Euler’s identity is a special case of Euler’s Formula. It’s known as “The Most Beautiful Equation in Math”. It hides a secret behind it. What it is?
If you think about the equation for a moment, it seems simple and straightforward. i is just another variable that can be added to any other number, right? At first glance, this may seem true but after careful thought one can see that something more interesting must be going on here. It turns out that there are many hidden treasures behind Euler’s identity which make it one of the most beautiful equations in mathematics history!
The late great physicist Richard Feynman called in his lectures “our jewel” and “the most remarkable formula in mathematics.”
Its beauty is fundamentally the surprise of coincidences, why are such a “protagonist” guys all in the same movie? Best crossover ever, from math perspective.
Crossover of Math
Euler’s identity is a identity that relates the five most important numbers in math: e, π, i, 1 and 0. These five numbers are all linked to each other through this formula!
An identity in mathematics is an equality that reveals two things that looked different as actually are the same.
Let’s meet the actors:
The number 0, is the most fundamental of all numbers. It’s a number that has no size and can’t be used to measure anything. When mathematicians talk about this number, they usually refer to it as “the additive identity”. In other words, if we add up any set of numbers with zero, the result will be equal to those same numbers.
The number 1, is basically the base of natural numbers. All natural numbers are defined as a sum of ones. That is for sum operator, for multiplication/division it is the multiplicative identity. If we multiply any number by one, the result will be equal to that same number.
The number e. By itself being an irrational number gives a shape of mysterious, however, beyond its math definition is known as the “exponential” or “natural” base of logarithms. It’s also used in calculus to describe how fast things change over time. We can interpret it as the derivative identity of calculus.
The imaginary number i. As an irrational number wasn’t sufficient to have, meet the last level group, imaginary number i. The imaginary number i, is the square root of negative one, and it’s what lies at the heart of i. It gets its name from being a number that cannot be physically realized, but that can be used to help solve problems in math.
Euler’s formula
Let’s to the movie, next…
Euler’s identity is actually a special case of Euler’s formula, Euler’s identity is a single point of Euler’s formula. It is analogous to be a child of many, where Euler’s formula is the parent. So, you could figure out what is behind.
This formula works for any value of theta, while Euler’s identity is just where theta is equal to pi.
Phasors (vector’s rotations)
A visual way of view we could see Euler’s formula as a complex number or a geometric vector. It turns out that Euler's number to the power of an imaginary number offers a rotation in the complex plane. That is what Euler's formula means with cosine and sine explicitly.
Explore it by yourself on GeoGebra:
De Moivre’s roots
Going forward, rotations is not exclusive of the Euler’s formula, it could be applied for any complex number. But you should be aware of the magnitude of the vector. In other words, you should rewrite those complex numbers in a such way that magnitude and the direction are separated. The fastest process to map complex numbers from cartesian to polar is applying natural log. Let’s make an example with vector v=(3,5):
Euler’s formula works as a power of a complex number, where in this case that complex number is e^i, which is the vector (cos(1),sin(1)) approx to (0.54,0.84) on the complex plane.
In essence, increasing the power of a complex number is equivalent to rotating it (and scaling if the magnitud is different than the unitary). So, what is the inverse process?
Power and roots of complex numbers
Complex numbers actually were introduced exploring roots of polinomial equations. There was a feeling than the number of roots of a polynomial equation is equal to the degree of that polynomial equation. But why there is no solution for x²=-1, there is where the imaginary number i was introduced.
So, we follow with the previous vector (3,5) as an example, and let’s power it 11.3 times, then let’s take fourth root of the result.
Here we use 2pi as a mapping from radians to revolutions, where 0.85 means 85% of a revolution while each 1 is 100%, thus a revolution per unity.
The result is we increase the magnitude to e^(19.88), while rotating it to give it one revolution and end at 85% of the next revolution.
Let’s take now the fourth root of that, what means split the angle in 4 (and the exponential magnitude, but let look only the angle as we are on rotations).
So if we want to find the square of a complex number we can simply divide its angle by 2, if we want 4 root then divide by 4…
De Moivre’s and the nth Root Theorem
We could conclude this as power of n is multiplying the angle n times while Nth root is splitting the angle by n. e.g. square root is taking the middle of the angle.
Conclusion
Now, how can we explain the beauty of Euler’s identity? It might be better explained as a question: what is behind this beautiful equation?
The answer lies in the depth concept of rotation. Rotations are often easy to understand because you can imagine them as oscillations.
In math it plays an important rol as the link between algebra an geometric, more complex subjects like topology could be join together with this linking. Also most of the solutions of differential equations are like e^(lambda*t).
The Discrete Fourier Transform:
In physics this rotation capability allows to describe more naturally rotations in real systems, the more remarcable equation using this as rotations is Schrödinger’s equation.
Schrödinger’s equation: In this clip of YouTube you could confirm the rotation by i in Schrödinger’s equation:
Interesting videos:
e^(iπ) in 3.14 minutes, using dynamics | DE5 — YouTube
Euler’s Formula & Euler’s Identity — Proof via Taylor Series — YouTube
Resources:
Coolman, R. (2015, July 1). Euler’s Identity: ‘The Most Beautiful Equation’. Livescience. https://www.livescience.com/51399-eulers-identity.html
(2021, May 4). Fórmula de Euler — Wikipedia, la enciclopedia libre. Es. https://es.wikipedia.org/wiki/F%C3%B3rmula_de_Euler
Sahely, Bichara. (2022). EULER’S FORMULA IS THE KEY TO UNLOCKING THE SECRETS OF QUANTUM PHYSICS.
Thibault, K. (2020, October 1). Euler’s Formula: A Complete Guide | Math Vault. Mathvault. https://mathvault.ca/euler-formula/#Hyperbolic_Functions
M. (2014, January 4). The Fourier Transform, explained in one sentence (Revolutions). Blog. https://blog.revolutionanalytics.com/2014/01/the-fourier-transform-explained-in-one-sentence.html
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(2019, July 7). e^(iπ) in 3.14 minutes, using dynamics | DE5. Youtube. https://www.youtube.com/watch?v=v0YEaeIClKY&ab_channel=3Blue1Brown
(2019, January 24). Euler’s Formula & Euler’s Identity — Proof via Taylor Series. Youtube. https://www.youtube.com/watch?v=cJaKaSCPg8c&t=274s&ab_channel=FlammableMaths
(2015, December 24). 12:29 / 15:45 e to the pi i for dummies. Youtube. https://www.youtube.com/watch?v=-dhHrg-KbJ0&t=746s&ab_channel=Mathologer
Maussa, A. (2022, August 4). Euler’s formula (phasor). Youtube. https://www.youtube.com/watch?v=yuSGaBeoAF4&ab_channel=AlfredoMaussa
