Euler's identity


In mathematics, Euler's identity 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 an exemplar of mathematical beauty, as it shows a profound connection between the most fundamental numbers in mathematics.
Fandom math:Euler's_identity

Mathematical beauty

Euler's identity is often cited as an example of deep mathematical beauty. Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:
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". Paul Nahin, a professor emeritus at the University of New Hampshire who wrote a book dedicated to Euler's formula and its applications in Fourier analysis, said Euler's identity is "of exquisite beauty".
Mathematics writer Constance Reid has said that Euler's identity is "the most famous formula in all mathematics". Benjamin Peirce, a 19th-century American philosopher, mathematician, and professor at Harvard University, after proving Euler's identity during a lecture, said that 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".
A 1990 poll of readers by The Mathematical Intelligencer named Euler's identity the "most beautiful theorem in mathematics". In a 2004 poll of readers by Physics World, Euler's identity tied with Maxwell's equations as the "greatest equation ever".
At least three books in popular mathematics have been published about Euler's identity:

Explanations

Imaginary exponents

Euler's identity asserts that is equal to −1. The expression is a special case of the expression, where is any complex number. In general, is defined for complex 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 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,
where the inputs of the trigonometric functions sine and cosine are given in radians.
In particular, when,
Since
and
it follows that
which yields Euler's identity:

Geometric interpretation

Any complex number can be represented by the point on the complex plane. This point can also be represented in polar coordinates as where is the absolute value of, and is the argument of . 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 = 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 is multiplied by it has the effect of rotating 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.

Generalizations

Euler's identity is also a special case of the more general identity that the th roots of unity, for, add up to 0:
Euler's identity is the case where.
A similar identity also applies to quaternion exponential: let be the basis quaternions; then,
More generally, let 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 equal to 1.

History

Euler's identity is a direct result of Euler's formula, first published in his monumental 1748 work of mathematical analysis, Introductio in analysin infinitorum, but 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.
Robin Wilson writes: