Euler's identity
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In mathematics, Euler's identity is the following equation:
- <math>e^{i \pi} = -1 \,\!<math>
sometimes expressed as:
- <math>e^{i \pi} + 1 = 0 \,\!<math>
presumably in order to use the fundamental numbers 0 and 1 (see below).
The equation appears in Leonhard Euler's Introduction, published in Lausanne in 1748. In this equation, e is the base of the natural logarithm, <math>i<math> is the imaginary unit (an imaginary number with the property i ² = -1), and <math> \pi <math> 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
- <math>e^{ix} = \cos x + i \sin x \,\!<math>
for any real number <math>x<math>. If we set <math>x = \pi<math>, then
- <math>e^{i \pi} = \cos \pi + i \sin \pi \,\!<math>
and since cos(π) = −1 and sin(π) = 0 by definition, we get
- <math>e^{i \pi} = -1 \,\!<math>
Perceptions of the identity
Benjamin Peirce, 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, as well as many others, 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 <math>\pi <math> is fundamental in trigonometry, <math>\pi <math> is a constant in a world which is Euclidean, or on small scales of non-Euclidean geometry (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 <math>e<math> 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 <math>dy / dx = y<math> with initial condition <math>y(0) = 1<math> is <math>y = e^x<math>).
- The imaginary unit <math>i<math> (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.
From a very naïve point of view, the result can also be considered remarkable because <math>e^{\pi}\approx 23<math> is a somehow big positive number, while inserting a "little" i gives <math>e^{i \pi} = -1<math>, a rather small negative number. (It is of course well known that only the real part of the exponent of a real number has an effect on the absolute value).
References
- Feynman, Richard P. The Feynman Lectures on Physics, vol. I - part 1. Inter European Editions, Amsterdam (1975)
External links
- 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)ca:Identitat d'Euler
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