Euler's identity

From Wikipedia, the free encyclopedia

For other uses, see List of topics named after Leonhard Euler.
The exponential function ez can be defined as the limit of (1+z/N)N, as N approaches infinity. Here, we take z=iπ, and take N to be various increasing values from 1 to 100. The computation of (1+iπ / N)N is displayed as N repeated multiplications in the complex plane, with the final point being the actual value of (1+iπ / N)N. As N gets larger, it can be seen that (1+iπ / N)N approaches a limit of -1. Therefore eiπ=-1.
The exponential function ez can be defined as the limit of (1+z/N)N, as N approaches infinity. Here, we take z=iπ, and take N to be various increasing values from 1 to 100. The computation of (1+iπ / N)N is displayed as N repeated multiplications in the complex plane, with the final point being the actual value of (1+iπ / N)N. As N gets larger, it can be seen that (1+iπ / N)N approaches a limit of -1. Therefore eiπ=-1.

  Part of a series of articles on
The mathematical constant, e

Natural logarithm

Applications in: compound interest · Euler's identity & Euler's formula  · half-lives & exponential growth/decay

Defining e: proof that e is irrational  · representations of e · Lindemann–Weierstrass theorem

People John Napier  · Leonhard Euler

Schanuel's conjecture

In mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation

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

where

e\,\! is Euler's number, the base of the natural logarithm,
i\,\! is the imaginary unit, one of the two complex numbers whose square is negative one (the other is -i\,\!), and
\pi\,\! is pi, the ratio of the circumference of a circle to its diameter.

Euler's identity is also sometimes called Euler's equation.

Contents

[edit] Nature of the identity

Euler's identity is considered by many to be remarkable for its mathematical beauty. Three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:

Furthermore, in mathematical analysis, equations are commonly written with zero on one side.

[edit] Perceptions of the identity

A reader poll conducted by Mathematical Intelligencer named the identity as the most beautiful theorem in mathematics.[1] Another reader poll conducted by Physics World in 2004 named Euler's identity the "greatest equation ever", together with Maxwell's equations.[2]

The book Dr. Euler's Fabulous Formula [2006], by Paul Nahin (Professor Emeritus at the University of New Hampshire), is devoted to Euler's identity; it is 400 pages long. The book states that the identity sets "the gold standard for mathematical beauty."[3]

Constance Reid claimed that Euler's identity was "the most famous formula in all mathematics."[4]

Gauss is reported to have commented that if this formula was not immediately apparent to a student on being told it, the student would never be a first-class mathematician.[5]

After proving the identity in a lecture, Benjamin Peirce, a noted nineteenth century mathematician and Harvard professor, said, "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." [6]

Stanford mathematics professor Keith Devlin says, "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]

[edit] Derivation

Euler's formula for a general angle.
Euler's formula for a general angle.

The identity is a special case of Euler's formula from complex analysis, which states that

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

for any real number x. (Note that the arguments to the trigonometric functions sin and cos are taken to be in radians.) In particular, if

x = \pi,\,\!

then

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

Since

\cos \pi = -1 \, \!

and

\sin \pi = 0,\,\!

it follows that

e^{i \pi} = -1,\,\!

which gives the identity

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

[edit] Taylor Series Expansion

Graph of the Taylor series expansion of eiπ on the complex plane.
Graph of the Taylor series expansion of eiπ on the complex plane.

\begin{align} e^{i \pi} &{}= \left( 1 - \frac{\pi^2}{2!} + \frac{\pi^4}{4!} - \frac{\pi^6}{6!} + \cdots \right) + i\left( \pi - \frac{\pi^3}{3!} + \frac{\pi^5}{5!} - \frac{\pi^7}{7!} + \cdots \right) \\ &{}= \cos (\pi) + i\sin (\pi) \end{align}

The identity can be expressed as the Taylor series expansion of Euler's formula, where z=π, thereby expressing eiπ as the sum of an infinite series. This can be plotted on the complex plane, plotting vectors which are approximations of increasing accuracy, converging around the point (0,-1), with complex part = 0 and real part = -1. The i in each term gives each vector a quarter turn, creating a spiral of shorter and shorter straight-line segments around (0,-1). This diagram was first plotted by LWH Hull in the Mathematical Gazette in 1959.[8] It has since been reconstructed by Takahashi and McGhee.

[edit] Generalization

Euler's identity is a special case of the more general identity that the nth roots of unity, for n > 1, add up to 0:

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

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

[edit] Attribution

While Euler wrote about his formula relating e to cos and sin terms, there is no known record of Euler actually stating or deriving the simplified identity equation itself; moreover, the formula was likely known before Euler.[9] Thus, the question of whether or not the identity should be attributed to Euler is unanswered. (If so, then this would be an example of Stigler's law of eponymy).

[edit] Notes

  1. ^ Nahin, 2006, p.2–3 (poll published in summer 1990 issue).
  2. ^ Crease, 2004.
  3. ^ Cited in Crease, 2007.
  4. ^ Reid.
  5. ^ Derbyshire p.210.
  6. ^ Maor p.160 and Kasner & Newman p.103–104.
  7. ^ Nahin, 2006, p.1.
  8. ^ See "Convergence on the Argand Diagram" in Mathematical Gazette, Vol 43, No 345 (October 1959).
  9. ^ Sandifer.

[edit] References

  • Crease, Robert P., "The greatest equations ever", PhysicsWeb, October 2004.
  • Crease, Robert P. "Equations as icons," PhysicsWeb, March 2007.
  • Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (New York: Penguin, 2004).
  • Kasner, E., and Newman, J., Mathematics and the Imagination (Bell and Sons, 1949).
  • Maor, Eli, e: The Story of a number (Princeton University Press, 1998), ISBN 0-691-05854-7
  • Nahin, Paul J., Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills (Princeton University Press, 2006), ISBN 978-0691118222
  • Reid, Constance, From Zero to Infinity (Mathematical Association of America, various editions).
  • Sandifer, Ed, "Euler's Greatest Hits", MAA Online, February 2007.

[edit] See also