In mathematics, Euler's identity is the following equation, which is true by definition:
The equation appears in Leonhard Euler's Introduction, published in Lausanne in 1748. In this equation, e is the base of the natural logarithm, i is the imaginary unit (an imaginary number with the property i ² = -1), and π is Archimedes' constant pi (π, the ratio of the circumference of a circle to its diameter).
for any real number x. If we set x = π, then
and since cos(π) = −1 and sin(π) = 0, we get
Perceptions of the identity
Benjamin Pierce, after proving the formula in a lecture, said, "Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth."
- The number 0, the identity element for addition (for all a, a+0=0+a=a). See Group (mathematics).
- The number 1, the identity element for multiplication (for all a, a×1=1×a=a).
- The number π is fundamental in trigonometry, π is a constant in a world which is Euclidean, on small scales at least (otherwise, the ratio of the length of the circumference of circle to its diameter would not be a universal constant, i.e. the same for all circumferences).
- The number e is a fundamental in connections to the study of logarithms and in calculus (such as in describing growth behaviors, as the solution to the simplest growth equation dy / dx = y with initial condition y(0) = 1 is y = ex).
- The imaginary unit i (where i 2 = −1) is a unit in the complex numbers. Introducing this unit yields all non-constant polynomial equations soluble in the field of complex numbers (see fundamental theorem of algebra).
Furthermore, all the most fundamental operators of arithmetic are also present: equality, addition, multiplication and exponentiation. All the fundamental assumptions of complex analysis are present, and the integers 0 and 1 are related to the field of complex numbers.
In addition, the result is remarkable to most students learning it for the first time because it is so highly counter-intuitive. Consider that
The simple insertion of i changes the result dramatically.
- Feynman, Richard P. The Feynman Lectures on Physics, vol. I - part 1. Inter European Editions, Amsterdam (1975)
- Proof of Euler's Identity by Julius O. Smith III
- Proof of Euler's Identity for a Layman by Ian Henderson