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    

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

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

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:


(a + bı) + (a - bı) = 2a    

(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.