Euler's identity

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.

  Part of a series of articles on
The mathematical constant e
Euler's formula.svg

Natural logarithm · Exponential function

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 analytical mathematics, Euler's Identity, named for the Swiss-German mathematician Leonhard Euler, is the equality

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
\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

Nature of the identity

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

Furthermore, in algebra and other areas of mathematics, equations are commonly written with zero on one side of the equal sign.

The common perceptions of this identity

A poll of readers that was conducted by The Mathematical Intelligencer magazine named Euler's Identity as the "most beautiful theorem in mathematics".[1] Another poll of readers that was conducted by Physics World magazine in 2004 chose Euler's Identity as the "greatest equation ever", in a dead heat with the four Maxwell's equations of electromagnetism.[2]

An entire 400-page mathematics book, Dr. Euler's Fabulous Formula (published in 2006), written by Dr. Paul Nahin (a Professor Emeritus at the University of New Hampshire), is devoted to Euler's Identity. This monograph states that Euler's 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]

The mathematician Carl Friedrich Gauss was reported to have commented that if this formula was not immediately apparent to a student upon being told it, that student would never become a first-class mathematician.[5]

After proving Euler's Identity during a lecture, Benjamin Peirce, a noted American 19th Century philosopher/mathematician and a professor at Harvard University, stated 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." [6]

The Stanford University mathematics professor, Dr. Keith Devlin, 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."[7]

A Derivation of Euler's Identity

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 sine and cosine are taken to be in radians, and not in degrees.) In particular,

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.\,\!

Generalizations of Euler's Identity

Euler's Identity is actually 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.

In another field of mathematics, by using quaternion exponentiation, one can show that a similar identity also applies to quaternions:

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

Attribution

While Euler wrote about his formula that relates e with cosine and sine terms, in the field of complex numbers, there is no known record of Euler's actually stating or deriving the simplified identity equation itself. Furthermore, Euler's formula was probably known before the life of Euler.[8] (If so, then this usage would be an example of Stigler's law of eponymy.) Thus, the question of whether or not this identity should be attributed to Euler is unanswerable.

See also

Notes

  1. ↑ Nahin, 2006, p.2–3 (poll published in the summer 1990 issue of the magazine).
  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. ↑ Sandifer.

References