Deriving the world’s most Beautiful Equation
An equation considered so beautiful, that Professor Feynman called it ‘our jewel’ and ‘the most remarkable formula in mathematics’. So much was he dazed by its beauty that he went out of his way to derive it in one of his famous lectures at Caltech. I’m talking about the famous Euler’s Identity which is given below.
Why is this equation so beautiful? Everyone will have different answers. But for me its that it connects the most fundamental constants of mathematics, e (also called Euler’s or Napier’s constant; it is the base for the natural logarithms), i (the imaginary unit, which is the solution to the equation x²+1=0), π (also called the Archimedes constant, it is most commonly used in circles and radian angles.) and 1.
The derivation of this is quite simple, however, it is because it has a parent equation, Euler’s Formula, which is given below.
It is the derivation of this formula that is a bit difficult. But we shall derive it using simple substitution in the form of Taylor Series.
Before we dive deep into the derivation, we shall first understand what is a Taylor Series. This will be a bit complex. So if you do not want to read this, you can go ahead and skip to the part where I talk about Exponential series, it’s expansion and the expansion of sine and cosine functions.
Generally speaking, the Taylor series is an infinite sum of a function expressed using the derivatives of the function at a single point ‘a’.
In mathematics terms, it means:
Here, f’n(a) means the nth derivative of the function at the point a. a can be any point, and hence can be an imaginary or complex, or real number.
In many cases, you also have a binomial term in the multiplied to the sum. So the expansion now becomes:
where the (x-a)^n can be expanded to any value of n using Newton’s Binomial Theorem.
An exponential series is a series in the form of e^nx where n can take any real values. The most famous exponential function is e^x. But what is this e?
e is the final amount you would get if you lend someone $1, and then compounded interest on that $1 infinitely many times. In mathematics terms,
This definition was proposed by Jacob Bernoulli, and the value of e was found by Leonard Euler, who found out that e ≈ 2.718….. . Just like π, its an irrational number that goes on forever.
Lets come back to the exponential function. The value of e^x was first verified by Taylor, and it goes hand in hand with the series. The Taylor Series expansion of the exponential function is:
Series of Sine and Cosine
Sine and Cosine, otherwise known as sin and cos, are the two fundamental trigonometric functions.
Because they are periodic functions(i.e they repeat their values after a certain amount of time), they are used extensively for studying wave phenomena, simple harmonics and other periodic motions.
The sine of any angle can be easily found using the Taylor series. The Taylor expansion of sin(x) is:
The cosine of any angle can be easily found using the same way. The Taylor Expansion of cos(x) is:
Now let us get into the derivation.
We need to prove that:
We will prove that the lift hand side is equal to the right hand side of the equation.
Now, let us take the expansion of e^x
Now put x=ix
Now, calculating the exponents, we get:
using the fact that i²=-1 and i³=-i, we arrive at:
Separating the real and the imaginary parts, we get:
which will be our left hand side.
Let us now take a look at the right hand side.
since cos(x) is purely real, the expansion remains the same,
now let us calculate the expansion of i sin(x). Since i is multiplied into sin (x), the resultant expansion will be multiplied to i as well.
now let us add these two expansions.
which is now equal to the expansion of e^ix.
This proves that the right hand side = left hand side.
Hence we have proved that
putting x=π, we get:
we know that cosπ=-1 and sinπ=0
so we get:
which is the desired result!
Thanks for reading this. Have a good day everyone!