### 4.4 Complex Numbers

There are two supported formats for complex numbers: rectangular and
polar. The default format is rectangular, displayed in the form
‘`(``real`,`imag`)’ where `real` is the real part and
`imag` is the imaginary part, each of which may be any real number.
Rectangular complex numbers can also be displayed in ‘`a`+`b`i’
notation; see Complex Formats.

Polar complex numbers are displayed in the form
‘`(``r``;``theta``)`’
where `r` is the nonnegative magnitude and
`theta`
is the argument or phase angle. The range of
`theta`
depends on the current angular mode (see Angular Modes); it is
generally between *-180* and *+180* degrees or the equivalent range
in radians.

Complex numbers are entered in stages using incomplete objects.
See Incomplete Objects.

Operations on rectangular complex numbers yield rectangular complex
results, and similarly for polar complex numbers. Where the two types
are mixed, or where new complex numbers arise (as for the square root of
a negative real), the current *Polar mode* is used to determine the
type. See Polar Mode.

A complex result in which the imaginary part is zero (or the phase angle
is 0 or 180 degrees or ‘`pi`’ radians) is automatically converted to a real
number.