Euler's identity ties e, i, and π in one equation
Euler's identity equates e raised to i times π, plus one, with zero—or writes the same relation as e to the i-π equals minus one. Named for Leonhard Euler, it is the π case of his formula and is prized as beauty linking five fundamental constants.
The pieces are e, base of natural logs; i, satisfying i squared equals minus one; π, circle circumference over diameter; and the additive and multiplicative identities zero and one. Addition, multiplication, and exponentiation each appear once. Writers call the compact form an exemplar of deep elegance because disparate definitions suddenly share a sentence.
Writing the relation as an expression equal to zero follows a habit common across several branches of mathematics. Admirers have been effusive. Stanford's Keith Devlin likened it to a Shakespearean sonnet, while the writer Constance Reid named it the best-known formula in all of mathematics. Benjamin Peirce, the nineteenth-century Harvard professor, proved it before a class and then confessed that it was paradoxical and that nobody knew what it meant.
Readers of The Mathematical Intelligencer voted it the most beautiful theorem in mathematics in a 1990 poll, and a 2004 Physics World survey had it sharing the title of greatest equation with Maxwell's equations of electromagnetism. At least three popular books are devoted to it: Paul Nahin's Dr. Euler's Fabulous Formula in 2011, David Stipp's A Most Elegant Equation in 2017, and Robin Wilson's Euler's Pioneering Equation in 2018. Nahin, an emeritus professor at the University of New Hampshire, is also among its quoted champions. Behind the fame, making sense of e to a complex power requires extending a real definition, such as the limit of one plus z over n raised to the n.
Source: Euler's identity
