By the way, we use ‘i’ because it’s stands for “imaginary.” That would be a very good description of a number like √(-1)! But, trust me, ‘i’ is very real (shouldn't that be "'i' am very real?" Never Mind!), which is to say, it truly does exist. Actually, “real” and “imaginary” numbers are two sub-classes of something called “complex” numbers. That’s not too surprising after what you’ve read so far! But I digress. In fact ε^iπ  is a complex number.

These numbers are just mind boggling. They have “magical” properties.(1)

If you take ε and raise it to the power of i multiplied by π, the result is a very reasonable , easy to understand and to WRITE, -1 ! Wow! How does that work?

Well, grasshopper. Trust me. It does. I was first told about this mystery when I was 18 years old in my first year of college calculus. I did not understand how this worked until many years later.

But IT IS SO COOOOOOOL!

1 "Any sufficiently advanced technology is indistinguishable from magic." Arthur C. Clarke, “Profiles of the Future”, 1961 (Clarke’s third law)