# Euler's identity

In mathematics,**Euler's identity**, a special case of Euler's formula, is the following:

*Introduction*, published in Lausanne in 1748. In this equation,

*e*is the base of the natural logarithm, is the imaginary unit (an imaginary number with the property

*i*² = -1), and is Archimedes' constant pi (π, the ratio of the circumference of a circle to its diameter).

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

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2 References 3 External link |

## Perceptions of the identity

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 is fundamental in trigonometry, 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 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 with initial condition is ).
- The imaginary unit (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).

In addition, the result is remarkable to most students learning it for the first time because it is so highly counterintuitive. Consider that

- while

*i*where

*i*

^{2}= −1 changes the result dramatically.

## References

- Feynman R.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
- Proof of Euler's Identity for a Layman'' by Ian Henderson