Mathematics Zone: Understanding the most beautiful equation in Mathematics

Euler was one of the most influential and prolific mathematicians in history. He had published over 800 papers and 20 books, making him the greatest contributor in mathematics. Referred as the Mozart of Mathematics, Euler left hardly any area of Mathematics untouched, contributing to various field like mathematical analysis, number theory, mechanics and hydrodynamics, cartography, fluid dynamics and topology. In this article, we'll try to understand the most beautiful equation in all of mathematics:

e^{i π} + 1 = 0

It connects the five most important constants of mathematics and three most important mathematical operations - addition, multiplication and exponentiation. So, how did Euler arrived at this result?

The Euler's constant e is defined as

lim_{} {(1 + \frac{1}{n})}^{n}

as n approaches infinity.It's approximate value is equal to 2.71828. In his most influential work, Introductio in analysin infinitorum, Euler defined the function e^x in analysis as:

e^{x} = lim_{} {(1 + \frac{x}{n})}^{n}

as n tends to infinity. So, we get:

e^{x} = 1 + \frac{x}{1!} + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + . . .

This is the known series for

e^{x}

Euler's brilliant mathematical mind replaced the real variable x with ix were i =

\sqrt{- 1}

So, we get:

e^{i x} = 1 + \frac{i x}{1!} + \frac{{(i x)}^{2}}{2!} + \frac{(i {x)}^{3}}{3!} + . .

We know that square of i is equal to -1. So, replacing subsequent values for

i^{3}, i^{4}, i^{5} . . .

, we get:

e^{i x} = 1 + \frac{i x}{1!} - \frac{x^{2}}{2!} - \frac{i x^{3}}{3!} + .

On separating real and imaginary parts, we have:

e^{i x} = (1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + . . .) + i (x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + . . .)

So, we got two trigonometric series of

c o s x

and

s i n x

respectively. Hence,

e^{i x} = \cos x + i \sin x

If we put,

x = π

, we get:

e^{i π} = - 1 a s \cos π = - 1

and

\sin π = 0

e^{i π} + 1 = 0

So, this was the story of the creation of most beautiful equation in mathematics. I would like to end this article by a quote from professor Keith Devlin - "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's Equation reaches down into the very depths of existence."

Mathematics Zone

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Thursday, April 21, 2016

Understanding the most beautiful equation in Mathematics

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