Euler’s equation explained, part 1

Posted on Monday 25 August 2008 at 7:35 pm by Jacob Aron
In Mathematics

In the very first post on Just a Theory I mentioned Euler’s equation, considered by many to be the most beautiful equation in the whole of mathematics. I decided to share with you everything I know about this wonderful equation, e + 1 = 0. Let’s start with some of the more unfamiliar elements.

π: You’ve probably met π, written Pi and pronounced “pie”, before and perhaps you remember that it is the number you get when dividing the circumference of a circle by its diameter. This neat little Wikipedia animation demonstrates the principle. Pi has an infinite number of digits because it is an irrational number; this means that it cannot be represented by a simple fraction such as 1/2 or 365/789. I’m hopeless at remembering digits of Pi and normally stick to 3.14.

i: What is the square root of 4? In other words, what number must you multiply by itself to get 4? The answer is of course 2. What about the square root of -4? It can’t be -2, because -2 is also the square root of 4 – since multiplying two negative numbers results in a positive number. The answer is 2i, because i is defined as the square root of -1. i is an imaginary number – but that doesn’t mean it’s just made up. For example, many problem in electrical engineering can only be expressed using i.

e: Another irrational number like π, e is the base of the natural logarithm. What that means isn’t really important in this context – just think of it as another important mathematical constant like π. The numerical value of e is approximately 2.718.

Exponentiation: You should be familiar with the concept of raising one number to the power of another, such as 23 = 2 x 2 x 2 = 8. This is exponentiation. What does the strange beast e represent then? If you are trying to multiply e by itself times you might be left scratching your head, but all will become clear as Euler’s equation is explained.

What with it being a bank holiday, I’ll give you a break and leave it there for today. Hopefully you’ve understood these building blocks, and your ready to tackle the full equation tomorrow. If not, leave a message in the comments and let me know if you need further clarification.

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