kmath Posted March 14, 2013 Share Posted March 14, 2013 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 Link to comment Share on other sites More sharing options...
imatfaal Posted March 14, 2013 Share Posted March 14, 2013 [latex]e^{ix} = \cos x +i \sin x[/latex] and the beautiful identity when x is set to pi Link to comment Share on other sites More sharing options...
Nehushtan Posted March 14, 2013 Share Posted March 14, 2013 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. Link to comment Share on other sites More sharing options...
Amaton Posted March 18, 2013 Share Posted March 18, 2013 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]. Link to comment Share on other sites More sharing options...
staffingby Posted April 1, 2013 Share Posted April 1, 2013 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 Link to comment Share on other sites More sharing options...
kmath Posted April 1, 2013 Author Share Posted April 1, 2013 Thanks everyone! Link to comment Share on other sites More sharing options...
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