On Complex Numbers XVII (Euler's Formula I)

"There is a famous formula, perhaps the most compact and famous of all formulas developed by Euler from a discovery of de Moivre: It appeals equally to the mystic, the scientist, the philosopher, the mathematician." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)

"Such a close connection between trigonometric functions, the mathematical constant 'e', and the square root of -1 is already quite startling. Surely, such an identity cannot be a mere accident; rather, we must be catching a glimpse of a rich, complicated, and highly abstract mathematical pattern that for the most part lies hidden from our view." (Keith Devlin, "Mathematics: the Science of Patterns", 1994)

"[…] and unlike the physics or chemistry or engineering of today, which will almost surely appear archaic to technicians of the far future, Euler’s formula will still appear, to the arbitrarily advanced mathematicians ten thousand years hence, to be beautiful and stunning and untarnished by time." (Paul J Nahin, "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills", 2006)

"But the number i is special for a decidedly different sort of reason - it’s math’s version of the ugly duckling. [...] The geometric interpretation of e^iπ is rich with emblematic potential. You could see its suggestion of a 180-degree spin as standing for a soldier’s about-face, a ballet dancer’s half pirouette, a turnaround jump shot, the movement of someone setting out on a long journey who looks back to wave farewell, the motion of the sun from dawn to dusk, the changing of the seasons from winter to summer, the turning of the tide. You could also associate it with turning the tables on someone, a reversal of fortune, turning one’s life around, the transformation of Dr. Jekyll into Mr. Hyde (and vice versa), the pivoting away from loss or regret to face the future, the ugly duckling becoming a beauty, drought giving way to rain. You might even interpret its highlighting of opposites as an allusion to elemental dualities—shadow and light, birth and death, yin and yang." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Euler’s general formula, e^iθ = cos θ + i sin θ, also played a role in bringing about the happy ending of the imaginaries’ ugly duckling story. [...] Euler showed that e raised to an imaginary-number power can be turned into the sines and cosines of trigonometry." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Raising e to an imaginary-number power can be pictured as a rotation operation in the complex plane. Applying this interpretation to e raised to the 'i times π' power means that Euler’s formula can be pictured in geometric terms as modeling a half-circle rotation." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"So here’s the main reason that Euler’s formula is flabbergasting: the top five celebrity numbers of all time appear together in it with no other numbers. (In addition, it includes three primordial peers from arithmetic: +, =, and exponentiation.) This conjunction of important numbers, which sprang up in different contexts in math and thus would seem to be completely unrelated, is staggering, and it accounts for much of the hullabaloo about the equation." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"We've seen how it [Euler's identity] can easily be deduced from results of Johann Bernoulli and Roger Cotes, but that neither of them seem to have done so. Even Euler does not seem to have written it down explicitly - and certainly it doesn't appear in any of his publications - though he must surely have realized that it follows immediately from his identity [i.e. Euler's formula], e^ix = cos x + i sin x. Moreover, it seems to be unknown who first stated the result explicitly." (Robin Wilson, "Euler's Pioneering Equation: The most beautiful theorem in mathematics", 2918)

"Gentlemen, that is surely true, 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 is the truth." (Benjamin Peirce [in William E Byerly, "Benjamin Peirce: II. Reminiscences", The American Mathematical Monthly 32 (1), 1925]

"Mathematics is very much like poetry... what makes a good poem - a great poem - is that there is a large amount of thought expressed in very few words. In this insense formulas like e^iπ + 1 = 0 [...] are poems." (Lipman Bers)