The Most Beautiful Identity in Mathematics

So many times mathematical formulas seems to difficult to understand. For instance if we think about the formula of discrimination, let us write,

x_1 and x_2 are the roots of ax²+bx+c=0

However if we make an effort to solve that equation we don’t need to memorise its formula because we know the coefficients of x variable and the constant c. They are all real numbers so that it is a bit easy to understand to concept of that equation. Then there is a big question is appeared to us. What if we have two irrational numbers, one imaginary number and integer and they are equal to zero for the particular equation?

Here we met Euler’s Identity. It is given as

Euler’s Identity

If we analysed the equation we can see that;

e as a Euler constant , i.e, irrational number

π as a constant that is ratio of circle’s circumference to its diameter

i as a complex number

What a beautiful identity, isn’t it? But how can we derive that beauty? What Euler’s Identity really say?

To answer these questions firstly we need to understand complex numbers and complex plane. A complex plane can be constructed two coordinates as real part and imaginary part. Let us draw it.

Complex Plane

Here the y-coordinates from Cartesian Plane is called Imaginary part of complex plane and the x-coordinates from Cartesian Plane is called Real part of complex plane. Now look at the Cartesian Coordinates. How can we show the coordinates of the point that lies on the unit circle?

Here the point P(0.6,0.8) satisfies the equation of unit circle x²+y²=1

Now we will use some trigonometry. If we draw a line from origin to the point that lies on the circle we have cosine and sine such that;

If we understand the geometry of unit circle, complex numbers seems to be more clear. The question is how can we show a point that lies on the circle on complex plane?

Let us drive a point z= 1+i on the complex plane which is lies on the circle. Here i is the complex part of the point z and 1 is the real part of the point z

From the above trigonometric representation of the circle in the Cartesian coordinates we can derive that;

Clearly i has a one unit lenght in imaginary part and 1 has a one unit lenght in unit part. Then the radius r is equal to square root of 2. So what about α? For that question it is easy to compute α because tanα=1. On the first quadrant tanα=1 is just satisfied for

α= π/4 + 2nπ where n=0,1,2,…

Now we almost have done. We showed that complex numbers can written as z = x+ iy where x is the real part and iy is the imaginary part. So let’s put x= r.cosα and y= i.r.sinα. If we take the unit circle on complex plane such that the radius is equal to 1 then we get z= cosα +i.sinα. Now we reached third question. What if we take n-th power of z.

To answering that question we need to look at Taylor Series expansion. We know the expression of sinα and cosα in Taylor Series such that;

Taylor series expansion of sinα and cosα

Now looking for Taylor Series expansion of e^x;

Taylor series expansion of e^x

Put x=iα for that expression then we have;

Since i²= -1 then we have;

So we nearly done everything. Remember that our point z can be written as form z= cosα +i*sinα Now put the Taylor Series expansion for that point we get;

That is exactly the expression of e to the power of (iα) which we shown above. Here e to the power of (iα) is called exponential form of complex numbers.

So finally if we return to Euler’s Identity we have been focused on e to the power of (iα). Put α=π on the equation then we have;

Since cosπ =-1 and sinπ= 0 then we have e to the power of (iπ) is equal to -1.

So we proved that the most beautiful identity in mathematics such that;

QED