The complex numbers are the field of numbers of the form , where and are real numbers and i is the imaginary unit equal to the square root of , . When a single letter is used to denote a complex number, it is sometimes called an "affix." In component notation, can be written . The field of complex numbers includes the field of real numbers as a subfield.
The set of complex numbers is implemented in Mathematica as Complexes. A number can then be tested to see if it is complex using the command Element[x, Complexes], and expressions that are complex numbers have the Head of Complex.
Complex numbers are useful abstract quantities that can be used in calculations and result in physically meaningful solutions. However, recognition of this fact is one that took a long time for mathematicians to accept. For example, John Wallis wrote, "These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen) are reputed to imply that the Case proposed is Impossible" (Wells 1986, p. 22).
Through the Euler formula, a complex number
(1) z=x+iy
may be written in "phasor" form
z=|z|(cos *+isin *)=|z|e^i*
Complex addition
(a+bi)+(c+di)= (a+c)+i(b+d)
complex subtraction
(a+bi)-(c+di)= (a-c)-i(b-d)
complex multiplication
(a+bi)(c+di)=(ac-bd)+i(ad+bc)
and complex division
a+bi/c+di=(ac+bd)+i(bc-ad)/c^2+d^2
