The most beautiful equation
Euler’s Identity
This is rightfully said to be the most beautiful equation in Mathematics. The way in which it combines the most fundamental and significant numbers in Mathematics is simply aesthetic.
The Euler’s identity is a combination of 3 most important constants.
- e = 2.71828, which is the base of natural logarithm, also known as the the Euler’s number,
- π = 3.14159, which is the ratio of a circle’s circumference to its diameter, also known as the Archimedes constant,
- i = √-1, which denotes imaginary number.
So, how do these disparate symbols combine to reproduce this masterpiece ? Surprisingly, this is not as tough as it seems to be.
Let’s see.
If we have a look at the Taylor series for our exponential function, we have,
Now, let us also see the Taylor series of sin(x) and cos(x)
Doesn’t e^x seems to some kind of a combination of sin(x) and cos(x) ? It, indeed, is.
Let us see what e^ix gives us
Voila!!
So, now we know that
Let us trivially replace our x by pi.
And that brings us to our equation,
Here we go.
Let us see what does this represents.
In a quadratic plane, where x axis denotes the real plane and y axis represents the imaginary plane, let us take a point z, such that,
z = x + iy
Now, if the line connecting the origin to z makes an angle of theta with the x axis and is of the length r, then,
x = r cos θ
y = r sin θ
Therefore, z = r cos θ + i r sin θ
For r = 1,
z = cos θ + isin θ, which as we can see from above is the same as e^iθ
When θ = π,
As is clearly visible, adding 1 to it will give us zero.
Isn’t it just beautiful ?
