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.
|(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
|(a + bı) - (a - bı) = 2bı||
|(a + bı)×(a - bı) = a2 + b2||
| = ||
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.