Theorems
Euler's Identity
OY-ler, rhymes with boiler, not YOO-ler; named for Leonhard Euler
Also Known As Euler's Equation
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Euler's identity is the equation e to the power i times pi, plus 1, equals 0, linking five fundamental constants of mathematics: e, i, pi, 1, and 0. It is considered an exemplar of mathematical beauty, showing a profound connection between the most fundamental numbers in mathematics.
Facts
Disputed
Proof YearThe identity follows from Euler's formula, published in his 1748 Introductio in analysin infinitorum, though historical records suggest Euler may never have explicitly written down this exact identity in this form himself. StatementRaising the base of the natural logarithm, e, to the power of i times pi, and adding 1, gives 0. The equation uses addition, multiplication, and exponentiation exactly once each. 1 Cross-Tradition Connections
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Proved By
Why this is disputed. Euler may never have explicitly written down this exact identity in this form, though it follows directly from his 1748 formula.
Sources
1. Euler's Identity (Wikipedia)
Wikimedia Foundationlead paragraphQuote, lead paragraph
Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation
1. Euler's Identity (Wikipedia)
Wikimedia Foundationhistory and attribution sectionQuote, history and attribution section
historical records suggest Euler may never have explicitly written down this particular form
Euler (Wiktionary)
Wikimedia FoundationPronunciation section, EnglishQuote, Pronunciation section, English
/ˈɔɪlə(ɹ)/, (sometimes proscribed) /ˈjuːlə(ɹ)/
Dissenting Readings (1 dissenting reading)
Proved By: Leonhard Euler
Euler's 1748 formula, e raised to the power i theta equals cosine theta plus i sine theta, implies the identity at theta equals pi, but no surviving source shows Euler himself explicitly writing the compact equation e to the i pi plus 1 equals 0. The identity's attribution to Euler by eponym reflects that it follows directly from his formula, not a documented instance of Euler writing this exact equation himself.
A dissenting reading, from a reviewerEuler's Identity (Wikipedia), Wikimedia Foundation
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