origin returns it to its original position. Generalizations Euler's identity is also a special case of the more general identity that the th roots of unity, for , add up to 0: Euler's identity is the case where . In another field of mathematics, by using quaternion exponentiation, one can show that a similar identity also applies to quaternions. Let be the basis elements; then, In general, given real , , and such that , then, For octonions, with real such that , and with the octonion basis elements , History While Euler's identity is a direct result of Euler's formula, published in his monumental work of mathematical analysis in 1748, Introductio in analysin infinitorum, it is questionable whether the particular concept of linking five fundamental constants in a compact form can be attributed to Euler himself, as he may never have expressed it. Robin Wilson states the following. See also De Moivre's formula Exponential function Gelfond's constant Notes References Sources Conway, John H., and Guy, Richard K. (1996), The Book of Numbers, Springer Crease, Robert P. (10 May 2004), "The greatest equations ever", Physics World [registration required] Dunham, William (1999), Euler: The Master of Us All, Mathematical Association of America Euler, Leonhard (1922), Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus, Leipzig: B. G. Teubneri Kasner, E., and Newman, J. (1940), Mathematics and the Imagination, Simon & Schuster Maor, Eli (1998), : The Story of a number, Princeton University Press Nahin, Paul J. (2006), Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills, Princeton University Press Paulos, John Allen (1992), Beyond Numeracy: An Uncommon Dictionary of Mathematics, Penguin Books Reid, Constance (various editions), From Zero to Infinity, Mathematical Association of America Sandifer, C. Edward (2007), Euler's Greatest Hits, Mathematical Association of America External links Intuitive understanding of Euler's formula Exponentials Mathematical identities E (mathematical constant) Theorems in complex analysis Leonhard Euler de:Eulersche Formel#Eulersche Identit.C3.A4t pl:Wzór Eulera#Tożsamość Eulera