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College Algebra

Course: College Algebra>Unit 6

Lesson 4: Complex conjugates and dividing complex numbers

Dividing complex numbers review

Review your complex number division skills.

How do we divide complex numbers?

Dividing a complex number by a real number is simple. For example:
$\begin{array}{rl}\frac{2+3i}{4}& =\frac{2}{4}+\frac{3}{4}i\\ \\ & =0.5+0.75i\end{array}$
Finding the quotient of two complex numbers is more complex (haha!). For example:
$\begin{array}{rl}& \phantom{=}\frac{20-4i}{3+2i}\\ \\ & =\frac{20-4i}{3+2i}\cdot \frac{3-2i}{3-2i}\end{array}$
We multiplied both sides by the conjugate of the denominator, which is a number with the same real part and the opposite imaginary part. What's neat about conjugate numbers is that their product is always a real number. Let's continue:
$\begin{array}{rl}& =\frac{\left(20-4i\right)\left(3-2i\right)}{\left(3+2i\right)\left(3-2i\right)}\\ \\ & =\frac{52-52i}{13}\end{array}$
Multiplying the denominator $\left(3+2i\right)$ by its conjugate $\left(3-2i\right)$ had the desired effect of getting a real number in the denominator. To keep the quotient the same, we had to multiply the numerator by $\left(3-2i\right)$ as well. Now we can finish the calculation:
$\begin{array}{rl}& =\frac{52}{13}-\frac{52}{13}i\\ \\ & =4-4i\end{array}$

Problem 1
$\frac{4+2i}{-1+i}=$

Want to try more problems like this? Check out this exercise.

Want to join the conversation?

• What do I do when I have a problem like this : 3 / 2+i
The "3" is over the "2+i".

I don't know what to do and nothing will explain.
• Here's a hint: you need to "rationalize" the denominator. When you see 2+i in the denominator, what you really have is 2 + √(-1)

To rationalize the denominator, try this:

3/(2+i) · (2-i)/(2-i)
= 3/(2+√(-1)) · (2-i)/(2-√(-1))
...and so on.

Try working this out, and please let me know if you have any more questions.

Hope this helps!
• What would be a real world application where imaginary numbers would be involved in practical applications? Fractal geometry is excluded!
• Imaginary numbers are used in electrical engineering to describe AC voltages.
• Hello, The equation from a review question i did was: 6-6i/8+2i. The answer I got was: 60/68 -60i/68 BUT the professor's solution was: 9/17 -15i/17. How am I wrong? :(
• Multiply the numerator & denominator by the conjugate of 8+2i = 8-2i
(6-6i)(8-2i) / (8+2i)(8-2i)
Numerator: (6-6i)(8-2i) = 48 -12i -48i -12 = 36 - 60i
-- looks like you got +12, rather than -12. Here the details: -6i(-2i) = 12i^2 = 12(-1) = -12
Denominator: (8+2i)(8-2i) = 64 -16i +16i + 4 = 68
Put the pieces back together: 36/68 - 60i/68
Reduce the fractions by 4: 9/17 -15i/17
Hope this helps
• I'm very comfortable rationalizing the denominator, but am still confused as to the reason we do this. Is the reason simply because we're trying to simplify the quotient as much as possible and it's not "clean" to have complex terms in the denominator?
(1 vote)
• Rationalizing the denominator makes the denominator an integer. And, this makes it easier to do other math operations with the fraction. For example, if you need to add/subtract fractions, it is easier to find a common denominator working with integers than working with denominators that are irrational numbers.
• What do I do when I have a problem like (-5+1/2i). I can't multiply by the conjugate of the denominator without the denominator becoming 0. I nee help. Can Sal please post a video on this.
• How do you get 0? I'm assuming your denominator is (-5+1/2i). It's not clear from what you have written as you have no fraction to show the numerator vs denominator.

Anyway... The conjugate for (-5+1/2i) would be (-5-1/2i)
Simplify to get: 25-0.25i^2 = 25-0.25(-1) = 25+0.25 = 25.25
There is no 0.

Hope this helps.
(1 vote)
• It seems to follow from the proof that the product of a complex number and it's conjugate is a natural number; is that correct?
• Natural numbers are: 1, 2, 3, 4, 5, 6, ...
The product of a complex number and its conjugate would create a real number. The set of real numbers includes: natural numbers, whole numbers, integers, rational numbers and irrational numbers.
(1 vote)
• what if its just 12/5i? I'm not sure what to do.
• I believe you need to multiply by (-i)/(-i). This will eliminate the "i" in the denominator.
Hope this helps.
(1 vote)
• can you tell me the more power of i till 90power
(1 vote)
• What if the equation has more than 1i in the numerator?
(1 vote)
• Then you need to simplify the numerator first, by combining like terms and simplifying any i exponents you might have.
(1 vote)
• Hello everyone, Khan academy is great for learning, I appreciate!

I have a question:
determine the number z if u = 4 - 3i

a) z/u = 0,4+0,8i