teaching:topics:number:euler_number [Simonweb wikinotes]

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======Euler's number======
We give it a name, \(\ \large\rm e\ \) because it turns up in so many quite different mathematical contexts.

\(1,\,\pi,\,0,\,{\rm i}\) and \(\rm e\) are considered __the__ fundamental numbers. I have listed them in the order they became part of mathematics historically, there was around 4000 years of mathematical thought and development, all across the globe, between understanding the importance of the ratio `pi` and the number \(\rm e\) (ideas are easy to carry along trade routes, and merchants are of course rather interested in numbers!). The equation
\[\large{\rm e}^{{\rm i}\pi}+1=0\]
links all five --- __after__ we extend our number system to include the complex numbers by defining the number \(\,\rm i\,\) such that\(\quad {\rm i}^2=-1, \quad\)it is a special case (when `x=\pi`) of what we call **Euler's formula**
\[\large{\rm e}^{{\rm i}x}=\cos x + {\rm i}\sin x\]
which is at the heart of understanding what \(\ \large a^z\) must mean when `z` is a complex number. 

Euler lived in the mid 1700s, the idea of a 'function' was being developed. \(\ {\rm e}^x\), \(\ \ln x\,\) and trigonometry considered as functions were being explored --- partly through the idea of power series. Taylor was before him, after Newton, the ideas of calculus were being refined and extended --- it was becoming both more rigorous and more general as many generations of mathematicians worked on it. Euler's Master of Philosophy dissertation, 1723 at age 16, compared the philosophies of Descartes and Newton. He introduced the notation \({\rm f}(x)\) and developed the power series for the exponential function ...
\[\large{\rm e}^x=\sum_{n=0}^\infty\frac{x^n}{n!}\]
Which, using the properties of \(\rm i\) along with the power series for `cos x` and `sin x`, gave the formula named after him. There are other approaches to that formula, and this combination of different ways to get there shows how deeply it crosses many areas of mathematics --- bringing algebra, trigonometry, geometry and number together when we embrace complex numbers as our underlying set of numbers.

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