Simple Steps to Divide Complex Numbers (Animated)

Complex division involves fractions where both the numerator and denominator can contain imaginary numbers.The basic format of complex division looks like this: a plus b i divided by c plus d i.The numerator and denominator are both complex numbers, containing real and imaginary parts.While this may look similar to regular division, complex division requires additional steps to handle the imaginary parts.Let's review the important rules for complex division.First, we must write the division as a single fraction. Second, we cannot leave imaginary numbers in the denominator. And third, both real and imaginary parts must be completely simplified.Let's look at an example to better understand the format.We want to divide three plus two i by one minus four i.First, we write this as a fraction.Then we identify each part: a equals three, b equals two, c equals one, and d equals negative four.In the next section, we'll learn how to eliminate the imaginary part in the denominator using the complex conjugate.For any complex number c plus d i, its complex conjugate is c minus d i.When we multiply a complex number by its conjugate, we multiply both the numerator and denominator by c minus d i.This gives us a fraction with the product of these terms in both numerator and denominator.Let's focus on the denominator and see how c plus d i times c minus d i equals c squared plus d squared.First, multiply the first terms: c times c equals c squared.Next, multiply the outer terms: c times negative d i equals negative c d i.Then multiply the inner terms: d i times c equals c d i.Finally, multiply the last terms: d i times negative d i equals negative d squared i squared, which equals positive d squared since i squared is negative one.Let's see what this means geometrically. Here's our complex number and its conjugate on the complex plane.The complex conjugate is a reflection across the real axis. When we multiply these numbers, we get a real number equal to the square of the distance from the origin.This multiplication always results in a real number: c squared plus d squared.Since we multiply both numerator and denominator by the same value, we're effectively multiplying by one, which doesn't change the value of our fraction.Now that we've multiplied by the complex conjugate, let's simplify the numerator.First, multiply the terms in the numerator using the distributive property.Remember that i squared equals negative one, so negative b d i squared becomes positive b d.Now we can combine like terms. Group the real parts and the terms with i separately.Let's work through an example to see how this process works with actual numbers.Now it's your turn to practice! Try simplifying this complex fraction.Pause the video and try to solve it yourself. Remember to follow the same steps we just learned.Here's the solution. Let's reveal it step by step.Remember, the key to complex division is multiplying by the complex conjugate, then carefully combining like terms and simplifying.Thanks for learning about complex division with Spark.E!