Mathematics

Does the equation "e to the power i x pi = -1" have any physical meaning? Is there a meaning waiting to be discovered?

Comments

For those who do not know (I had to look it up...it's been a while!), this equation follows from a more general equation, known as Euler's Equation: exi = cos(x) + i sin x.

Complex analysis is applied in many parts of physical science, and it would be surprising if such a fundamental relation did not have some physical interpretation in, say, fluid mechanics. But I don't know to what extent complex exponents are in play there.

There is another question that arises simply within mathematics and that might have an interesting answer, namely: Is there some natural geometrical interpretation of Euler's equiation. I don't know the answer to that question, either.

Euler's equation is important in quantum mechanics. In quantum mechanics, the state of a particle is given by a function, and the formula for that function generally includes a term of the form eix, which, according to Euler's equation, is equal to cos(x) + i sin(x). The appearance of cos(x) and sin(x) here introduces an oscillatory term into the formula. This is why quantum mechanics predicts wave-like behavior of particles.

There is a geometrical interpretation of Euler's equation. The complex numbers are usually pictured as a plane. (See question 316 for more about this.) In this plane, the number eix always lies on a circle of radius 1, centered at 0. When x = 0, eix = e0 = 1, which is the rightmost point on the circle. As x increases, the point eix moves counterclockwise around the circle, completing a full revolution every time x increases by 2 pi.

Although Euler's equation has applications in science, I'm not sure I would call this a physical meaning. Others may disagree, but I don't think of mathematics as getting its meaning from applications. The meaning of Euler's equation is determined by the definition of exponentiation with complex numbers, and that definition is entirely internal to mathematics, and makes no reference to applications. But that definition leads to formulas like Euler's that turn out to be useful in science.

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