e^(ix) = cos x + i sin x, the general formula of which the famous equation is one case (1748)
Euler's formula, in the Introductio in analysin infinitorum
Science Β· Mathematics
Euler's identity
1748
Mathematics β analysis and complex numbers
The equation e^(iΟ) + 1 = 0, routinely voted the most beautiful in mathematics because it links five fundamental constants β the base of natural logarithms, the circle constant, the imaginary unit, one and zero β in a single line using addition, multiplication and exponentiation exactly once each. It is the special case of Euler's formula, e^(ix) = cos x + i sin x, published in the Introductio in analysin infinitorum of 1748, at x equal to half a turn. Richard Feynman called the formula "our jewel" and "the most remarkable formula in mathematics".
How to spot them
- βΈFive constants in one equation, each used once, summing to nothing
- βΈThe case x = Ο of a formula relating the complex exponential to cosine and sine
- βΈCalled 'our jewel' by Richard Feynman
- βΈBenjamin Peirce is said to have told his class it was absolutely paradoxical yet certainly true
- βΈRoutinely tops polls of the most beautiful equation in mathematics
Key facts
The equation containing five fundamental constants, each appearing exactly once
e, the circle constant, the imaginary unit, 1 and 0
Feynman's 'our jewel' and 'the most remarkable formula in mathematics' (1977)
From volume I, chapter 22 of the Feynman Lectures on Physics
Roger Cotes's logarithmic version of 1714, more than thirty years before the famous statement (1714)
Cotes gave the equivalent ix = log(cos x + i sin x); Newton called him a great genius
A Harvard professor's lecture remark that it is absolutely paradoxical and certainly true
Benjamin Peirce, as reported by his students
The result of raising e to the power of the circle constant times the square root of β1
β1, which is why adding 1 gives 0
Voted the most beautiful theorem in mathematics in a 1988 reader poll by The Mathematical Intelligencer (1988)
It came first, ahead of the polyhedron formula V β E + F = 2
Key ideas
- Euler's formula turns the complex exponential into rotation: e^(ix) traces the unit circle in the complex plane as x increases, so at x = Ο you have travelled half way round and arrived at β1.
- Because it relates exponentials to sine and cosine, the formula is the engine of Fourier analysis, signal processing and AC circuit theory.
Quick-fire facts
- β’The identity is e^(iΟ) + 1 = 0, the case x = Ο of e^(ix) = cos x + i sin x
- β’Euler published the formula in his Introductio in analysin infinitorum (1748), though Roger Cotes had an equivalent logarithmic statement in 1714
- β’The five constants are e, Ο, i, 1 and 0; the operations are addition, multiplication and exponentiation
- β’Richard Feynman called the formula 'our jewel' and 'the most remarkable formula in mathematics' in his Lectures on Physics
- β’The Harvard mathematician Benjamin Peirce is quoted as calling it 'absolutely paradoxical; we cannot understand itβ¦ but we have proved it, and therefore we know it must be the truth'
- β’It topped a 1988 poll of readers of The Mathematical Intelligencer and a 2004 Physics World poll of the greatest equations, tied with Maxwell's equations
- β’A related curiosity is that i to the power i is a real number, approximately 0.20788
- β’Euler (1707β1783) was born in Basel and worked in Saint Petersburg and Berlin; he went blind in later life and remained prodigiously productive
- β’The general identity e^(2Οi) = 1 expresses that a full turn returns you to the start
Drill it β 7 cards
1 / 7new card
Sources
- MacTutor History of Mathematics Archive, University of St Andrews β Leonhard Euler
- Richard P. Feynman β The Feynman Lectures on Physics, Volume I, Chapter 22