“Euler's identity”的意思、由来-开放百科全书

In mathematics, Euler's identity{{#tag:ref |The term "Euler's identity" (or "Euler identity") is also used elsewhere to refer to other concepts, including the related general formula {{math|e^ix {{=}} cos x + i sin x}},^[1] and the Euler product formula.^[2] |group=n}} (also known as Euler's equation) is the equality

Euler's identity is named after the Swiss mathematician Leonhard Euler. It is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics.

Mathematical beauty

Euler's identity is often cited as an example of deep mathematical beauty.^[3] Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:^[4]

Furthermore, the equation is given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.

Mathematics writer Constance Reid has opined that Euler's identity is "the most famous formula in all mathematics".^[7] 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".^[8]

A poll of readers conducted by The Mathematical Intelligencer in 1990 named Euler's identity as the "most beautiful theorem in mathematics".^[9] 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".^[10]

A study of the brains of sixteen mathematicians found that the "emotional brain" (specifically, the medial orbitofrontal cortex, which lights up for beautiful music, poetry, pictures, etc.) lit up more consistently for Euler's identity than for any other formula.^[11]

At least two books in popular mathematics have been published about Euler's identity. One is A Most Elegant Equation: Euler's formula and the beauty of mathematics, by David Stipp (2017). Another is Euler's Pioneering Equation: The most beautiful theorem in mathematics, by Robin Wilson (2018).

Explanations

Imaginary exponents

Fundamentally, Euler's identity asserts that

is equal to -1. The expression

is a special case of the expression

, where {{math|z}} is any complex number. In general,

is defined for complex {{math|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 {{math|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 {{math|x}},

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

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 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

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.

Generalizations

Euler's identity is also a special case of the more general identity that the {{mvar|n}}th roots of unity, for {{math|n > 1}}, add up to 0:

In another field of mathematics, by using quaternion exponentiation, one can show that a similar identity also applies to quaternions. Let {{math|{{mset|i, j, k}}}} be the basis elements; then,

In general, given real {{math|a₁}}, {{math|a₂}}, and {{math|a₃}} such that {{math|a₁² + a₂² + a₃² {{=}} 1}}, then,

For octonions, with real {{math|a_n}} such that {{math|a₁² + a₂² + ... + a₇² {{=}} 1}}, and with the octonion basis elements {{math|{{mset|i₁, i₂, ..., i₇}}}},

History

It has been claimed that Euler's identity appears in his monumental work of mathematical analysis published in 1748, Introductio in analysin infinitorum.^[12] However, it is questionable whether this particular concept can be attributed to Euler himself, as he may never have expressed it.^[13] Moreover, while Euler did write in the Introductio about what we today call Euler's formula,^[14] which relates {{mvar|e}} with cosine and sine terms in the field of complex numbers, the English mathematician Roger Cotes (who died in 1716, when Euler was only 9 years old) also knew of this formula and Euler may have acquired the knowledge through his Swiss compatriot Johann Bernoulli.^[13]

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 seems 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], {{nowrap|e^ix {{=}} cos x + i sin x}}. Moreover, it seems to be unknown who first stated the result explicitly….

Mathematical beauty

Explanations

Imaginary exponents

Geometric interpretation

Generalizations

History

See also

Notes

References

Sources

External links