Euler’s Identity explained and the hope of order within chaos.
Euler’s Equation gives me hope.
When I first saw it, it was written on a white board of my Calculus II class by a man I knew had written a lot of love letters in his life. He stepped back from the board after making quite the dramatically understated gesture, he pointed at it with his Expo marker and said,
“My God, that’s a sexy equation right there! Just really, pure elegance. “
I was hooked.
Euler’s equation connects the five most basic fundamental constants in the mathematical world in such a simple way that it just astounds and delights.
1 and 0 everyone knows. These are basic integers that refer to the presence of nothing or the presence of something. Everything about our counting and computing is based on 1s and 0s.
pi seems to have its own sex appeal all on its own. The ratio of any circle’s circumference to its diameter (3.1415….), this number is both constant and infinite and even has its own day, which I celebrate with as many round delicious food items as possible with other nerds. It’s infinite-ness makes it irrational. It’s simplicity makes it irresistible.
e is another irrational character, with a much less tantalizing source. Called Euler’s number, It falls out of a calculation of a line held tangent to a curve, and its true for any curve and line tangent to it anywhere. Sometimes extra nerdy folk refer to as the base of a natural logarithm. While true and constant, it is also infinite (2.718…) and therefore another irrational bugger.
i is one of the mysteries of math to me. It stands for the square root of -1 (√−1). Now, any person who knows that a square root is a number which multiplied by itself will reach the desired result, namely i²= −1, will see the problem here. There is no real number you can multiple by itself to get -1, because multiplying a negative by a negative will always produce a positive. So, our friend i is beyond irrational, it’s literally imaginary. It falls out of real equations based on real life engineering, but it’s indeed just an imaginary concept that lets mathematicians continue solving problems. (Don’t spend a lot of time on this one, it will drive you to smoking…