A complex number is a number of the form
*a* + *bi*
, where
*i* =
and
*a*
and
*b*
are real numbers. For example,
5 + 3*i*
,
- + 4*i*
,
4.2 - 12*i*
, and
- -
*i*
are all complex numbers.
*a*
is
called the real part of the complex number and
*bi*
is called the
imaginary part of the complex number. In the complex number
6 - 4*i*
, for
example, the real part is 6 and the imaginary part is
-4*i*
.

To add two complex numbers, add their real parts and add their imaginary parts:
(*a*
_{1} + *b*
_{1}
*i*) + (*a*
_{2} + *b*
_{2}
*i*) = (*a*
_{1} + *a*
_{2}) + (*b*
_{1} + *b*
_{2})*i*
.

*Examples*:

(12 + 6*i*) + (11 + 5*i*) = (12 + 11) + (6 + 5)*i* = 23 + 11*i*

(5 - 7*i*) + (4 + *i*) = (5 + 4) + (- 7 + 1)*i* = 9 - 6*i*
.

(2 - 4*i*) + (- 6 - 5*i*) = (2 - 6) + (- 4 - 5)*i* = - 4 - 9*i*
.

To subtract two complex numbers, subtract their real parts and subtract their
imaginary parts:
(*a*
_{1} + *b*
_{1}
*i*) - (*a*
_{2} + *b*
_{2}
*i*) = (*a*
_{1} - *a*
_{2}) + (*b*
_{1} - *b*
_{2})*i*
.

*Examples*:

(4 + 5*i*) - (2 + 3*i*) = (4 - 2) + (5 - 3)*i* = 2 + 2*i*
.

(3 - 7*i*) - (4 + 6*i*) = (3 - 4) + (- 7 - 6)*i* = - 1 - 13*i*

(- 4 + 2*i*) - (3 - 11*i*) = (- 4 - 3) + (2 - (- 11))*i* = - 7 + 13*i*

(6 - 9*i*) - (- 3 - 4*i*) = (6 - (- 3)) + (- 9 - (- 4))*i* = 9 - 5*i*

To multiply a complex number by a
scalar, multiply the
real part by the scalar and multiply the imaginary part by the scalar:
*c*(*a* + *bi*) = *ca* + *cbi*
.