In a 1988 poll of readers of the Mathematical Intelligencer, mathematicians voted it the most beautiful theorem in all of mathematics. The physicist Richard Feynman called it “the most remarkable formula in math.” The mathematician Keith Devlin wrote that “like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form, Euler’s equation reaches down into the very depths of existence.”
The equation is this: eiπ + 1 = 0
It contains five numbers, each fundamental to mathematics, and connects them in a relationship so compact and so unexpected that it has inspired awe for nearly three centuries. The five constants are:
- e (approximately 2.71828): the base of the natural logarithm, fundamental to calculus, compound interest, population growth, and radioactive decay
- i: the imaginary unit, defined as the square root of negative one, fundamental to complex number theory, electrical engineering, and quantum mechanics
- π (approximately 3.14159): the ratio of a circle’s circumference to its diameter, fundamental to geometry, trigonometry, and every branch of physics
- 1: the multiplicative identity, the number that when multiplied by any other number leaves it unchanged
- 0: the additive identity, the number that when added to any other number leaves it unchanged
These five numbers come from completely different areas of mathematics. That they should be related at all is surprising. That they should be related in so simple a form is, to mathematicians, genuinely mysterious.
The Man Behind the Formula
Leonhard Euler (1707 to 1783) was the most prolific mathematician in history. Born in Basel, Switzerland, he spent most of his career at the academies of St. Petersburg and Berlin, producing over 850 papers and dozens of books on every branch of mathematics known in his time. He introduced much of the notation that mathematicians still use today, including e for the base of the natural logarithm, i for the imaginary unit, f(x) for a function, and Σ for summation.
Euler’s productivity was legendary. He could write a mathematical paper in the time it took an assistant to prepare lunch. He continued working after losing his sight in 1771, performing complex calculations in his head and dictating his results. At the time of his death, he had so many unpublished papers that the St. Petersburg Academy continued publishing them for nearly fifty years.
Euler’s identity is a special case of a more general result known as Euler’s formula: eix = cos(x) + i·sin(x). This formula connects the exponential function with the trigonometric functions through complex numbers. When x = π, the formula gives eiπ = cos(π) + i·sin(π) = −1 + 0 = −1. Adding 1 to both sides yields the identity: eiπ + 1 = 0.
Why It Is Beautiful
Mathematicians have struggled to explain why Euler’s identity feels beautiful rather than merely correct. Several features contribute to its aesthetic power.
Economy: the equation uses each of the five fundamental constants exactly once, with no extraneous elements. It says precisely what it needs to say and nothing more. This compression of meaning into minimal form is a quality that mathematicians associate with elegance.
Surprise: the identity connects concepts that seem to have nothing to do with each other. What does the base of the natural logarithm have to do with the ratio of a circle’s circumference to its diameter? What does an imaginary number have to do with either? The identity reveals a hidden connection between areas of mathematics that appear entirely unrelated. This quality of unexpected connection is central to mathematical beauty.
Depth: the identity is not a shallow coincidence. It is the surface expression of a deep structural relationship between exponential functions, trigonometric functions, and complex numbers. Euler’s formula, the generalization from which the identity follows, is one of the most important tools in all of applied mathematics. It is used in electrical engineering, quantum mechanics, signal processing, and fluid dynamics. The beauty of the identity is inseparable from the depth and power of the mathematics it represents.
Understanding the Formula
Euler’s formula, eix = cos(x) + i·sin(x), can be understood geometrically. In the complex plane (where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers), the expression eix traces a circle of radius 1 as x varies. When x = 0, the point is at (1, 0). When x = π/2, the point is at (0, 1), or equivalently at the imaginary unit i. When x = π, the point is at (−1, 0).
This means that eiπ is simply the point on the unit circle that is diametrically opposite to the starting point. It is −1. The identity eiπ + 1 = 0 says nothing more (and nothing less) than this: if you start at the point 1 on the unit circle and travel halfway around (an angle of π radians), you arrive at −1.
Put this way, the identity seems almost obvious. The surprise is that the exponential function (which describes growth and decay) should be connected to circular motion through the imaginary unit. This connection is not obvious from the definitions of any of the individual concepts. It emerges only when they are combined in exactly the right way.
The Constants and Their Histories
Each of the five constants in Euler’s identity has its own rich history.
Zero was the last of the five to be recognized as a number. The concept of zero as a placeholder appeared in Babylonian mathematics, but zero as a number in its own right (with its own arithmetic properties) was developed in India around the 7th century CE and transmitted to Europe through Arab mathematicians.
One seems trivial but is philosophically subtle. The ancient Greeks debated whether 1 was a number at all (Euclid defined it as a “unit” rather than a number). Its role as the multiplicative identity makes it fundamental to all of arithmetic and algebra.
Pi has been studied since antiquity. Archimedes approximated it by inscribing and circumscribing polygons around a circle. Its irrationality was proved by Johann Lambert in 1761, and its transcendence (it is not the root of any polynomial with rational coefficients) was proved by Ferdinand von Lindemann in 1882.
e emerged from the study of compound interest in the 17th century. Jacob Bernoulli discovered the constant while investigating the limit of (1 + 1/n)n as n grows without bound. Euler gave it its name and studied its properties extensively.
i, the square root of −1, was introduced by Italian mathematicians in the 16th century as a tool for solving cubic equations. It was regarded as a “useful fiction” for centuries before Euler and Gauss demonstrated that complex numbers (numbers of the form a + bi) are a legitimate and indispensable part of mathematics.
Euler and the Mathematical Tradition
Euler’s identity sits at the intersection of nearly every major branch of mathematics: arithmetic (0 and 1), algebra (the equation itself), analysis (e and its properties), geometry (π and the circle), and number theory (the deep properties of these constants). It is a meeting point of traditions that stretch back thousands of years.
The geometric tradition that produced π begins with Euclid’s Elements, where the properties of circles and their relationship to straight lines are rigorously established. Kronecker Wallis’s edition, completing Oliver Byrne’s color-coded visual interpretation across all thirteen books, makes the geometric foundations of mathematics tangible and beautiful in their own right.
The algebraic and analytical traditions that produced e and i were developed by generations of mathematicians whose private notebooks reveal how these ideas evolved. Gauss’s Selected Visual Notebooks document the working methods of the mathematician who, after Euler, did the most to develop the theory of complex numbers. Gauss’s geometric interpretation of complex numbers as points in a plane (the same plane in which Euler’s formula traces its circle) was one of the key insights that made complex analysis possible.
More Than Beautiful
Euler’s identity is celebrated for its beauty, but its importance goes beyond aesthetics. The formula it derives from, eix = cos(x) + i·sin(x), is one of the most practically useful equations in all of science and engineering. It simplifies calculations in alternating current circuit analysis, quantum mechanics, signal processing, and control theory. Without it, modern electrical engineering and telecommunications would be far more difficult.
The identity also raises a philosophical question that mathematicians have debated for centuries: why is mathematics beautiful? Is mathematical beauty a property of the equations themselves, or is it a projection of the human mind onto abstract structures? Euler’s identity feels meaningful in a way that goes beyond its logical content. It feels as if the five constants were meant to be related in this way, as if the equation reveals something about the deep structure of mathematics itself.
Whether this feeling reflects a genuine property of mathematical reality or merely the limitations of human intuition is a question that mathematics alone cannot answer. But the feeling is real, and it has motivated mathematicians for centuries. Euler’s identity is the most compact expression of a conviction that drives all of mathematics: that beneath the apparent complexity of the world, there is a hidden simplicity, and that the language of mathematics is the key to finding it.
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