Euler's identity

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

The equation appears in Leonhard Euler's 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 i² = -1), and is Archimedes' constant pi (π, the ratio of the circumference of a circle to its diameter).

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. Here are some interesting properties of these constants:

• The number 0 is the identity element for addition (for all a, a+0=0+a=a). See Group (mathematics).
• The number 1 is the identity element for multiplication (for all a, a*1=1*a=a). Both 0 and 1 are elementary for counting and arithmetic.
• The number is a fundamental number for 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 number for logarithms. Also, is important in describing growth behaviors, as the solution to the simplest growth equation with initial condition is .
• Finally, the imaginary unit (where i² = -1) is a unit in the complex numbers, and is the simplest purely imaginary complex number. Introducing this unit yields all non-constant polynomial equations soluble in the field of complex numbers. (see Fundamental Theorem of Algebra).

The formula also involves the fundamental arithmetical operations of addition, multiplication and exponentiation.

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

for any real number . If we set , then

and since cos(π) = -1 and sin(π) = 0, we get

and

References

• Feynman RP - The Feynman Lectures on Physics, vol. I - part 1. Inter European Editions, Amsterdam (1975)