Math is Beautiful


Mathematics has always been beautiful to me. It has a certain purity in it when explaining the universe that allows no misunderstandings or objections, only truth. In my studies of mathematical series, I’ve finally came across Euler’s Formula and Euler’s Identity. The results of these concepts are truly profound and ought to be understood by every human being on this planet. For some reason, they are hidden away behind complex mathematics, but the logic behind it is rather simple. I wonder how education would change if concepts like these were presented to younger minds from the get-go. Would more people be interested in mathematics and thus, science? I digress…

Euler’s Formula
Euler’s formula can be derived:
1. Using the Maclaurin Series of cos(x) and sin(x).

cos(x) = 1 - x^2/(2!)+x^4/(4!)-x^6/(6!)+x^8/(8!)-x^10/(10!)...

sin(x) = x - x^3/(3!)+x^5/(5!)-x^7/(7!)+x^9/(9!)-x^11/(11!)...

2. Using the Maclaurin Series of e^x

e^x approx 1 + x^2/(2!)+x^3/(3!)+x^4/(4!)+x^5/(5!)+x^6/(6!)...

3. Using the imaginary unit for Maclaurin Series of e^ix

e^ix approx 1 + (ix)^2/(2!)+(ix)^3/(3!)+(ix)^4/(4!)+(ix)^5/(5!)+(ix)^6/(6!)...

By applying the rules of imaginary unit, i, the Maclaurin Series of e^ix can be rearranged to form…

e^ix approx (1 - x^2/(2!)+x^4/(4!)-x^6/(6!)+x^8/(8!)...) + i(x-x^3/(3!)+x^5/(5!)-x^7/(7!)+x^9/(9!)...)

4. Therefore, e^ix = cos(x)+isin(x)

This has plenty of further uses in math, but taking a ‘poetic’ dive into math, let us plug in x=pi

5. e^ix with x=pi




This is Euler’s Identity.

What is profound about this? Well, you are relating Euler’s number (e), an important mathematical constant in finance (namely compound interest), with pi which relates a circle’s circumference to its diameter, with imaginary unit i, which is commonly used in engineering disciplines to find roots of polynomials. You relate these three unique constants of mathematics with the two most simple numbers of one and zero. You are connecting vast areas of life together in one beautiful and simple mathematic equation.

If this doesn’t make you feel emotional in any way, then I don’t know what will. This tells us that the universe, in some way, is connected.

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