# Online Encyclopedia

# Euler's identity

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).

The identity is a special case of Euler's formula from complex analysis, which states that

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."

It was called "*the most remarkable formula in mathematics*" by Richard Feynman. Feynman found this formula remarkable because it links some very fundamental mathematical constants:

- 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*d**y*/*d**x*=*y*with initial condition*y*(0) = 1 is*y*=*e*^{x}). - 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

- while

The simple insertion of *i* changes the result dramatically.

## References

- Feynman, Richard P.
*The Feynman Lectures on Physics*, vol. I - part 1. Inter European Editions, Amsterdam (1975)

## External link

*Proof of Euler's Identity*by Julius O. Smith III http://ccrma-www.stanford.edu/~jos/mdft/Proof_Euler_s_Identity.html*Proof of Euler's Identity for a Layman*by Ian Henderson http://board.c8world.com/index.php?showtopic=2202