eπi + 1 = 0 is one of the more interesting mathematical equations since it takes 3 different number concepts, the natural log raised to pi times i, the imaginary constant. Simplifying this expression leads to a simple answer of -1. Now I am pretty nerdy, but this seems pretty amazing to me.
- Three of the most important numbers can be put in an expression to equal -1
- How a number raised to a power can be a negative
- How the proof is done with elementary functions
Because of all this, the concept of Math has intrigued the last few days, so I thought I would put my own explanation up on here and spread some math knowledge to all.
First off, I am going to explain what each of the numbers involved is.
e is the base of the natural logarithm. e can be expressed two different ways. The typical definition is the following:
However, it can also be defined as:
Both of these will lead to a series which goes on to infinity and will lead to an irrational number:
π is the ratio of a circle's circumference to it's diameter. This is an irrational number and is equal to:
i is defined as the following:
Because you cannot typically take the square root of a negative number, the idea of i or the imaginary digit was introduced. For example, the square root of -9 would then be 3i.
Using the definition:
Now using the definition of i:
Rearraning the equation, we get:
Using the following substitutions, from the Taylor Series: