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e to the pi times i

I like math, but might not know as much about it as some real math nuts.  I was reading the web comic XKCD, which has a lot of math jokes.  I read this comic titled e to the pi times i (language warning) which says if you take e to the power of ? times i or e^(?*i) gives you negative one.  That seemed really odd to me.  First thing I did was put the formula into Google and see if it gave the same answer, and it did.  The math really bends my mind, so I wanted to share it with everyone else.

So lets look at the constants in the formula:

  • ?: Most people are familiar with ? (or pi), which is the ratio to a circle to its diameter.  It allows you to move between the diameter of a circle and its circumference.  It is an irrational number, meaning it cannot be represented completely in decimal notation.  It is approximately 3.14159265. . . .
  • i:Is the imaginary result of the square root of negative one, or i^2 = -1.  Since a negative number multiplied by a negative number results in a positive number, without imaginary numbers, this would impossible.
  • e: The one less people are familiar with is is Euler’s number or e.  It is the base of the natural logarithm.  It is also irrational with the approximate value of 2.71828 18284 59045 23536. . .

Mathematically speaking ?, e and i are considered some of the most important constants along with 0 and 1.  Pretty exciting eh?

So it turns out this formula is called Euler’s identity.  I still don’t understand how it works though.  According to Carl Friedrich Gauss, since this formula is not immediately apparent to to me (as a student), I will never be a first-class mathematician.  That is OK.  I just enjoy math as a hobby right now.

2 Comments

  1. Juan

    I also was perplexed at first with this beautiful identity. To understand it first you have to understand the Euler first derived e^(xi)= cos(x) isin(x).
    He arrived at this by using the accepted taylor polynomial for e^x= 1 x x^(2)/(2!) x^(3)/3! x^(4)/4 and so on. So, being as confident in his abilities as he was, Euler simply let x = xi. Then, as you probably noticed now, he let x = pi and got the extraordinary definition.

  2. ActualRandy

    I wasn’t familiar with that equation either, and I have a BS in math! But there are a ton of topics in math, and I only was exposed to some of them. Since Euler’s Identity is part of algebra, and I never took non-abstract algebra in college, it never came up. Looks cool – thanks!

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