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?

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.

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 please and thank U.... 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-...

Remember to use brackets z/u = 0.4+0.8i # I presume there was some keyboard issue don't use "," in place of a decimal point => z = u(0.4+0.8i)

Main content

Course: Precalculus > Unit 3

Lesson 3: Complex conjugates and dividing complex numbers

Dividing complex numbers review

Google Classroom

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} \frac{2 + 3 i}{4} & = \frac{2}{4} + \frac{3}{4} i \\ = 0.5 + 0.75 i \end{aligned}

Finding the quotient of two complex numbers is more complex (haha!). For example:

\begin{aligned} \frac{20 - 4 i}{3 + 2 i} \\ = \frac{20 - 4 i}{3 + 2 i} \cdot \frac{3 - 2 i}{3 - 2 i} \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} = \frac{(20 - 4 i) (3 - 2 i)}{(3 + 2 i) (3 - 2 i)} \\ = \frac{52 - 52 i}{13} \end{aligned}

Multiplying the denominator

(3 + 2 i)

by its conjugate

(3 - 2 i)

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

(3 - 2 i)

as well. Now we can finish the calculation:

\begin{aligned} = \frac{52}{13} - \frac{52}{13} i \\ = 4 - 4 i \end{aligned}

Want to learn more about dividing complex numbers? Check out this video.

Check your understanding

Problem 1

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

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

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ZardinTheTreeAtlas
Posted 5 years ago. Direct link to ZardinTheTreeAtlas's post “What do I do when I have ...”
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.
Button navigates to signup pageComment on ZardinTheTreeAtlas's post “What do I do when I have ...”
(11 votes)
Answer
- Polina Vitić
  Posted 5 years ago. Direct link to Polina Vitić's post “Here's a hint: you need t...”
  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!
  Comment on Polina Vitić's post “Here's a hint: you need t...”
  (28 votes)
Matthew Johnson
Posted 4 years ago. Direct link to Matthew Johnson's post “What would be a real worl...”
What would be a real world application where imaginary numbers would be involved in practical applications? Fractal geometry is excluded!
Button navigates to signup pageComment on Matthew Johnson's post “What would be a real worl...”
(6 votes)
Answer
- KLaudano
  Posted 4 years ago. Direct link to KLaudano's post “Imaginary numbers are use...”
  Imaginary numbers are used in electrical engineering to describe AC voltages.
  Comment on KLaudano's post “Imaginary numbers are use...”
  (15 votes)
Kyra
Posted 7 years ago. Direct link to Kyra's post “Hello, The equation from ...”
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? :(
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Kim Seidel
  Posted 7 years ago. Direct link to Kim Seidel's post “Multiply the numerator & ...”
  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
  Comment on Kim Seidel's post “Multiply the numerator & ...”
  (3 votes)
Tatiana
Posted a year ago. Direct link to Tatiana's post “I'm very comfortable rati...”
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?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Kim Seidel
  Posted a year ago. Direct link to Kim Seidel's post “Rationalizing the denomin...”
  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.
  Button navigates to signup page
  (7 votes)
HJKNAPP
Posted 6 months ago. Direct link to HJKNAPP's post “What do I do when I have ...”
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.
Button navigates to signup pageComment on HJKNAPP's post “What do I do when I have ...”
(2 votes)
Answer
- Kim Seidel
  Posted 6 months ago. Direct link to Kim Seidel's post “How do you get 0? I'm as...”
  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)
  Your denominator becomes (-5)^2-(1/2i)^2
  Simplify to get: 25-0.25i^2 = 25-0.25(-1) = 25+0.25 = 25.25
  There is no 0.
  
  Hope this helps.
  Comment on Kim Seidel's post “How do you get 0? I'm as...”
  (1 vote)
Maximus
Posted a year ago. Direct link to Maximus's post “It seems to follow from t...”
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?
Button navigates to signup pageComment on Maximus's post “It seems to follow from t...”
(2 votes)
Answer
- Kim Seidel
  Posted a year ago. Direct link to Kim Seidel's post “Natural numbers are: 1, 2...”
  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.
  Button navigates to signup page
  (1 vote)
Grace Wyatt
Posted 5 years ago. Direct link to Grace Wyatt's post “what if its just 12/5i? I...”
what if its just 12/5i? I'm not sure what to do.
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(2 votes)
Answer
- Kim Seidel
  Posted 5 years ago. Direct link to Kim Seidel's post “I believe you need to mul...”
  I believe you need to multiply by (-i)/(-i). This will eliminate the "i" in the denominator.
  Hope this helps.
  Button navigates to signup page
  (1 vote)
Maryam shoukat
Posted 3 years ago. Direct link to Maryam shoukat's post “can you tell me the more ...”
can you tell me the more power of i till 90power
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(1 vote)
Answer
connerking2
Posted 5 years ago. Direct link to connerking2's post “What if the equation has ...”
What if the equation has more than 1i in the numerator?
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(1 vote)
Answer
- Rae Riddle
  Posted 5 years ago. Direct link to Rae Riddle's post “Then you need to simplify...”
  Then you need to simplify the numerator first, by combining like terms and simplifying any i exponents you might have.
  Button navigates to signup page
  (1 vote)
poetrade314
Posted 2 years ago. Direct link to poetrade314's post “Hello everyone, Khan acad...”
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
please and thank U....

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-...
Button navigates to signup pageButton navigates to signup page
(0 votes)
Answer
- cossine
  Posted 2 years ago. Direct link to cossine's post “Remember to use brackets ...”
  Remember to use brackets
  
  z/u = 0.4+0.8i # I presume there was some keyboard issue don't use "," in place of a decimal point
  
  => z = u(0.4+0.8i)
  Button navigates to signup page
  (2 votes)