THE MOST BEAUTIFUL EQUATION IN MATHEMATICS- EULER'S IDENTITY

We have come across many equations and formulas in mathematics, but nothing can match the beauty and elegance of Euler's identity.

Before we dig deep into Euler's identity, we need to know some basic fundamental 'constants' in mathematics. Namely, the Euler's number, pi and imaginary unit, represented by, e, 𝝅 and i, respectively.

Euler's number is the base for natural logarithms and its value is approximately equal to 2.71828

Pi, our age-old friend, is basically associated with circles and its value is equal to, 3.14

Imaginary unit, i is a factor that we use mainly in the field of complex numbers and its value is equal to √-1 .

Portrait 1: Leonhard Euler

Euler's Identity related these three fundamental constants in mathematics. You can clearly observe from our previous observation that both  e and 𝝅, have non-terminating decimal expansion. Both these constants are irrational in nature. 

The real beauty of Euler's Identity pertains to this fact that when, e is raised to i𝝅, you get -1.

That is,

Just look at its beauty, when an irrational number, e is raised to another system i𝝅 containing an irrational number 𝝅 we get a rational number, -1. 

Moreover, when we look at this equation we see that, three fundamental constants, e, i and 𝝅, which were thought to have no relationship between themselves in the past where unified to together to yield us an unexpected answer, like -1. 

Also just imagine, two constants, e and 𝝅 having non-terminating decimal expansion are related in this equation and we get a perfect terminating rational number, -1 which we never would have even expected would be the result. 

Portrait 2: Jakob Bernoulli. He discovered the mathematical constant, 'e' .

 

PROOF FOR EULER'S IDENTITY

''What can be asserted without proof, can be dismissed without proof''

-Christopher Hitchens

We know from our mathematics classes that complex numbers can be expressed in terms of sine and cosine. This method of representation is called the 'Polar form' or 'Polar representation'.

The polar form, of a given complex numbers,

z = r(cosθ + i sinθ)                      .....(1)

where, 

z is a complex number of the form, x + iy

r = √x² + y² 

'i' is the imaginary unit which equal, √-1

As a special case of 'Euler's formula', we get the following result,

           .....(2)    

Substitute, x = 𝝅 in (2), we get

          .....(3)

We know that cos𝝅 is -1.

How?

cos𝝅 = cos (2.90 - 0)

            

         = - cos 0

         = -1

Also we know that sin𝝅 is 0.

How?

sin𝝅 = sin (2.90 - 0)

         = + sin 0

         = 0

If you find this this theoretical calculation difficult you can find the values of basic angles i.e. multiples of 90⁰ directly from the graphs of sine and cosine given below,

Sine graph:

Plot 1: sine graph

From this graphical representation it is clear that, sin 180 ⁰ = 0. (here 180 ⁰ corresponds to '𝝅' radian measure)

Cosine graph:

Plot 2: cosine graph

From this graphical representation it is clear that, cos 180 ⁰ = -1 (here 180 ⁰ corresponds to '𝝅' radian measure)

Substituting these values in (3), we get

This implies that,

Three quantities, with nothing in common, used in different fields are related in a simple equation of an 'inch' long. Euler's identity, is the best example of a true mathematical beauty. In Euler's identity, we make use of three fundamental operations in mathematics at the same time, namely addition, multiplication and exponentiation, also we have in this equation, the additive identity that is zero and the multiplicative identity that is one.

In short equation (3), summarises almost all the fundamental ideas in mathematics, no equation can replace its beauty and clarity of thought, for it is elegant in all respects. 

If, Einstein's Energy-Mass Equivalence is the 'world's most famous equation', then without doubt, Euler's Identity, is the 'world's most beautiful equation'...

For additional information regarding Euler's Identity and its detailed proof do watch the following video.

Video 1: A explanatory video on 'Euler's Identity'

REFERENCES:

1. Wikipedia: The Free Encyclopaedia

2. NCERT Class XI Textbook in Mathematics

3. An Introduction to the Analysis of Infinitesimals (1748), by Leonhard Euler

*All the media published in this blog belongs to their original creators and 'Knowledge Through Science' does not claim any right over it.