Complex Numbers (part 2) Dividing complex numbers. Complex conjugates.
Complex Numbers (part 2)
⇐ Use this menu to view and help create subtitles for this video in many different languages. You'll probably want to hide YouTube's captions if using these subtitles.
- So far we've learned what a complex number is; we've even
- learned how to graph it.
- And we learned how to add, subtract, and multiply it.
- Where I left off in the last video was, how do we divide
- two complex numbers.
- So I said, let's say I have one complex numbers, zone.
- And that equals a plus b i.
- And I want to divide that by ztwo.
- Which is, c plus d i.
- So let me ask you a question.
- And I touched on this in the last video.
- And let me do it in a different color over here.
- We know that a plus b times a minus b is equal to a
- squared minus b squared.
- And you can multiply it out, in case you're not sure.
- Remember, it's just a times a plus b times a minus
- a times b, plus a times a, and you'll get this.
- But you know how to do this, anyway.
- There's a review of it, if you need to do it.
- So, given that, what is c plus d?
- What happens if we do something very similar
- with a complex number?
- If we say c plus d i times c minus d i.
- Well, in this case a is c.
- And b is d i, right?
- So this is just going to be equal to c squared
- minus d i squared.
- d i squared.
- And that equals c squared minus d squared i squared.
- And that equals c squared minus d squared.
- And i squared is negative one, right?
- So this is going to be multiplied by negative one, so
- it cancels out this negative.
- So you get c squared plus d squared.
- That's interesting.
- When I multiply a complex number times this other number,
- which is very similar to it, but it's kind of the imaginary
- part, goes in the other direction.
- When I multiply the two, I get a completely real number.
- All of the i's disappear.
- And, in general, this number -- if we call this -- well, in our
- example this was ztwo, so if we say that ztwo equals c plus d i,
- the quantity c minus d i is called its conjugate.
- And that's just good terminology to know.
- And the sign for conjugate is that line over the top.
- So the conjugate of ztwo is c minus d i.
- Or you could say, the conjugate of c minus d i
- is equal to c plus d i.
- Or you could say it the other way around.
- The conjugate of c plus d i is equal to c minus d i.
- And notice, we're just switching the direction in
- the imaginary -- along the imaginary axis, when we take
- the conjugate of something.
- With that said, let me erase that and go back
- to our original problem.
- Because the conjugate is the tool we're going
- to use to divide this.
- So we know when we multiply an imaginary number times its
- conjugate, we get a real number.
- And we know, also, if we multiply -- we can multiply
- anything by one, and we get the same number.
- So let's multiply the numerator and denominator of this
- expression by the conjugate of the denominator.
- So let me do that.
- So the conjugate of the denominator is going
- to be c minus d i.
- So c minus d i over c minus d i.
- So this was c plud d i, so this is its conjugate.
- And so what do we get?
- So in the numerator, we get a c -- I don't want to run out of
- space, I always do -- a c, so a times c, minus a d i, minus a d
- i -- these i's are looking funny -- this is an i.
- Plus b c i; plus b c i.
- And then the last term, we have a plus b minus b.
- So it's minus b d i squared.
- Minus b d i squared.
- All of that.
- And this is a plus b times a minus b.
- So it's equal to a squared minus b squared.
- So this is going to be equal to -- and it this will become
- second nature to you after a while, but you might want
- to just multiply it out.
- This equals c squared plus d squared.
- And don't take my word for it.
- Actually, algebraically, multiply this out and just
- realize you can only add real parts to real parts and
- imaginary parts to imaginary parts.
- So let me simplify that.
- That equals -- let's see, the real parts.
- This is real, a c.
- And this is minus b d i squared.
- So the i squared is minus one.
- So it switches the sign here, so it becomes plus b d.
- And we can get rid of d i.
- So the real parts are, a c plus b d.
- That's that, and that.
- And then the imaginary parts are plus -- this one's
- positive, so I'll put one first -- b c minus a d i, all of that
- over c squared plus d squared.
- And that still might not look like a complex number to you.
- But then we can separate them out and we could say well,
- that equals a c plus b d over c squared plus d squared.
- And that's the real part.
- Plus b c minus a d over c squared plus d squared.
- And that times i, and that's the imaginary part.
- So you can't merge, when you're adding and subtracting, the
- real part to the imaginary part.
- But you can most definitely scale an imaginary
- number by a real number.
- And that's essentially what we're doing.
- We're multiplying one over c squared plus d
- squared times this.
- So, division might seem a little complicated when I
- write it all in variables.
- But let me give you an example and you will hopefully see that
- it -- with real numbers, and -- not real numbers, with
- actual numbers, I should be careful with what I say.
- Let's say I have one plus two i.
- And I want to divide that by, I don't know, let's divide it by,
- I'm going to pick a random number.
- two plus threei.
- And so what do we do?
- We multiply it times the conjugate of the denominator.
- two minus threei over -- over itself, right?
- Because then we're not changing the number.
- This is just one, this simplifies to one.
- It equals -- the bottom, we can multiply it out.
- But hopefully it's second nature to you.
- It equals four plus nine, right?
- Because that's just a squared plus b squared.
- Well, I mean, it's a squared minus b squared, but then the
- i's, when you multiply, and it becomes a negative number.
- Try it out if you don't believe me.
- And then the top, we get one times two is two.
- one times minus threei is minus threei.
- And you have twoi times two, which is plus fouri.
- And then you have twoi times minus threei.
- So that's minus six.
- Minus sixi squared.
- Well, what does i squared equal?
- That equals negative one.
- So negative one times negative six.
- Get rid of the i squared and this becomes a positive.
- So then, what are our real parts?
- Our real parts are two and six.
- so two plus six is eight.
- And what are our imaginary parts?
- Minus threei plus fouri.
- So that's just plus onei, right?
- Minus three plus four is positive one.
- So it's just plus onei.
- Over thirteen.
- Or we could write that as -- if we wanted to write that in the
- traditional complex form -- is eight / thirteen plus one over thirteeni.
- So when I divided one complex number by another, I got
- another complex number.
- And an interesting exercise for you to do is, pick some
- random complex numbers.
- Plot them out on complex plane, and see what happens when you
- multiply them, when you divide them, when you add them,
- when you subtract them.
- And when you scale them.
- Or when you take the conjugate.
- And that'll give you a better intuition of what's going
- on with these numbers.
- Anyway, I will see you in the next video.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
Have something that's not a question about this content?
This discussion area is not meant for answering homework questions.
Share a tip
When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...
Have something that's not a tip or feedback about this content?
This discussion area is not meant for answering homework questions.
Discuss the site
For general discussions about Khan Academy, visit our Reddit discussion page.
Flag inappropriate posts
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
- disrespectful or offensive
- an advertisement
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
- a tip or feedback in Questions
- a question in Tips & Feedback
- an answer that should be its own question
about the site