How To Intuitively Understand Euler’s Identity?

Euler’s identity is popularly known as the most beautiful equation in mathematics amongst enthusiasts and professionals alike. Yet, there exists an air of inaccessibility surrounding it. For anyone looking at it from the outside, the following questions might arise:

1. Why is Euler’s identity considered so beautiful?

2. How can I understand the mathematics behind Euler’s identity more intuitively?

In this essay, I will focus on answering both of these questions. To this end, I will skip explanations for miniscule mathematical details. Instead, I will focus only on the information necessary for understanding the identity intuitively.

At the end of the essay, you should be able to appreciate the sense of mathematical beauty associated with Euler’s identity and intuitively grasp the mathematics behind it. Let’s get started.

This essay is supported by Generatebg

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Why is Euler’s Identity Considered So Beautiful?

Let us take a look at Euler’s identity first:

It contains three of the fundamental arithmetic operations, and each of these occurs exactly once:

1. Addition (a + b)

2. Multiplication (a * b)

3. Exponentiation (a^b)

Furthermore, the equation also links five of the most fundamental mathematical constants/numbers:

1. The number ‘0’ — the additive identity

2. The number ‘1’ — the multiplicative identity

3. The number ‘π’ (3.14159…) — the fundamental circle constant

4. The number ‘e’ (2.71828…) — the growth identity (for more details, check this essay out)

5. The number ‘i’ — the imaginary unit of the complex numbers

Imagine a mathematician seeing the most fundamental tools of their trade being strung together each just once to produce a profound statement that radiates deep meaning. It is akin to a musical piece where all the fundamental notes occur exactly once to produce the most pleasurable symphony. It is “simplicity”; it is “deep meaning”; it is “beauty”; it is “elegance”.

It is all that. But hang on a minute! If it is all that, how come many people from outside the world of mathematics are unable appreciate it? Right, let’s move on to answering that question.

Why is Euler’s Identity Not Appreciable for Outsiders?

The short answer to this question is: Baggage!

You see, the typical outsider is familiar with the operations of addition and multiplication. However, exponentiation is not everyone’s cup of tea. As fundamental as it is to mathematics, not everyone uses it in day-to-day life (at least not explicitly) nor is everyone familiar with its application.

On the contrary, the typical outsider is familiar with the numbers ‘0’ and ‘1’. Some are familiar with the constant ‘π’, but for most, it is just something that occurs in the area formula for a circle. Most never get to learn the deeper applications of π.

The constant ‘e’ and the imaginary number ‘i’ are by far the entities with the most baggage. They rarely appear in explicit forms in real-life applications, but are silently hidden in the technical architecture. They also require deeper levels of abstraction for any outsider to understand. Therefore, most non-math folks are turned off by equations containing these terms. They remain as silent heroes that save the day without getting the credit they deserve.

In music, a fan can appreciate a pleasant symphony even without understanding the technicality of each note. However, with mathematics, this is not the case; at least, not out of the box. Unless Euler’s identity is expressed in a graphical form (or another easily consumable form), it remains elusive and inaccessible to the typical outsider.

The good news is that with the development of graphical mathematical engines, mathematical beauty is getting more and more accessible to anyone who can appreciate art.

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How to Intuitively Understand Euler’s Identity?

Let us now try to establish an intuitive understanding for the math behind Euler’s identity. A good place to start is with imaginary numbers.

Covering Basics

If you are not familiar with imaginary numbers, just know that they originate from an intriguing question: Can the square of a number result in a negative number? The answer: normally, not. The “normally” here refers to the real numbers as we know it.

However, if we “imagined” special numbers that have this property, they turn out to be incredibly useful in mathematics. So much so, that some mathematicians argue that imaginary numbers really do exist outside of our imagination. This is because of the fact that they are so useful in describing certain real-world phenomena.

The unit imaginary number is ‘i’ (= √(-1)) — the same ‘i’ found in Euler’s identity.

Now that we have covered what ‘i’ is, let us move on to a graphical representation of real numbers along the X-axis and imaginary numbers along the Y-axis in a cartesian plane known as the complex plane.

It turns out that imaginary numbers are especially useful when they are combined with real numbers. Such a combination is known as a complex number (which takes the form (x + iy)). With this is as the basis, it is not hard to see that any real number can also be expressed as a complex number as follows:

Rotating Vectors

The key to an intuitive understanding of Euler’s identity is to consider complex numbers as vectors from the origin of the complex plane. Such vectors can be expressed in polar coordinates as follows:

Let us now consider a vector from ‘0’ to ‘1’. Since this vector lies on the positive real axis, it encloses an angle (initially) of zero radians. Furthermore, its magnitude is equal to 1.

Now suppose that we rotate this vector by π radians (or 180° — we consider anticlockwise rotation positive), we end up with a vector that is pointing at ‘-1’ on the negative real axis. Consequently, its magnitude is also equal to 1. The corresponding equation is given as follows:

The result is Euler’s identity. All that it is trying to say (intuitively, speaking) is:

If we rotate a unit vector lying on the positive real axis of the complex plane by π radians (or 180°), the result is equal to -1.

Similar results also apply to multiples of π. For instance, if we rotate the unit vector lying on the positive real axis by 2π radians (or 360°), we end up at the same place we started from. Consequently, the result is equal to 1.

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Final Thoughts

Usually, challenges emerge in understanding Euler’s identity because we invoke trigonometric functions to explain the mathematics behind it.

What I have tried to do here is to bypass the trigonometric functions and directly give an intuition for rotation of the unit vector along the positive real axis on the complex plane.

I hope that this enables you as the reader to begin appreciating the simplicity, beauty, and elegance behind Euler’s identity.

If you still feel that Euler’s identity is not your cup of tea, worry not. As with music and art, different people perceive things differently (subjectivity), and it is perfectly all right to not like something that other people like.

In my opinion, a taste for beauty and elegance should not be acquired. Instead, it should emerge naturally out of one’s own life experience(s)!

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Further reading that might interest you: Why Do We Really Use Euler’s Number For Growth? and What Is So Special About 69! ?