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Euler's IdentityIn mathematics, Euler's identity is the following equation: -
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 i ² = -1), and 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 . If we set , then -
and since cos(π) = −1 and sin(π) = 0 by definition, 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 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).
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 considering 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)
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