- Multiplication of complex numbers is accomplished in a manner similar to the multiplication of binomials. We can use either the distributive property or the FOIL [multiplying First, Outer, Inner, and Last terms together] method to perform the multiplication operation.
- For the sake of simplicity, we can split the multiplication of complex numbers into three different cases.
1) Multiplying a complex number by a real number
- , where is any real number.
2) Multiplying a complex number by a pure imaginary number
- , where is a pure imaginary number.
3) Multiplying two complex numbers
- It is difficult to divide a complex number by another complex number because we cannot divide anything by an imaginary number. The division is only possible when the fraction has a real-number denominator.
- So, the first step towards division must be to eliminate the complex part in the denominator. To do so, we multiply the numerator and the denominator by the complex conjugate of the denominator.
- Recall that the complex conjugate of is .
- Suppose we want to divide by , where . We first write the division as a fraction, find the complex conjugate of the denominator, and multiply it by the numerator and denominator of the fraction.
Example 1: Find the product of and .
Example 2: Compute .
Example 3: Express in the standard form of a complex number.
Example 4: Find the quotient of and .
Example 5: Simplify .
Solution: The denominator can be written as . So, the complex conjugate of the denominator will be or simply .
- Complex number multiplication is carried out similarly to the multiplication of binomials.
- In complex number division, we first multiply the numerator and denominator by the complex conjugate of the denominator and then simplify it until we reach the standard form .
- The complex conjugate of is .
- It should be noted that all real numbers are complex numbers.
- The complex conjugate of any pure imaginary number is negative of the same number—for example, the complex conjugate of is .