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Precalculus

Unit 3: Lesson 3

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{aligned} \dfrac{2+3i}{4}&=\dfrac{2}{4}+\dfrac{3}{4}i \\\\ &=0.5+0.75i \end{aligned}
Finding the quotient of two complex numbers is more complex (haha!). For example:
\begin{aligned} &\phantom{=}\dfrac{20-4i}{3+2i} \\\\ &=\dfrac{20-4i}{3+2i}\cdot\dfrac{3-2i}{3-2i} \end{aligned}
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{aligned} &=\dfrac{(20-4i)(3-2i)}{(3+2i)(3-2i)} \\\\ &=\dfrac{52-52i}{13} \end{aligned}
Multiplying the denominator left parenthesis, 3, plus, 2, i, right parenthesis by its conjugate left parenthesis, 3, minus, 2, i, right parenthesis 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 parenthesis, 3, minus, 2, i, right parenthesis as well. Now we can finish the calculation:
\begin{aligned} &=\dfrac{52}{13}-\dfrac{52}{13}i \\\\ &=4-4i \end{aligned}

Problem 1
start fraction, 4, plus, 2, i, divided by, minus, 1, plus, i, end fraction, equals

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
• how would you do 8/3i ?
(1 vote)
• Same way you would do it up there. You will multiply -(3i) on both the numerator and denominator, making
(8 * -3 i)/(-9i²)

-9i² is going to equal to -9, since i² is -1, and -9*-1 = 9.

so right now = (8 * -3i)/9
8*-3i = -24 i

BUT 24i/9 can be simplified to 8i/3.

Other way to solve it (simpler way) is to do that same process, but just multiply by i on both numerator and denominator.

P.S: If you are confused to how why this works, just think of the equation instead as (8 + 0i) / (3i + 0).

Hope that this helps you!
• 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.
• x2 − 2x + 15 = 0
Have you tried it? What did you get?
(1 vote)
• If we have a complex number such as 2+7i/2-8i, what would be the conjugate then?
(1 vote)
• Look at the denominator. The conjugate would be 2+8i
Hope this helps.
• 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

I did so > z / 4-3i = 0,4+0,8i

z = 0,4+0,8i (4-3i) but something is wrong I wanna understanding everything in every part-...
(1 vote)
• You have (0.4+0.8i)(4-3i), then you need to FOIL or double distribute this to get the simplified answer. 0.4(4-3i)+0.8i(4-3i)=1.6-1.2i+3.2i-2.4i^2, but i^2=-1, 1.6+2.4-1.2i+3.2i gives 4+2i.