Starting at e⁰ = 1, travelling at the velocity i relative to one's position for the length of time π, and adding 1, one arrives at 0. (The diagram is an Argand diagram)

Part of a series of articles on The mathematical constant, e Leonhard Euler (1707 - 1783) is the eponym of all of the topics listed below. ... Image File history File links EulerIdentity2. ... Image File history File links EulerIdentity2. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... Natural logarithm Image File history File links Euler's_formula. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... Applications in Compound interest · Euler's identity & Euler's formula  · Half lives & Exponential growth/decay Compound interest refers to the fact that whenever interest is calculated, it is based not only on the original principal, but also on any unpaid interest that has been added to the principal. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... Half-Life For a quantity subject to exponential decay, the half-life is the time required for the quantity to fall to half of its initial value. ... In mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the functions current size. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ... Defining e Proof that e is irrational  · Representations of e · Lindemann–Weierstrass theorem In mathematics, the series expansion of the number e can be used to prove that e is irrational. ... The mathematical constant e can be represented in a variety of ways as a real number. ... In mathematics, the Lindemann–Weierstrass theorem states that if α1,...,αn are algebraic numbers which are linearly independent over the rational numbers, then are algebraically independent over the algebraic numbers; in other words the set has transcendence degree n over . ... People John Napier  · Leonhard Euler For other people with the same name, see John Napier (disambiguation). ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Schanuel's conjecture Schanuels conjecture is that given any set of n complex numbers which have linear independence over the rational numbers, the set (up to twice the size) has transcendence degree of at least n over the rationals. ...

In mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation Analysis has its beginnings in the rigorous formulation of calculus. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ...

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. e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... A ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. ...

Nature of the identity

Euler's identity is remarkable for its mathematical beauty. Three basic arithmetic functions are present exactly once: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants: An example of beauty in method - a simple and elegant proof of the Pythagorean theorem. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... In mathematics, multiplication is an elementary arithmetic operation. ... “Exponent” redirects here. ... A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ...

• The number 0, the additive identity.
• The number 1, the multiplicative identity.
• The number π, which is ubiquitous in trigonometry, geometry of Euclidean space, and mathematical analysis.
• The number e, the base of natural logarithms, which occurs widely in mathematical analysis.
• 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. For other senses of this word, see zero or 0. ... This article is about the number one. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Analysis has its beginnings in the rigorous formulation of calculus. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... The natural logarithm is the logarithm to the base e, where e is equal to 2. ... Analysis has its beginnings in the rigorous formulation of calculus. ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... This article is about the concept of integrals in calculus. ...

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] For thermodynamic relations, see Maxwell relations. ...

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] Constance Bowman Reid is the author of several biographies of mathematicians and popular books about mathematics. ...

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] Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ...

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] For others with a similar name, see Benjamin Pierce. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Harvard University is a private university in Cambridge, Massachusetts, USA, and a member of the Ivy League. ...

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] Stanford redirects here. ... Keith Devlin is an English mathematician and writer. ...

Derivation

Euler's formula for a general angle.

The identity is a special case of Euler's formula from complex analysis, which states that Image File history File links Euler's_formula. ... Image File history File links Euler's_formula. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ...

for any real number x. (Note that the arguments to the trigonometric functions sin and cos are in taken to be in radians.) In particular, if In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... In mathematics and physics, the radian is a unit of angle measure. ...

then

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: In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ...

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] Thus, the question of whether or not the identity should be attributed to Euler is unanswered.

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.

See also

• Exponential function
• Gelfond's constant