Complex Numbers - Video Tutorials & Practice Problems

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1

concept

Introduction to Complex Numbers

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So we've worked with real numbers like three and imaginary numbers like two I separately. But you're often going to see expressions that have these two numbers together. So something like three plus two, I and these are actually called complex numbers when I have both a real number and an imaginary number added together. Now, complex numbers are gonna be really important for us throughout this course and have a ton of different uses. And while they might sound a little complicated, at first, I'm gonna walk you through what they are and how we use them step by step. So let's get started. So a complex number has a standard form of A plus B I. So in this complex number, A plus bi A is called the real part of the number. And B is called the imaginary part because it's multiplying I are imaginary unit. Now, it's important to know that B by itself is the imaginary part. It's not the whole term B I. So when I'm identifying the real and the imaginary part in the number I have up here, this three plus two I three is going to be the real part of my number and then two is going to be the imagine part, just the two by itself. Now let's look at a couple different examples of complex numbers and identify the real and imaginary parts of each of them. So first I have this four minus three, I, so looking at this number, the real part A is gonna be this four because it's out there by itself, it's not multiplying my imaginary unit. This is going to be my real part A then B, so I want to look for what is multiplying I my imaginary unit. And in this case, it is negative three. Now it's important to look at everything that's multiplying our imaginary unit. So if it's a negative number, if it's a square root, if it's a combination of a number and a square root, I want to get everything that's multiplying my imaginary unit. So in this case, B is going to be a negative three. That's what's multiplying my imaginary unit. I let's look at another example. So here I have zero plus seven, I now if I look for the real part of my number, what is not multiplying my imaginary unit? I have this zero. So that means that A, my real part is going to be zero, then B my imaginary part, the part that's multiplying my imaginary unit, I, in this case is going to be positive seven. Now you might look at this number and think, couldn't you just write that number as seven I, that zero isn't really doing anything. And you're right, I could just write this as seven I, but we still need to know that if we looking at this as a complex number, it still has a real part, it's just zero. So let's look at another example. So I have two plus zero I over here. So what do you think the real part of this number is, well, since this two is out here by itself, it's not multiplying my imaginary unit two is going to be the real part of my number A then looking at B so the amount part, the part that is multiplying my imaginary unit in this case is just zero. So again, you might be looking at this number thinking, isn't that just a real number? Couldn't I just write that as two? And you're right again, I could just write this as two. But remember if we're looking at this as a complex number, it still has an imaginary part, it's just zero. So that's all for this one. And I'll see you in the next video.

2

Problem

Problem

Identify the real and imaginary parts of each complex number. $-4-9i$

A

$a = -9, b = -4$

B

$a = -4, b = -9$

C

$a = 4, b = 9$

D

$a = -4, b = 9$

3

Problem

Problem

Identify the real and imaginary parts of each complex number. $3+2i\sqrt3$

A

$a=3,b=2\sqrt3$

B

$a = 3, b = 2$

C

$a=2\sqrt3,b=3$

D

$a=3,b=\sqrt3$

4

Problem

Problem

Write the complex number in standard form. $\frac{9+\sqrt{-16}}{3}$

A

$3+4i$

B

$9+16i$

C

$3+\frac43i$

D

$\frac{13i}{3}$

5

concept

Adding and Subtracting Complex Numbers

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3m

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Everyone now that we know what complex numbers are, we can perform operations on them like addition and subtraction. Now I know you may be thinking that this is just yet another new skill you're going to have to learn. But adding and subtracting complex numbers is actually exactly the same as adding and subtracting algebraic expressions. So we're just gonna take a skill that we already know and apply it to our complex numbers. Let's go ahead and jump in. So just like with algebraic expressions, when you add and subtract complex numbers, you're just going to combine like terms. So when I'm given algebraic expressions to add, let's say four plus eight X plus two plus three X, then I just need to remove my parentheses and then combine any like terms I have. So here I have four and two, both constants that combine to six and then I have eight X and three X, both X terms that combine to 11 X. So when I'm working with algebraic expressions, I have some constants. And so terms with variables that get combined to give me my final solution. When I'm working with complex numbers, I still have constants. But instead of terms with a variable, I have terms with an I, with my imaginary unit. So I'm really just gonna treat that I as though it is a variable. So let's look at adding these complex numbers. So here I have four plus eight, I plus two plus three, I, so let's go ahead and get rid of all of those parentheses. So I have four plus eight, I plus two plus three. I and now I can just combine like terms. So I have constants four and two, which are going to combine to give me a six and then I have my imaginary terms. Eight I and positive three I which are going to combine to give me positive 11. I. Now one thing that we need to consider whenever we're adding or subtracting complex numbers is that we always want to express our final answer in standard form. Now remember standard form for complex numbers is just A plus B I. So I got six plus 11 I here, which is already in that standard form. So my final answer is six plus 11. I let's take a look at subtracting some complex numbers. So again, I have four plus eight I, but now minus two plus three. I, so let's go ahead and remove those parentheses. So I have four plus eight I, but now since I'm subtracting, I need to make sure that I distribute that negative into my parentheses here. So I'm left with a minus two now and a minus three, I now I can simply combine like terms as I did before. So again, my constant four and then negative two, these will combine to give me a positive two and then I have A I and negative three. I, so my A I and negative three, I are going to combine to give me a positive five. I, now again, we want to make sure that our answer is in standard form here A plus B I and it is. So my final answer is two plus five I and that's all there is to adding and subtracting complex numbers. I'll see you in the next video.

6

Problem

Problem

Find the difference. Express your answer in standard form. $\left(2+8i\right)-\left(4-i\right)$

A

$-2+9i$

B

$6+7i$

C

$2+7i$

D

$2-9i$

7

Problem

Problem

Find the sum. Express your answer in standard form. $5\left(4+7i\right)+6\left(3-2i\right)$

A

$7+5i$

B

$38+23i$

C

$2+47i$

D

$7+9i$

8

concept

Multiplying Complex Numbers

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Everyone. So not only can we add and subtract complex numbers, but we can also multiply them. Now, multiplication might seem a bit more intimidating than adding or subtracting. But we're actually still going to multiply our complex numbers. The same way we multiply algebraic expressions. We're again gonna take a skill that we already know and apply it to our complex numbers. And don't worry, I'm still going to walk you through it step by step. So let's go ahead and take a look. Now, algebraic expressions are always multiplied one of two ways. So complex numbers will be the same, we either distribute or we use foil. Now, when we either distribute or foil, we're always going to end up with an I squared term. Now, we know that I squared is just equal to negative one. So I'm going to use that in order to simplify my final answer. So let's go ahead and just jump right into an example. Looking at my first expression over here, I have three I times seven minus two. I now my very first step is going to be to either distribute or foil and looking at my expression here, I have this three I by itself. So since it's just one term of multiplying this other complex number, I'm going to go ahead and distribute. That seems like the best choice here. So I'm going to distribute this three I into my seven and my negative two I in order to get 21 I and then three I times negative two I is going to give me negative six I squared. Make sure when you're multiplying an I by an I, you get I squared. So step number one is done and I can go ahead and move on to step two, which is to apply the fact that I squared equals negative one. So looking at my expression here, I have 21 I minus six I squared. So I have my I squared right here and I need to take this whole term and simplify it using I squared equals negative one. So this negative six I squared just becomes negative six times negative one and negative six times negative one is just positive six. So I can bring my 21 I down here and I have 21 I plus six. So I've completed step two and step three is to combine my like terms. Now, looking over here, I don't have any like terms that need to get combined. So step three is also done. And this is my final answer, but I wanna make sure that I express my answer in standard form. So here at 21 I plus six and I know that standard form is A plus B I. So I'm gonna go ahead and flip this around in order to get six plus 21 I, and this is gonna be my final solution. So let's go ahead and look at another example. So over here I have negative six plus two, I times three plus four. I let's go ahead and start with step one, which is either to distribute or foil. Now, since I have two complex numbers that each have two terms in them, it looks like foiling is be my best option for step one. So let's go ahead and foil. So I'm gonna start with my first term. So negative six times three is gonna give me negative 18 and then I have my outside terms negative six times four, which is gonna give me or times four I which is gonna give me negative 24. I, then I have my inside terms. Two I times three is gonna give me positive six I and then lastly my last terms which is two I times four. I now this since I'm multiplying two I terms, I'm gonna end up with an I squared. This is gonna give me plus eight I squared. OK. So we have completed step number one. Let's go ahead and move on to step two, which is to apply that I squared equals negative one. Now, I definitely have an I squared term over here, I have this eight I squared. So I'm gonna go ahead and simplify this whole term into eight times negative one. Now eight times negative one is just negative eight. So I have applied my I squared equals negative one and I can go ahead and pull all of my other terms down as well. So I can bring my negative 18 and then I have both of my I terms negative 24 I and positive six. I, so now step three is to combine all of my like terms and I have some like terms that need to get combined here. So negative 18 and negative eight are going to combine to negative 26. And then my other like terms are these I terms negative 24 and positive six I which are going to combine to give me a negative 18. I. So my final answer I've completed step three. I've combined my light terms. I have negative 26 minus 18. I and I of course want to check that this is in standard form A plus B I and it is so I'm good to go and this is my final answer. That's all for this video. I'll see you in the next one.

9

Problem

Problem

Perform the indicated operation. Express your answer in standard form. $\left(3+8i\right)^2$

A

$-55+48i$

B

$9+64i$

C

$24i$

D

$9+24i$

10

Problem

Problem

Find the product. Express your answer in standard form. $2i\left(9-4i\right)\left(6+5i\right)$

A

$8+18i$

B

$54-20i$

C

$54-40i$

D

$-42+148i$

11

concept

Complex Conjugates

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5m

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Hey, everyone, after learning how to multiply complex numbers, you may think that dividing complex numbers is up next. And you're right. But before we learn how to divide complex numbers, we actually need to learn about something called the complex conjugate. Now, the complex conjugate might seem a little abstract or maybe even a little useless at first. But I'm going to show you exactly what the complex conjugate is and how we're going to use it in a way that will help us to divide complex numbers. So let's go ahead and get started. Now, in order to find the complex conjugate of some complex number, we simply need to reverse the sign of the imaginary part of our complex number. So if I have some complex number A plus B I, I'm going to look at that imaginary part. So in this case, positive B and I'm simply going to reverse that sign. So the complex conjugate of A plus B I is A minus B I and this is going to work in the reverse direction as well. So if I wanted to find the conjugate of A minus BI, I would again look at that imaginary part and simply reverse the sign. So the complex conjugate of A minus B I is simply A plus B I. So let's take a look at a couple of examples and identify some complex conjugates. So looking at my first example here I have one plus two, I now I want to look at that imaginary part of my number. So in this case, positive two and simply reverse that sign. So the real part of my number is going to stay the same that doesn't get changed. So I'll have one and then that positive two is going to reverse sign into a negative two. So the complex conjugate of one plus two, I is one minus two, I let's look at another example. So if I have one minus two, I, which is just the answer that we just got the last one. And I want to find that complex conjugate. I, again, I'm just going to look at that imaginary part. So here I have a negative two and I'm gonna reverse that sign. So again, my real part stays the same. I still have one and then my negative two is going to reverse sign into a positive two. And of course, my eye on the end there. So the complex conjugate of one minus two, I is one plus two. I, and you might notice that that's just the reverse of what we did in part. A, let's take a look at one more example here. So here I have negative of one plus two. I now I just again, want to look at that imaginary part in this case, positive two and reverse that sign. Now, the real part of our number is going to say the same even if it's negative. So this negative one is going to stay a negative one and then my positive two, since that's my imaginary part is going to reverse sign to negative two. So the complex conjugate of negative one plus two, I is negative one minus two. I now what do you think will happen if I take a complex number and it's conjugate and I multiply them by each other? Well, let's take a look. So if I have a complex number, so here I have two plus three, I and I multiplied by its conjugate to minus three I, I'm going to need to foil. So let's go ahead and do that. So I need to take my first term two times two and that will give me four. And then my outside terms two times negative three. I that's going to give me a negative six I and then my inside terms three I times two is going to give me a positive six I and then of course my last terms positive three I times negative three, I is gonna give me negative nine I squared. Now, whenever we multiply complex numbers, remember we need to look for that I squared term. So here I have negative nine I squared and this just becomes negative nine times negative one, which we know negative nine times negative one is just positive nine. And then I can bring down all of my other terms. So my four comes back down and then I have negative six I positive six I. So looking at this, you might notice that I have a negative six I and a positive six I in the middle there. And you might notice that these are going to cancel out. So if I have minus six, I plus six, I, those are gone, those are going to get canceled. So I'm just left with four plus nine. So the like terms I need to combine four and nine are going to combine to give me 13. So two plus three I my complex number times its conjugate to minus three. I gave me a real number and this is going to happen anytime I multiply complex conjugates. So multiplying complex conjugates by foiling is always going to give me a real number. Now, this is going to be really useful for us when we're dividing complex numbers, which we'll see in the next video. Now, something else that you might have noticed here is that this four is really just whoa is really just the a term of my complex number squared. So this four is really just a squared and then my nine that I have on the end here, end here is just my B term three squared as well. So whenever I take complex conjugates and multiply them by each other A plus B I times A minus B I, I'm really just going to get A squared plus B squared. Now, this is not something that you have to memorize, but it can be a helpful shortcut. If you do remember it, that's all for complex conjugates. I'll see you in the next video.

12

Problem

Problem

Find the product of the given complex number and its conjugate. $4-5i$

A

$16$

B

$41$

C

$25$

D

$20$

13

Problem

Problem

Find the product of the given complex number and its conjugate. $-7-i$

A

$50$

B

$14$

C

$49$

D

$1$

14

concept

Dividing Complex Numbers

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4m

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Now that we know how to add subtract and multiply complex numbers, let's learn how to divide them. Well, whenever I divide by a complex number, I'm always going to end up with a fraction that has an I in the bottom. So I'll have something, doesn't really matter what in the numerator divided by a complex number that has a term with I in it. Now looking at this fraction, I'm not really sure how to simplify it and come to a solution that actually makes any sense. So this I and the denominator is actually bad. I don't know how to deal with it and I want to get rid of it in any way that I can. So how am I going to take this complex denominator and turn it into something real? Well, we just learned that if we have a complex number and multiply it by its conjugate that we're simply left with a real number. So that's actually going to completely solve our problem and give us something that we know how to work with. So I'm going to go ahead and walk you through how to use the complex conjugate to divide complex numbers and come to a solution. Let's go ahead and jump right into an example. So I want to find the quotient of these numbers 3/1 plus two. I now, the first thing I want to do is get rid of that I in the denominator. So my very first step is actually gonna be to multiply both the top and the bottom by my complex conjugate of the bottom. Now, since the bottom is the number that I want to get rid of, I want to get rid of the eye and that I'm gonna go ahead and multiply by the complex conjugate. Now, the complex conjugate of one plus two, I, I'm just gonna take that imaginary part and flip the sign. This is gonna be one minus two. I now if I'm multiplying the bottom of my fraction by that, I need to also multiply the top of my fraction as well so that I'm not really changing the value of anything. So let's go ahead and expand this out. So with my numerator, I can take this three and distribute it into that complex conjugate, which will give me three minus six. I. Now with the denominator, since I have two complex numbers, I can go ahead and use foil here. So my first term is that one times one, which will give me one, my outside term, one times negative to I will give me negative two. I then my inside term two I times one, gives me positive two I and my last term two, I times negative two, I will give me negative four I squared. Now, we know that whenever we're left with an I squared term, we can further simplify that. So this negative four I squared is really just negative four times negative one, which we know is just positive four. So let's go ahead and rewrite our fraction and simplify it as far as we can. So the numerator is still just three minus six. I and then my denominator here, let's look at what terms we have left. Well, I have this one and then I have minus two, I plus two. I, but I know that those middle terms are just going to cancel out. So I don't have to worry about them and then I have this plus four. So I'm simply just left with one plus four and one plus four is just five. So my denominator here is five. OK. So we have completed step number one, we have multiplied the top and bottom by our complex conjugate and simplified as much as we can. Now let's move on to step number two, which is actually to expand our fraction further into the real and imaginary parts. So I want to take the real part and I want to split it from my imaginary part. So looking at my fraction over here, I can just take the numerator and split it keeping that denominator on both terms. So I will really just end up with 3/5 minus 6/5. I, so I have expanded my fraction into my real and imaginary part. Step two is done. Finally, step three, which is to simplify our fraction to our lowest terms. So I have 3/5 minus six fifths I, and this is actually already in its lowest terms. So that means that this is just my solution. And I've completed step three, I'm completely done and I have my solution. That's all there is to dividing complex numbers. Let me know if you have any questions.

15

Problem

Problem

Find the quotient. Express your answer in standard form.

$\frac{6+i}{4-2i}$

A

$\frac{11}{10}+\frac45i$

B

$\frac65+\frac45i$

C

$\frac{11}{10}-\frac45i$

D

$22+16i$

16

Problem

Problem

Find the quotient. Express your answer in standard form.