Today, we're going to talk about dividing complex numbers. So we'll start by reviewing complex numbers, and we'll review conjugates, which we'll use to divide our complex numbers. And then, we'll do an example.
So let's start by reviewing complex numbers. Complex numbers are in the form a plus bi, where a is the real part of a complex number, b is the imaginary part of a complex number, and i is the imaginary unit.
We define i to be equal to the square root of negative 1, and we say that it's an imaginary number. And the reason that we say it's an imaginary number is because no real number squared can equal a negative number.
So let's start by reviewing conjugates. Remember, the conjugate of a binomial is just another binomial with the opposite sign between the two terms. And we use complex conjugates when we're dividing complex numbers, so that we don't have a complex number in the denominator of a fraction.
So let's start by looking at reviewing what the conjugates of these two binomials would be. So for the first example, negative 4 plus 5i, I know the conjugate will be another binomial with just the opposite sign between the terms. So instead of plus, I'll have minus. So the conjugate will be negative 4 minus 5i.
Here, I have a minus sign between my two terms, so my conjugate will have a plus sign. So the conjugate of 3 minus i is 3 plus i. So now, let's see how we can use these complex conjugates to divide complex numbers.
So let's do an example dividing complex numbers. I've got 2 plus 4i, divided by 3 minus 5i. I'm going to divide these two complex numbers using complex conjugates. And that process is very similar to the one that you use when you are rationalizing a denominator using conjugates.
So I'm going to divide these two complex numbers, first by identifying the complex conjugate of my denominator. The complex conjugate of 3 minus 5i is 3 plus 5i. And I'm going to multiply by 3 plus 5i in the numerator and the denominator of my fraction.
So multiply by 3 plus 5i in the numerator and in the denominator. And I do that because multiplying by 3 plus 5i in the numerator and 3 plus 5i in the denominator is the same as multiplying by a fraction that is equal to 1, which does not change our original expression.
So when I multiply these two fractions together, I'm going to multiply the two expressions in the numerators. I'm going to do that using FOIL.
So multiplying my two first terms, 2 times 3, that will give me 6. My outside two terms, 2 and 5i, multiplied will give me 10i. My inside two terms, 4i and 3, multiplied will give me 12i. And my last two terms, 4i and 5i, multiplied will give me 20i squared.
And my denominator is going to multiply again, using FOIL. So 3 times 3 will give me 9. My outside two terms, 3 and 5i, will give me 15i. My inside two terms, negative 5i and 3, will give me negative 15i. And my last two terms, negative 5i and positive 5i, will give me negative 25i squared.
So I'm going to simplify this, first by combining my like terms. In my numerator, I have 10i and 12i. I'm going to add them together, and that will give me 22i. Similarly in my denominator, I have two like terms, 15i positive and a negative 15i. When I combine these together, they're going to cancel each other out.
And this will always happen when you're multiplying by a complex conjugate. You'll always have two terms that will cancel each other out. So I'll be left with 9 and a negative 25i squared.
The second thing I can do to simplify this is substituting my i squared with a negative 1. I know that i squared is equal to negative 1, so I'm going to replace both i squareds with negative 1. So I'll have 6 plus 22i, and then plus 20 times negative 1, over 9 minus 25 times negative 1.
Now I'm going to simplify both in the numerator and the denominator by multiplying by that negative 1. So this will become plus negative 20, and this will become minus negative 25.
Now, I can combine in the numerator my two constants. 6 plus a negative 20 is going to give me negative 14. So I'll negative 14 plus 22i over-- 9 minus negative 25 will give me 34.
And now, I can separate my fraction into two separate fractions, one being the real part of this complex number, and one being the imaginary part. So separating this out, this becomes negative 12/34 plus 22i/34. And I can reduce both of these fractions. So this will become negative 7/17 plus 11/17i for my final answer.
So let's go over our key points from today. Complex numbers consist of a real part and an imaginary part. The square root of negative 1 is imaginary, because no real number squared results in a negative number.
Conjugates are used when dividing complex numbers to eliminate a complex number from the denominator. And simplify expressions with an i squared term by substituting negative 1 for i squared, multiplying by the coefficient, and writing as a real number with the opposite sign.
So I hope that these key points and examples helped you understand a little bit more about dividing complex numbers. Keep using your notes and keep on practicing, and soon, you'll be a pro. Thanks for watching.
the conjugate of a binomial is a binomial with the opposite sign between its terms