Английская Википедия:Euler's identity

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Шаблон:Short description Шаблон:Other uses Шаблон:E (mathematical constant) In mathematics, Euler's identity (also known as Euler's equation) is the equality <math display=block>e^{i \pi} + 1 = 0</math> where

<math>e</math> is Euler's number, the base of natural logarithms,

<math>i</math> is the imaginary unit, which by definition satisfies <math>i^2 = -1</math>, and

<math>\pi</math> is pi, the ratio of the circumference of a circle to its diameter.

Euler's identity is named after the Swiss mathematician Leonhard Euler. It is a special case of Euler's formula <math>e^{ix} = \cos x + i\sin x</math> when evaluated for <math>x = \pi</math>. 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 that Шаблон:Pi is transcendental, which implies the impossibility of squaring the circle.

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 number 0, the additive identity
• The number 1, the multiplicative identity
• The [[pi|number Шаблон:Mvar]] (Шаблон:Mvar = 3.1415...), the fundamental circle constant
• The [[e (mathematical constant)|number Шаблон:Math]] (Шаблон:Math = 2.718...), also known as Euler's number, which occurs widely in mathematical analysis
• The [[Imaginary unit|number Шаблон:Math]], the imaginary unit such that <math>i^2=-1</math>

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". 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".

Mathematics writer Constance Reid has opined that Euler's identity is "the most famous formula in all mathematics". 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".

A poll of readers conducted by The Mathematical Intelligencer in 1990 named Euler's identity as the "most beautiful theorem in mathematics". 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".

At least three books in popular mathematics have been published about Euler's identity:

• Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills, by Paul Nahin (2011)
• A Most Elegant Equation: Euler's formula and the beauty of mathematics, by David Stipp (2017)
• Euler's Pioneering Equation: The most beautiful theorem in mathematics, by Robin Wilson (2018).

Explanations

Imaginary exponents

Файл:ExpIPi.gif

In this animation Шаблон:Mvar takes various increasing values from 1 to 100. The computation of Шаблон:Math is displayed as the combined effect of Шаблон:Mvar repeated multiplications in the complex plane, with the final point being the actual value of Шаблон:Math. It can be seen that as Шаблон:Mvar gets larger Шаблон:Math approaches a limit of −1.

Fundamentally, Euler's identity asserts that <math>e^{i\pi}</math> is equal to −1. The expression <math>e^{i\pi}</math> is a special case of the expression <math>e^z</math>, where Шаблон:Math is any complex number. In general, <math>e^z</math> is defined for complex Шаблон:Math by extending one of the definitions of the exponential function from real exponents to complex exponents. For example, one common definition is:

<math>e^z = \lim_{n\to\infty} \left(1+\frac z n \right)^n.</math>

Euler's identity therefore states that the limit, as Шаблон:Math approaches infinity, of <math>(1 + i\pi/n)^n</math> is equal to −1. This limit is illustrated in the animation to the right.

Файл:Euler's formula.svg

Euler's formula for a general angle

Euler's identity is a special case of Euler's formula, which states that for any real number Шаблон:Math,

<math>e^{ix} = \cos x + i\sin x</math>

where the inputs of the trigonometric functions sine and cosine are given in radians.

In particular, when Шаблон:Math,

<math>e^{i \pi} = \cos \pi + i\sin \pi.</math>

Since

<math>\cos \pi = -1</math>

and

<math>\sin \pi = 0,</math>

it follows that

<math>e^{i \pi} = -1 + 0 i,</math>

which yields Euler's identity:

<math>e^{i \pi} +1 = 0.</math>

Geometric interpretation

Any complex number <math>z = x + iy</math> can be represented by the point <math>(x, y)</math> on the complex plane. This point can also be represented in polar coordinates as <math>(r, \theta)</math>, where r is the absolute value of z (distance from the origin), and <math>\theta</math> 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 <math>(r \cos \theta, r \sin \theta)</math>, implying that <math>z = r(\cos \theta + i \sin \theta)</math>. According to Euler's formula, this is equivalent to saying <math>z = r e^{i\theta}</math>.

Euler's identity says that <math>-1 = e^{i\pi}</math>. Since <math>e^{i\pi}</math> is <math>r e^{i\theta}</math> for r = 1 and <math>\theta = \pi</math>, 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 <math>\pi</math> radians.

Additionally, when any complex number z is multiplied by <math>e^{i\theta}</math>, it has the effect of rotating z counterclockwise by an angle of <math>\theta</math> 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 <math>\pi</math> radians around the origin has the same effect as reflecting the point across the origin. Similarly, setting <math>\theta</math> equal to <math>2\pi</math> yields the related equation <math>e^{2\pi i} = 1,</math> 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 Шаблон:Mvarth roots of unity, for Шаблон:Math, add up to 0:

<math>\sum_{k=0}^{n-1} e^{2 \pi i \frac{k}{n}} = 0 .</math>

Euler's identity is the case where Шаблон:Math.

In another field of mathematics, by using quaternion exponentiation, one can show that a similar identity also applies to quaternions. Let Шаблон:Math be the basis elements; then,

<math>e^{\frac{1}{\sqrt 3}(i \pm j \pm k)\pi} + 1 = 0. </math>

In general, given real Шаблон:Math, Шаблон:Math, and Шаблон:Math such that Шаблон:Math, then,

<math>e^{\left(a_1i+a_2j+a_3k\right)\pi} + 1 = 0. </math>

For octonions, with real Шаблон:Math such that Шаблон:Math, and with the octonion basis elements Шаблон:Math,

<math>e^{\left(a_1i_1+a_2i_2+\dots+a_7i_7\right)\pi} + 1 = 0. </math>

History

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, 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 states the following. Шаблон:Quote

See also

• De Moivre's formula
• Exponential function
• Gelfond's constant

References

Sources

• Conway, John H., and Guy, Richard K. (1996), The Book of Numbers, Springer Шаблон:ISBN
• Crease, Robert P. (10 May 2004), "The greatest equations ever", Physics World [registration required]
• Dunham, William (1999), Euler: The Master of Us All, Mathematical Association of America Шаблон:ISBN
• Euler, Leonhard (1922), Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus, Leipzig: B. G. Teubneri
• Kasner, E., and Newman, J. (1940), Mathematics and the Imagination, Simon & Schuster
• Maor, Eli (1998), Шаблон:Mvar: The Story of a number, Princeton University Press Шаблон:ISBN
• Nahin, Paul J. (2006), Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills, Princeton University Press Шаблон:ISBN
• Paulos, John Allen (1992), Beyond Numeracy: An Uncommon Dictionary of Mathematics, Penguin Books Шаблон:ISBN
• Reid, Constance (various editions), From Zero to Infinity, Mathematical Association of America
• Sandifer, C. Edward (2007), Euler's Greatest Hits, Mathematical Association of America Шаблон:ISBN
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• Шаблон:Cite journal
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External links

• Intuitive understanding of Euler's formula