Euler's identity

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The exponential function ez can be defined as the limit of (1 + z/N)N, as N approaches infinity, and thus eiπ is the limit of (1 +iπ/N)N. In this animation N takes various increasing values from 1 to 100. The computation of (1 + iπ/N)N is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of (1 +iπ/N)N. It can be seen that as N gets larger (1 +iπ/N)N approaches a limit of −1.

In mathematics, Euler's identity[n 1] (also known as Euler's equation) is the equality

${\displaystyle e^{i\pi }+1=0}$

where

e is Euler's number, the base of natural logarithms,
i is the imaginary unit, which satisfies i2 = −1, and
π 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 considered an example of mathematical beauty.

Explanation

Euler's formula for a general angle

Euler's identity is a special case of Euler's formula from complex analysis, which states that for any real number x,

${\displaystyle e^{ix}=\cos x+i\sin x\,\!}$

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

In particular, when x = π, or one half-turn (180°) around a circle:

${\displaystyle e^{i\pi }=\cos \pi +i\sin \pi .\,\!}$

Since

${\displaystyle \cos \pi =-1\,\!}$

and

${\displaystyle \sin \pi =0,\,\!}$

it follows that

${\displaystyle e^{i\pi }=-1+0i,\,\!}$

which yields Euler's identity:

${\displaystyle e^{i\pi }+1=0.\,\!}$

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]

(Note that both π and e are transcendental numbers.)

Furthermore, the equation is 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."[5] 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".[6]

The mathematics writer Constance Reid has opined that Euler's identity is "the most famous formula in all mathematics".[7] And Benjamin Peirce, a noted American 19th-century 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]

Writer and journalist John Derbyshire, in his book Prime Obsession, quips, "Gauss is supposed to have said -- and I wouldn't put it past him -- that if this was not immediately apparent to you on being told it, you would never be a first-class mathematician."[9] Somewhat countering that, the mathematician and mathematics-historian Harold Edwards, in his review of Derbyshire's book, writes, "If Gauss really said that, it was not his finest moment: but it sounds more like something someone who lacked the prerequisites to even be a second or third class mathematician, or even a first class forger, might have put in his mouth."[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]

Generalizations

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:

${\displaystyle \sum _{k=0}^{n-1}e^{2\pi ik/n}=0.}$

Euler's identity is the case where n = 2.

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

${\displaystyle e^{{\frac {(i\pm j\pm k)}{\sqrt {3}}}\pi }+1=0.\,}$

In general, given real a1, a2, and a3 such that ${\displaystyle {a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}=1}$, then,

${\displaystyle e^{(a_{1}i+a_{2}j+a_{3}k)\pi }+1=0.\,}$

For octonions, with real an such that ${\displaystyle {a_{1}}^{2}+{a_{2}}^{2}+\dots +{a_{7}}^{2}=1}$ and the octonion basis elements {i1, i2,..., i7}, then,

${\displaystyle e^{(a_{1}i_{1}+a_{2}i_{2}+\dots +a_{7}i_{7})\pi }+1=0.\,}$

History

It has been claimed that Euler's identity appears in his monumental work of mathematical analysis published in 1748, Introductio in analysin infinitorum.[13] However, it is questionable whether this particular concept can be attributed to Euler himself, as he may never have expressed it.[14] (Moreover, while Euler did write in the Introductio about what we today call "Euler's formula",[15] which relates e with cosine and sine terms in the field of complex numbers, the English mathematician Roger Cotes also knew of this formula and Euler may have acquired the knowledge through his Swiss compatriot Johann Bernoulli.[14])

Notes and references

Notes

1. The term "Euler's identity" (or "Euler identity") is also used elsewhere to refer to other concepts, including the related general formula eix = cos x + i sin x,[1] and the Euler product formula.[2]

References

1. Dunham, 1999, p. xxiv.
2. {{#invoke:citation/CS1|citation |CitationClass=encyclopaedia }}
3. Template:Cite news
4. Paulos, p. 117.
5. Nahin, 2006, p. 1.
6. Nahin, 2006, p. xxxii.
7. Maor p. 160 and Kasner & Newman pp. 103–104.
8. Derbyshire, p. 210.
9. http://www.jstor.org/discover/10.2307/27641971?uid=3739256&uid=2&uid=4&sid=21104323990401
10. Nahin, 2006, pp. 2–3 (poll published in the summer 1990 issue of the magazine).
11. Crease, 2004.
12. Conway and Guy, pp. 254–255.
13. Sandifer, p. 4.
14. Euler, p. 147.