What is Euler's Formula?

Euler's Formula is a profound mathematical equation that bridges the realms of complex numbers and trigonometry, revealing a stunning relationship: e^(iθ) = cos(θ) + i*sin(θ). This elegant expression, named after the Swiss mathematician Leonhard Euler, is a cornerstone in mathematics, connecting seemingly disparate concepts with simplicity and beauty. How does this formula unlock the secrets of complex analysis? Let's dive deeper.
H.R. Childress
H.R. Childress

The 18th-century Swiss mathematician Leonhard Euler developed two equations that have come to be known as Euler's formula. One of these equations relates the number of vertices, faces, and edges on a polyhedron. The other formula relates the five most common mathematical constants to each other. These two equations ranked second and first, respectively, as the most elegant mathematical results according to "The Mathematical Intelligencer."

Euler's formula for polyhedra is sometimes also called the Euler-Descartes theorem. It states that the number of faces, plus the number of vertices, minus the number of edges on a polyhedron always equals two. It is written as F + V - E = 2. For example, a cube has six faces, eight vertices, and 12 edges. Plugging into Euler's formula, 6 + 8 - 12 does, in fact, equal two.

Scientist with beakers
Scientist with beakers

There are exceptions to this formula, because it only holds true for a polyhedron that does not intersect itself. Well-known geometrical shapes including spheres, cubes, tetrahedra, and octagons are all non-intersecting polyhedra. An intersecting polyhedron would be created, however, if someone were to join two of the vertices of a non-intersecting polyhedron. This would result in the polyhedron having the same number of faces and edges, but one fewer vertice, so it is obvious that the formula is no longer true.

On the other hand, a more general version of Euler's formula can be applied to polyhedra that intersect themselves. This formula is often used in topology, which is the study of spatial properties. In this version of the formula, F + V - E is equal to a number called Euler's characteristic, which is often symbolized by the Greek letter chi. For example, both the donut-shaped torus and the Mobius strip have an Euler's characteristic of zero. Euler's characteristic can also be less than zero.

The second Euler's formula includes the mathematical constants e, i, Π, 1, and 0. E, which is often called Euler's number and is an irrational number that rounds to 2.72. The imaginary number i is defined as the square root of -1. Pi (Π), the relationship between the diameter and circumference of a circle, is approximately 3.14 but, like e, is an irrational number.

This formula is written as e(i*Π) + 1 = 0. Euler discovered that if Π was substituted for x in the trigonometric identity e(i*Π) = cos(x) + i*sin(x), the result was what we now know as Euler's formula. In addition to relating these five fundamental constants, the formula also demonstrates that raising an irrational number to the power of an imaginary irrational number can result in a real number.

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Discussion Comments


That seems like it would be a good formula to use in a classroom, since kids of a certain age would be interested in that kind of "trick".

It would also be a good way to help them prove their answers.


@bythewell - As famous mathematicians go, Euler was actually fairly prosaic, at least in his personal habits and proclivities. He was welcome at court in his native land and he was well known and respected in his time.

He was very productive as well. I don't know how many formulas can actually be directly attributed to him, but I know he contributed to a lot of different fields and he has a diagram named after him that is used in logic and philosophy.

There is one story about him which is a bit quirky, but it's probably not true. The story goes that the queen was being bothered by an atheist who was influencing people at court. She called on Euler to call him out and Euler announced that he had discovered a formula that proved that God exists.

Of course, what he showed the man was gibberish, but the atheist didn't know any mathematics, so he accepted it at face value.


I'm always kind of fascinated by the lives of mathematicians. They often seem to be such strange folk, and I'm not sure if that's because focusing on mathematics makes you strange or if a certain kind of person is drawn towards becoming a mathematician.

I don't actually know all that much about Euler though, except that he's a very famous mathematician. I'm surprised that the polyhedral formula and the other one are the only two associated with his name.

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