The complex numbers are the Field of numbers of the form , where and are Real
Numbers and *i* is the Imaginary Number equal to . When a single letter is
used to denote a complex number, it is sometimes called an ``Affix.'' The Field of complex numbers includes
the Field of Real Numbers as a Subfield.

Through the Euler Formula, a complex number

(1) |

(2) |

(3) |

(4) |

(5) | |||

(6) |

The Powers of complex numbers can be written in closed form as follows:

(7) |

The first few are explicitly

(8) | |||

(9) | |||

(10) | |||

(11) |

(Abramowitz and Stegun 1972).

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 16-17, 1972.

Arfken, G. *Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 353-357, 1985.

Courant, R. and Robbins, H. ``Complex Numbers.'' §2.5 in
*What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.*
Oxford, England: Oxford University Press, pp. 88-103, 1996.

Morse, P. M. and Feshbach, H. ``Complex Numbers and Variables.'' §4.1 in
*Methods of Theoretical Physics, Part I.* New York: McGraw-Hill, pp. 349-356, 1953.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Complex Arithmetic.'' §5.4 in
*Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.* Cambridge, England:
Cambridge University Press, pp. 171-172, 1992.

© 1996-9

1999-05-26