If you read "The History of a Number" by Eli Maor, you will realize that the main thing is the exponential function, denoted as exp, which can simply be described as a function that is its own derivative. "
Another way to say this, is that exp is a solution to the simple differential equation y′=y . As such, it is a building block for solutions to differential equations of all kinds.
Differential equations describe how the change in some quantity relates to the quantity itself. They describe how the universe works at all levels - from the most microscopic and fundamental, to the macrosopic. This includes purely abstract mathematical concepts which have no direct ties to phenomena in the physical universe.
So it is no surprise that the function that is the building block for solving differential equations comes up very often.
Because the function is so important, we want to know more about it. One question of interest is – what is the value of exp(1) ? This is useful, because one of the properties of exp (which we can prove using the definition we started with) is that exp(x+y)=exp(x)exp(y) . Using this, we can show that exp(n)=exp(1)n for every integer n (where taking a power is a simple repeated multiplication). In other words, knowing the value of the function at 1 allows us to find its value for every integer. So we give the value of exp(1) a name. The name we choose is e.
That’s what e is – the value of the exponential function at 1. The importance of e can be understood by understanding the importance of the exponential function, which itself can be understood by understanding the importance of differential equations.
You can also read more about this on M. Rosenfeld's
blog.
