0,2

If one considers rotations in the complex plane the complex number i can be thought of as a rotation by 90 degrees counterclockwise. This also corresponds to half of pi. If one raises the number e (2.718...) to the angle in question, (that angle corresponding to the complex number i, or half of pi), the result is an number 4.81047738... which can be thought of as corresponding (but not being equivalent to) the complex number, i.

The next 90 degree sweep, which corresponds to i^2, is 23.1406..., and is the known constant, e^pi. Continuing around the circle, we find the numbers i^3 = 111.31..., i^4 = 535.4... etc. Truncating these numbers to integers yields the series, 0, 4, 23, 111, etc. I came across this sequence as I tried to find a real number that corresponded to the complex number 'i' in the complex plane.

Frank Ayres, Jr., Robert E. Moyer and Murray R. Spiegel, "Essential Trigonometry and Algebra for University Students" (?) (1999), p. 3.

Roger Penrose, The Road to Reality (?) (2005), p. 88 (figure 5.3)

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics,Euler's Famous Identity.

e^(i*Pi)+1=0 <==> i=e^(i*Pi/2). Therefore i^n can imply e^(n*i*Pi/2).

Table[ Floor@ Exp[n*Pi/2], {n, 0, 21}] (* Robert G. Wilson v *)

Cf. A042972.

Sequence in context: A038381 A082970 A017973 this_sequence A015532 A024050 A038723

Adjacent sequences: A124504 A124505 A124506 this_sequence A124508 A124509 A124510

nonn

Zacariaz Martinez (euneirophrenia(AT)gmail.com), Dec 27 2006

Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 31 2006