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In mathematics, Euler's identity [note 1] (also known as Euler's equation) is the equality
where
Euler's identity is named after the Swiss mathematician Leonhard Euler. It is a special case of Euler's formula when evaluated for . Euler's identity is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in a proof [3] [4] that π is transcendental, which implies the impossibility of squaring the circle.
Euler's identity is often cited as an example of deep mathematical beauty. [5] Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants: [6]
The equation is often given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.
Stanford University mathematics professor Keith Devlin has said, "like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence". [7] And Paul Nahin, a professor emeritus at the University of New Hampshire, who has written a book dedicated to Euler's formula and its applications in Fourier analysis, describes Euler's identity as being "of exquisite beauty". [8]
Mathematics writer Constance Reid has opined that Euler's identity is "the most famous formula in all mathematics". [9] And Benjamin Peirce, a 19th-century American philosopher, mathematician, and professor at Harvard University, after proving Euler's identity during a lecture, stated that the identity "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". [10]
A poll of readers conducted by The Mathematical Intelligencer in 1990 named Euler's identity as the "most beautiful theorem in mathematics". [11] In another poll of readers that was conducted by Physics World in 2004, Euler's identity tied with Maxwell's equations (of electromagnetism) as the "greatest equation ever". [12]
At least three books in popular mathematics have been published about Euler's identity:
Fundamentally, Euler's identity asserts that is equal to −1. The expression is a special case of the expression , where z is any complex number. In general, is defined for complex z by extending one of the definitions of the exponential function from real exponents to complex exponents. For example, one common definition is:
Euler's identity therefore states that the limit, as n approaches infinity, of is equal to −1. This limit is illustrated in the animation to the right.
Euler's identity is a special case of Euler's formula, which states that for any real number x,
where the inputs of the trigonometric functions sine and cosine are given in radians.
In particular, when x = π,
Since
and
it follows that
which yields Euler's identity:
Any complex number can be represented by the point on the complex plane. This point can also be represented in polar coordinates as , where r is the absolute value of z (distance from the origin), and is the argument of z (angle counterclockwise from the positive x-axis). By the definitions of sine and cosine, this point has cartesian coordinates of , implying that . According to Euler's formula, this is equivalent to saying .
Euler's identity says that . Since is for r = 1 and , this can be interpreted as a fact about the number −1 on the complex plane: its distance from the origin is 1, and its angle from the positive x-axis is radians.
Additionally, when any complex number z is multiplied by , it has the effect of rotating z counterclockwise by an angle of on the complex plane. Since multiplication by −1 reflects a point across the origin, Euler's identity can be interpreted as saying that rotating any point radians around the origin has the same effect as reflecting the point across the origin. Similarly, setting equal to yields the related equation which can be interpreted as saying that rotating any point by one turn around the origin returns it to its original position.
Euler's identity is also a special case of the more general identity that the nth roots of unity, for n > 1, add up to 0:
Euler's identity is the case where n = 2.
A similar identity also applies to quaternion exponential: let {i, j, k} be the basis quaternions; then,
More generally, let q be a quaternion with a zero real part and a norm equal to 1; that is, with Then one has
The same formula applies to octonions, with a zero real part and a norm equal to 1. These formulas are a direct generalization of Euler's identity, since and are the only complex numbers with a zero real part and a norm (absolute value) equal to 1.
While Euler's identity is a direct result of Euler's formula, published in his monumental work of mathematical analysis in 1748, Introductio in analysin infinitorum , [16] it is questionable whether the particular concept of linking five fundamental constants in a compact form can be attributed to Euler himself, as he may never have expressed it. [17]
Robin Wilson states the following. [18]
We've seen how it [Euler's identity] can easily be deduced from results of Johann Bernoulli and Roger Cotes, but that neither of them seem to have done so. Even Euler does not seem to have written it down explicitly – and certainly it doesn't appear in any of his publications – though he must surely have realized that it follows immediately from his identity [i.e. Euler's formula], eix = cos x + i sin x. Moreover, it seems to be unknown who first stated the result explicitly....
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation ; every complex number can be expressed in the form , where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number ,a is called the real part, and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world.
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x, one has
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are generally expressed in either degrees or radians.
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