Two complex numbers are equal if and only if their real parts are equal
and their imaginary parts are equal. For any complex numbers
a + bı
and
c + dı
, the following rules are true for the four basic operations.
operations
(a + bı) + (c + dı) = (a + c) + (b + d )ı


(a + bı)  (c + dı) = (a  c) + (b  d )ı


(a + bı)×(c + dı) = (ac  bd )+ (ad + bc)ı


fraca+bıc+dı =


The technique for dividing complex numbers involves multiplying the denominator
by its complex conjugate. The complex conjugate of a complex number
a + bı
is
a  bı
. Complex conjugates are useful and special for the
following reasons:
conjugates
(a + bı)  (a  bı) = 2bı


(a + bı)×(a  bı) = a
^{2} + b
^{2}


=


When complex conjugates are added or multiplied, the sum or product is real. So
when the denominator of a fraction is a complex number, you can multiply the
numerator and the denominator by the complex conjugate of the denominator to get
a fraction with a real denominator.
The complex conjugate is an important tool for simplifying expressions with
complex numbers. Another important fact about complex conjugates is that when a
complex number is the root of a polynomial with real coefficients, so is its
complex conjugate. We'll take a closer look in the next section.