The Beautiful Equation: How Euler United All Mathematics

The Equation That Contains Everything

Let us calculate the most beautiful relationship in all of mathematics. Consider the expression $e^{i\pi} + 1 = 0$. Why does this simple arrangement of symbols captivate the mathematical mind? It is not merely an equation; it is a unification. Here we see five fundamental constants, each born from a distinct domain of thought, brought together into a single, elegant statement of truth.

First, we have $e$, the base of natural logarithms, the constant of growth and analysis. Then $i$, the imaginary unit, the square root of $-1$, a concept that once seemed impossible to algebraists. We have $\pi$, the ratio of a circle’s circumference to its diameter, the ancient constant of geometry. And finally, the additive identity $0$ and the multiplicative identity $1$, the foundations of arithmetic itself. Analysis, algebra, geometry, and arithmetic—disciplines that for centuries seemed separate—are here revealed to be inextricably linked.

How did we arrive at this unity? It was not through philosophical speculation, but through pragmatic calculation and the innovation of notation. By standardizing the symbols we use—by giving names to these constants and treating them with the same rigorous rules as any other number—we allow the mathematics to speak for itself. The beauty of this equation lies in its inevitability. When we follow the logic of the complex plane, when we extend the definitions of exponentials and trigonometry, this result emerges not as a surprise, but as a necessary truth. It is a puzzle that solves itself once the pieces are correctly named.

This equation is the gateway to a broader realization: that mathematics is a single, universal language. The boundaries we draw between “number theory” and “geometry” or “calculus” and “physics” are artificial. In my work, I have found that methods from one domain often illuminate the darkest corners of another. To understand the unity of mathematics, we must look at how these connections are forged—through rotation, through counting, and through the optimization of nature itself.

Rotation Revealed Through Notation

To understand the depth of $e^{i\pi} + 1 = 0$, we must first look at the relationship between exponential functions and trigonometry. For a long time, logarithms of negative numbers were a source of confusion and contradiction. But by introducing the imaginary unit $i$ into our exponents, we unlock a geometric interpretation that clarifies everything.

Consider the formula I developed: $e^{ix} = \cos(x) + i\sin(x)$. This is not just a formula; it is a bridge. On the left, we have the exponential function, the language of growth and decay. On the right, we have sine and cosine, the language of oscillation and waves. The imaginary unit $i$ acts as the translator. When we visualize complex numbers in the plane—with real numbers on one axis and imaginary numbers on the other—multiplication by a complex number becomes a geometric operation.

Specifically, multiplying by a complex number in polar form corresponds to a rotation and a scaling. If we take a number with magnitude $1$ and angle $\theta$, represented as $e^{i\theta}$, and multiply it by another, the angles simply add. This reveals that complex multiplication is, at its heart, rotation. The notation $e^{i\theta}$ captures this perfectly because exponents add when bases are multiplied: $e^{i\alpha} \cdot e^{i\beta} = e^{i(\alpha+\beta)}$. The algebra of exponents mirrors the geometry of rotation.

Now, let us apply this to our beautiful equation. If we set $x = \pi$, we are rotating by $\pi$ radians—exactly 180 degrees. Starting from $1$ on the real axis, a rotation of 180 degrees lands us precisely at $-1$. Thus, $e^{i\pi} = -1$. Rearranging this gives $e^{i\pi} + 1 = 0$. The mystery dissolves into geometry. We are not performing some mystical operation; we are simply turning halfway around the circle. The notation $e, i, \pi$ allowed us to see this connection, transforming a problem of algebra into a clear geometric truth. This is the power of notation: it clarifies thought, making the complex simple and the hidden visible.

The Totient Function: Counting Across Domains

Just as we connected analysis and geometry, we can also bridge number theory and algebra. Consider the problem of “breakability” of numbers. How do we measure the structure of an integer? I introduced the totient function, $\phi(n)$, to count the number of positive integers less than or equal to $n$ that are coprime to $n$.

Two numbers are coprime if their greatest common divisor is $1$. For example, let us calculate $\phi(8)$. The integers less than 8 are 1, 2, 3, 4, 5, 6, 7, 8. Which of these share no factors with 8 other than 1? The number 1 is coprime. 2 shares a factor of 2. 3 is coprime. 4 shares 2. 5 is coprime. 6 shares 2. 7 is coprime. 8 shares 8. So the coprime integers are {1, 3, 5, 7}. There are four of them. Thus, $\phi(8) = 4$.

This function becomes incredibly powerful when we look at prime numbers. For a prime number $p$, every integer less than $p$ is coprime to it, because a prime has no factors other than 1 and itself. Therefore, the calculation is trivial: $\phi(p) = p - 1$. This simple observation is the key to unlocking the structure of modular arithmetic.

When we move to composite numbers, the function is multiplicative. If $m$ and $n$ are coprime, then $\phi(mn) = \phi(m)\phi(n)$. This allows us to calculate $\phi(n)$ for any number if we know its prime factorization. For a number $n = p \times q$ where $p$ and $q$ are primes, $\phi(n) = (p-1)(q-1)$. This calculation connects the discrete world of counting integers to the algebraic structure of modular rings.

This leads us to a generalization of Fermat’s Little Theorem. Fermat stated that $a^{p-1} \equiv 1 \pmod p$ for a prime $p$. I extended this to all numbers: $a^{\phi(n)} \equiv 1 \pmod n$ whenever $a$ and $n$ are coprime. This theorem—Euler’s Theorem—shows that the totient function governs the cycles of modular exponentiation. It connects the properties of prime numbers to the behavior of powers in modular arithmetic. Just as $e^{i\pi}$ connects geometry and algebra, $\phi(n)$ connects counting and algebraic structure. We see again that mathematical truth is interconnected; the properties of numbers are not isolated facts but part of a grander system.

When Physics Becomes Calculus

The unity of mathematics extends beyond pure numbers into the physical world. We often think of physics as the study of forces and motion, and calculus as the study of change. But these too are one. In my work on dynamics, I developed what are now known as the Euler-Lagrange equations, which are founded on the principle of least action.

The principle of least action states that nature is efficient. When a system moves from one state to another, it follows a path that minimizes (or makes stationary) a quantity called the “action.” The action is the integral of the Lagrangian—the difference between kinetic and potential energy—over time. To find the path that minimizes this integral, we use the calculus of variations.

Here, we are not just solving for a number or a variable; we are solving for a function, a path through space and time. The Euler-Lagrange equations provide the condition that this optimal path must satisfy. What is remarkable is that this same mathematical structure applies universally. Whether we are describing the motion of a pendulum, the path of a light ray, or the dynamics of a fluid, the mathematics is the same.

This shows the versatility of the calculus. We can translate a physical problem—“how does this object move?”—into a calculus problem—“what function minimizes this integral?”. By solving the calculus problem, we discover the physical law. The distinction between the physical reality and the mathematical description blurs. The universe appears to operate according to mathematical principles of optimization. My work in this area was driven by the conviction that nothing takes place in the world whose meaning is not mathematical. By applying the tools of calculus to the problems of physics, we reveal the underlying order of nature.

Let Us Calculate: The Universal Method

Throughout my life, I have published over 800 papers, covering nearly every branch of mathematics. Some may see this as a lack of focus, but I see it as the only way to truly understand the subject. Mathematics is not a collection of separate rooms; it is a vast, open palace. To understand one part, you must explore the others.

The common thread in all my work is the belief in systematic calculation and the power of notation. When we define our terms clearly—when we write $f(x)$ for a function, $\Sigma$ for a sum, $i$ for the imaginary unit—we free our minds to see the patterns. Notation is not just a shorthand; it is a tool for discovery. It allows us to manipulate complex ideas as if they were simple objects.

And we must not be afraid to calculate. Before we can prove a theorem rigorously, we must often discover it pragmatically. We calculate specific cases, we look for patterns, we trust the logic of the operations. The truth of $e^{i\pi} + 1 = 0$ was not found by staring at axioms, but by manipulating series and integrals, by following the calculation where it led.

This is the lesson I offer: do not compartmentalize your knowledge. The methods of number theory can solve problems in algebra; the tools of calculus can explain the physical world. The totient function, the exponential formula, the principle of least action—these are all manifestations of the same universal truth. Mathematics is the language of that truth. So, let us not be idle. Let us explore, let us connect, and above all, let us calculate.