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A 1990 poll of readers by The Mathematical Intelligencer named Euler's identity the "most beautiful theorem in mathematics". [11] In a 2004 poll of readers by Physics World, Euler's identity tied with Maxwell's equations (of electromagnetism) as the "greatest equation ever". [12]
Visual proof of the Pythagorean identity: for any angle , the point (,) = (, ) lies on the unit circle, which satisfies the equation + =.Thus, + =. In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables ...
Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". [2] When x = π, Euler's formula may be rewritten as e iπ + 1 = 0 or e iπ = −1, which is known as Euler's identity.
[11] [12] The technique of approximating sums by integrals (specifically using the integral test or by direct integral approximation) is fundamental in deriving such results. This specific identity can be a consequence of these approximations, considering:
Heine's identity; Hermite's identity; Lagrange's identity; Lagrange's trigonometric identities; List of logarithmic identities; MacWilliams identity; Matrix determinant lemma; Newton's identity; Parseval's identity; Pfister's sixteen-square identity; Sherman–Morrison formula; Sophie Germain identity; Sun's curious identity; Sylvester's ...
An identity is an equation that is true for all possible values of the variable(s) it contains. Many identities are known in algebra and calculus. In the process of solving an equation, an identity is often used to simplify an equation, making it more easily solvable. In algebra, an example of an identity is the difference of two squares:
Consequently, from the equation for the unit circle, + =, the Pythagorean identity. In the figure, the point P has a negative x -coordinate, and is appropriately given by x = cos θ , which is a negative number: cos θ = −cos(π − θ ) .
Comment: The proof of Euler's four-square identity is by simple algebraic evaluation. Quaternions derive from the four-square identity, which can be written as the product of two inner products of 4-dimensional vectors, yielding again an inner product of 4-dimensional vectors: (a·a)(b·b) = (a×b)·(a×b).