Euler’s Identity

THE MOST BEAUTIFUL EQUATION IN MATH

Math.

So many thoughts and associations with just among these 4 letters. Some interpret it as a wonderful system of natural synchronization, while others may count it as something immense and incapable of realization.


Scientifically speaking, Google claims Math to be:

“…the abstract science of number, quantity, and space, either as abstract concepts (pure mathematics), or as applied to other disciplines such as physics and engineering (applied mathematics).”

 

Intellect my friends, intellect.


Consequently, it is quite plausible that the world of Math may start from the sum/subtraction of standard integers (1, 2, 3…) and end with complicated integrals (Newton memes come in).

However, have you ever wondered whether once, there would be this perfect, idyllically structured equation, where simple numbers could replace the artworks of Leonardo Da Vinci in a mathematical sense? Well here it is young scientists and amateurs:


Euler’s identity


 

Back in 1988, a Mathematical Intelligencer poll voted Euler’s identity as the most beautiful feat of all of mathematics. In one mystical equation, Euler had merged the most amazing numbers of mathematics.

Why is it called something so poignant? doesn’t it look like a regular boring algebra class problem?

The point here is that the uniqueness lies in the use of the 3 basic arithmetic operations: addition, multiplication and exponentiation. Besides that, it also involves 5 fundamental mathematical constants:

  • The Number 0
  • The Number 1
  • The Number π= 3.1415… (“…and that’s the way, uh-huh, uh-huh, I like it…” A.K.A. Night From the Museum)
  • The Number e= 2.718…
  • The Number $i =\sqrt {-1}$, the imaginary unit of complex numbers

 

But where does it come from and what does it mean?


 

As I mentioned above, $i =\sqrt {-1}$. This might seem shocking because negative numbers are not supposed to have square roots. However, if we simply decree that $-1$ does have a square root and call it $i$, then we can build a whole new class of numbers, called the complex numbers. Complex numbers have the form $x+iy,$ where $x$ and $y$ are ordinary real numbers (for the complex number $i$ we have $x=0$ and $y=1$). Note that a real number can also be viewed as a complex number. The number $-1$, for example, is a complex number with $x=-1$ and $y=0$.

Just like a real number is represented by a point on a number line, a complex number $z$ is represented by a point on the plane. To the complex number $z=x+iy$ we associate the point with coordinates $(x,y)$.

Thus, knowing the basic rules of trigonometry, we can assume that:

Now let’s head back to the part where we mentioned the complex number “z” which was equal to $z=x+iy$. While uniting the equation with trigonometry, we may achieve:

And it happens to be that for real numbers “r” and angle “θ” (approved by using power series):

As:

 


TADA!

There goes math’s rare beauty


 

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Written by Shohruh Artaman
A typical writer illustrating typical principles
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  1. Damn, that’s a nice article

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  2. yes, it is a nice article. I knew that Artaman is a good speaker, but today I am here discovering another side of him. Keep up the good work!

    Reply

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