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Euler's identity is often cited as an example of deep mathematical beauty. Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:
The number 0, the additive identity.
The number 1, the multiplicative identity.
The number π (π = 3.141...), the fundamental circle constant.
The number e (e = 2.718...), a.k.a. Euler's number, which occurs widely in mathematical analysis.
The number i, the imaginary unit of the complex numbers.
Furthermore, the equation is given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.
Stanford University mathematics professor Keith Devlin has said, "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". And Paul Nahin, a professor emeritus at the University of New Hampshire, who has written a book dedicated to Euler's formula and its applications in Fourier analysis, describes Euler's identity as being "of exquisite beauty".
Mathematics writer Constance Reid has opined that Euler's identity is "the most famous formula in all mathematics". And Benjamin Peirce, a 19th-century American philosopher, mathematician, and professor at Harvard University, after proving Euler's identity during a lecture, stated that the identity "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".
A poll of readers conducted by The Mathematical Intelligencer in 1990 named Euler's identity as the "most beautiful theorem in mathematics". In another poll of readers that was conducted by Physics World in 2004, Euler's identity tied with Maxwell's equations (of electromagnetism) as the "greatest equation ever".
A study of the brains of sixteen mathematicians found that the "emotional brain" (specifically, the medial orbitofrontal cortex, which lights up for beautiful music, poetry, pictures, etc.) lit up more consistently for Euler's identity than for any other formula.
At least three books in popular mathematics have been published about Euler's identity: