<< Chapter < Page | Chapter >> Page > |
Now, let’s multiply two complex numbers. We can use either the distributive property or more specifically the FOIL method because we are dealing with binomials. Recall that FOIL is an acronym for multiplying First, Inner, Outer, and Last terms together. The difference with complex numbers is that when we get a squared term, it equals
Given two complex numbers, multiply to find the product.
Dividing two complex numbers is more complicated than adding, subtracting, or multiplying because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator to write the answer in standard form We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of is For example, the product of and is
The result is a real number.
Note that complex conjugates have an opposite relationship: The complex conjugate of is and the complex conjugate of is Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.
Suppose we want to divide by where neither nor equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.
Multiply the numerator and denominator by the complex conjugate of the denominator.
Apply the distributive property.
Simplify, remembering that
The complex conjugate of a complex number is It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.
Find the complex conjugate of each number.
Notification Switch
Would you like to follow the 'College algebra' conversation and receive update notifications?