What makes e so special?

[kmath]

Hello everyone,

 

For as long as I have known about the constant [latex]e[/latex], I have been in awe of its many uses and at times strange properties. Now I know its definition as a limit and as a series, but I have never quite understood just what makes [latex]e[/latex] so special, beyond the fact that it helps us solve problems. So my question is: what makes [latex]e[/latex] so significant to the overall study of mathematics?

 

I hope that makes sense. I look forward to others perspective on this.

 

Thanks

[imatfaal]

[latex]e^{ix} = \cos x +i \sin x[/latex]

 

and the beautiful identity when x is set to pi

 

[Nehushtan]

Well, the most obvious special property is that the function [latex]e^x[/latex] is its own derivative and antiderivative (indeed the only such functions are given by [latex]f(x)=Ae^x[/latex] where [latex]A[/latex] is a constant). This property makes it extremely useful in many areas of mathematics, for example techniques for solving differential equations.

The trigometric and hyperbolic functions for complex numbers are related to those for real numbers via [latex]e[/latex]. Many other important properties of [latex]e[/latex] are listed in the Wikipedia article on it.

[Amaton]

As the notion says, it's not so much about [math]e[/math] itself, but rather [math]e[/math] raised to a power. And from this, we see the obvious importance of the exponential function and the natural logarithm.

 

This is also neat I think. Consider [math]e^k[/math] where [math]k[/math] can be any imaginary number. If we look at values of [math]e^k[/math] for all imaginary numbers, we see that these values form a unit circle in the complex plane. This is in my opinion one of the more beautiful and fundamental ideals regarding [math]e[/math].

[staffingby]

hey, is the unique number with the property that the area of the region bounded by the hyperbola , the x-axis,

and the vertical lines and is 1. In other words

[kmath]

Thanks everyone!