Euler's identity

The exponential function e^z can be defined as the limit of (1 + z/N)^N, as N approaches infinity, and thus e^iπ 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 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 mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation

where

is Euler's number, the base of the natural logarithm,

is the imaginary unit, one of the two complex numbers whose square is negative one (the other is ), and

is pi, the ratio of the circumference of a circle to its diameter.

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

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:

• The number 0.
• The number 1.
• The number π, which is ubiquitous in trigonometry, geometry of Euclidean space, and mathematical analysis (π ≈ 3.14159).
• The number e, the base of natural logarithms, which also occurs widely in mathematical analysis (e ≈ 2.71828).
• The number i, imaginary unit of the complex numbers, which contain the roots of all nonconstant polynomials and lead to deeper insight into many operators, such as integration.

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

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]

Claims related to cognition

Where Mathematics Comes From culminates in a perspective from the cognitive science of mathematics, a case study of Euler's identity as an example - the authors George Lakoff and Rafael E. Núñez argue that this identity reflects a cognitive structure peculiar to humans or to their close relatives, the hominids. Variations of that argument were hinted at by "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" in which Eugene Wigner argued to "abandon the idealization that the level of human intelligence has a singular position on an absolute scale. In some cases it may even be useful to consider the attainment which is possible at the level of the intelligence of some other species." In other words, he suggested that mathematics would not only be perceived differently, it would possibly be quite different in concept for beings of greater or lesser intelligence - more or less what Lakoff and Nunez argued more explicitly forty years later. Rather than refer to beauty, status or absolute truth, Lakoff, Nunez and Wigner all thought advanced mathematics - and thus the identity which is so basic to it - reflected something about specifically human (or primate or mammal or biped) intelligence that might or might not be shared by creatures of very different bodies and minds. Kurt Goedel had even earlier argued that an alien tradition of mathematics might not necessarily rely so much on the consistency or proof concepts used in modern Western mathematics, and that even basic ideas (like the identity) might be different, particularly if there was no foundational reason outside mathematics to believe them. To these thinkers, either the identity shapes the human mind, or the human mind shapes the identity, or they have a symbiotic relationship, one that would explain why beauty or truth are so readily seen in the identity by mathematicians, who are human, and perhaps also why some feel compelled to write bad poetry about it.

Derivation

Euler's formula for a general angle.

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

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

Since

and

it follows that

which gives the identity

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:

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

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.^[8] (If so, then this would be an example of Stigler's law of eponymy.) Thus, the question of whether or not the identity should be attributed to Euler is unanswered.

See also

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

Notes

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.