CONJUGATES AND SPECIAL PRODUCTS
Two complex numbers that are alike exceptfor the sign of their imaginary parts are called CONJUGATE COMPLEX NUMBERS. For example, 3 + 5i and 3 - 5i are conjugates. Either number is the conjugate of the other.
If one complex number is known, the conjugate can be obtained immediately by changingthe sign of the imaginary part. The conjugate of -8 + is -8 - The conjugate of - is
The sum of two conjugate complex numbersis a real number, as illustrated by the following:
Product of Two Conjugates
The product of two conjugate complex numbers is a real number. Multiplying two conjugates is equivalent to finding the product of the sum and difference of two numbers.
Consider the following examples:
Squaring a Complex Number
Squaring a complex number is equivalent to raising a binomial to the second power. Forexample :
DIVISION OF COMPLEX NUMBER
When dividing by a pure imaginary, the denomfnator may be rationalized and the problemthus simplified by multiplying both numerator and denominator by the denominator. Thus,
Division of complex numbers can be accomplished by multiplying the numerator and denominator by the number that is the conjugate of the denominator. This process is similar to the process of rationalizing a denominator inthe case of real numbers that are irrational.
As an example, consider
The denominator is 3 + i. Its conjugate is3 - i. Multiplying numerator and denominator by 3- i gives
Practice problems. Rationalize the denominators and simplify: