Complex Numbers - Online Tutor, Practice Problems & Exam Prep

Complex numbers, expressed in the form a + bi, consist of a real part (a) and an imaginary part (b). Operations such as addition, subtraction, multiplication, and division can be performed similarly to algebraic expressions. The complex conjugate, a - bi, is crucial for division, allowing the elimination of the imaginary unit in the denominator. Multiplying a complex number by its conjugate results in a real number, specifically a² + b². Understanding these concepts is essential for further studies in complex number applications.

Introduction to Complex Numbers

So we've worked with real numbers like 3 and imaginary numbers like 2i separately, but you're often going to see expressions that have these two numbers together, so something like 3+2i. And these are actually called complex numbers when you have both a real number and an imaginary number added together. Now, complex numbers are going to 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 going to 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+bi. So in this complex number, a+bi, a is called the real part of the number, and b is called the imaginary part because it's multiplying i, our imaginary unit. Now it's important to know that b by itself is the imaginary part. It's not the whole term bi. So when I'm identifying the real and the imaginary part of the number I have up here, this 3+2i, 3 is going to be the real part of my number. And then 2 is going to be the imaginary part, just the 2 by itself.

Now, let's look at a couple of different examples of complex numbers and identify the real and imaginary parts of each of them. So first, I have this 4 − 3i. So looking at this number, the real part, a, is going to be this 4 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 3. 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 3. That's what's multiplying my imaginary unit i.

Let's look at another example. So here I have 0+7i. Now if I look for the real part of my number, what is not multiplying my imaginary unit, I have this as 0. So that means that, a, my real part is going to be 0. Then, b, my imaginary part, the part that's multiplying my imaginary unit i, in this case is going to be positive 7. Now you might look at this number and think, couldn't you just write that number as 7i? That zero isn't really doing anything. And you're right. I could just write this as 7i. But we still need to know that if we're looking at this as a complex number, it still has a real part. It's just 0.

So let's look at another example. So I have 2+0i over here. So what do you think the real part of this number is? Well, since this 2 is out here by itself, it's not multiplying my imaginary unit. 2 is going to be the real part of my number, a. Then looking at b, so the imaginary part, the part that is multiplying my imaginary unit, in this case, is just 0. So, again, you might be looking at this number thinking, isn't that just a real number? Couldn't I just write this as 2? And you're right. Again, I could just write this as 2. But remember, if we're looking at this as a complex number, it still has an imaginary part. It's just 0.

So that's all for this one, and I'll see you in the next video.

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. $a = -4, b = 9$

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. $a=3,b=\sqrt3$

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

• A. $3+4i$
• B. $9+16i$
• C. $3+\frac43i$
• D. $\frac{13i}{3}$

Adding and Subtracting Complex Numbers

Hey, 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 going to take a skill that we already know and apply it to our complex numbers. Let's go ahead and jump in.

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 \(4 + 8x + 2 + 3x\), then I just need to remove my parentheses and then combine any like terms I have. So here I have 4 and 2, both constants that combine to 6, and then I have \(8x\) and \(3x\), both x terms that combine to \(11x\). So when I'm working with algebraic expressions, I have some constants and some 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 going to treat that i as though it is a variable. So let's look at adding these complex numbers. Here I have \(4 + 8i + 2 + 3i\). So let's go ahead and get rid of all of those parentheses. So I have \(4 + 8i + 2 + 3i\). And now I can just combine like terms. So I have constants 4 and 2, which are going to combine to give me a 6. And then I have my imaginary terms, \(8i\) and \(+3i\), which are going to combine to give me \(+11i\).

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 \(a + bi\). So I got \(6 + 11i\) here, which is already in that standard form. So my final answer is \(6 + 11i\).

Let's take a look at subtracting some complex numbers. So, again, I have \(4 + 8i\), but now minus \(2 + 3i\). So let's go ahead and remove those parentheses. So I have \(4 + 8i\). 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 2 now and a minus 3i. Now I can simply combine like terms as I did before. So, again, my constants, 4 and then negative 2, these will combine to give me a positive 2. And then I have \(8i\) and negative \(3i\). So my \(8i\) and negative \(3i\) are going to combine to give me a positive \(5i\).

Now, again, we want to make sure that our answer is in standard form here, \(a + bi\), and it is. So my final answer is \(2 + 5i\). And that's all there is to adding and subtracting complex numbers. I'll see you in the next video.

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. $2-9i$

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. $7+9i$

Multiplying Complex Numbers

Hey, 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. So we're again going to 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 3i times 7 minus 2i. Now my very first step is going to be to either distribute or foil. And looking at my expression here, I have this 3i by itself. So since it's just one term, I'm multiplying this other complex number, I'm gonna go ahead and distribute. That seems like the best choice here. So I'm gonna distribute this 3i into my 7 and my negative 2i in order to get 21i. And then 3i times negative 2i is gonna give me negative 6 I2. Make sure when you're multiplying an I by an I, you get I squared. So step number 1 is done, and I can go ahead and move on to step 2, which is to apply the fact that I2 = -1. So looking at my expression here, I have 21i minus 6 I squared. So I have my I squared right here, and I need to take this whole term and simplify it using I2 = -1. So this negative 6 I squared just becomes negative 6 times negative 1 and negative 6 times negative 1 is just positive 6. So I can bring my 21i down here, and I have 21i + 6. So I've completed step 2. And step 3 is to combine my like terms. Now looking over here, I don't have any like terms that need to get combined. So step 3 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 21i + 6, and I know that standard form is a + bi. So I'm gonna go ahead and flip this around in order to get 6 + 21i, and this is gonna be my final solution.

So let's go ahead and look at another example. So over here, I have negative 6 + 2i times 3 + 4i. Let's go ahead and start with step 1, 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 gonna be my best option for step 1. So let's go ahead and foil. So I'm gonna start with my first term, so negative 6 times 3 is gonna give me negative 18. And then I have my outside terms, negative 6 times 4, which is gonna give me or times 4 I, which is gonna give me negative 24i. Then I have my inside terms, 2i times 3 is gonna give me positive 6i. And then lastly, my last terms, which is 2i times 4i. Now this, since I'm multiplying 2i terms, I'm gonna end up with an I squared. This is gonna give me plus 8 I2. Okay. So we have completed step number 1. Let's go ahead and move on to step 2, which is to apply that I2 = -1. Now I definitely have an I squared term over here. I have this 8 I squared. So I'm gonna go ahead and simplify this whole term into 8 times negative one. Now 8 times negative one is just negative 8. So I have applied my I squared = negative 1, 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 24i and positive 6i. So now step 3 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 8 are going to combine to negative 26 and then my other like terms are these I terms, negative 24 and positive 6 I, which are gonna combine to give me a negative 18 I. So my final answer, I've completed step 3. I've combined my like terms. I have negative 26 minus 18 I. And I, of course, wanna check that this is in standard form a + bI 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.

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

• A. $-55+48i$
• B. $9+64i$
• C. $24i$
• D. $9+24i$

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. $-42+148i$

Complex Conjugates

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+bi, 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 bi is a minus bi. 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 bi is simply a+bi. 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 1+2i. Now I want to look at that imaginary part of my number, so in this case, positive 2, 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 a 1. And then that positive 2 is going to reverse sign into a negative 2. So the complex conjugate of 1+2i is 1 minus 2i. Let's look at another example. So if I have 1 minus 2i, which is just the answer that we just got a last one, and I want to find that complex conjugate, I, again, am just going to look at that imaginary part. So here I have a negative 2, and I'm going to reverse that sign. So, again, my real part stays the same. I still have 1. And then my negative 2 is going to reverse sign into a positive 2 And of course, my I on the end there. So the complex conjugate of 1 minus 2i is 1+2i. 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 1+2i. Now I just again want to look at that imaginary part, in this case, positive 2, and reverse that sign. Now the real part of our number is going to stay the same even if it's negative. Negative. So this negative one is going to stay a negative one. And then my positive 2, since that's my imaginary part, is going to reverse sign to negative 2. So the complex conjugate of negative 1+2i is negative 1 minus 2i.

Now what do you think will happen if I take a complex number and its 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 2+3i, and I multiply it by its conjugate, 2 minus 3i, I'm going to need to foil. So let's go ahead and do that. So I need to take my first term, 2 times 2, and that will give me 4. And then my outside terms, 2 times negative 3i, that's going to give me negative 6i. And then my inside terms, 3i times 2 is going to give me a positive 6i. And then, of course, my last terms, positive 3i times negative 3i is going to give me negative 9i². Now whenever we multiply complex numbers, remember, we need to look for that i² term. So here I have negative 9i², and this just becomes negative 9 times negative 1, which we know negative 9 times negative 1 is just positive 9. And then I can bring down all of my other terms. So my 4 comes back down, and then I have negative 6i, positive 6i. So looking at this, you might notice that I have a negative 6i and a positive 6i in the middle there, and you might notice that these are going to cancel out. So if I have minus 6i, plus 6i, those are gone. Those are going to get canceled. So I'm just left with 4+9. So the like terms I need to combine, 13, are going to combine to give me 13. So 2+3i, my complex number times its conjugate, 2 minus 3i,2+3i,2-3i gave me a real number. And this is going to happen any time 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 4 is really just, woah, is really just the a term of my complex number squared. So this 4 is really just a². And then my 9 that I have on the end here end here is just my b term, 3 squared, as well. So whenever I take complex conjugates and multiply them by each other, a+bi times a minus bi, I'm really just going to get a² plus b². 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.

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

• A. $16$
• B. $41$
• C. $25$
• D. $20$

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

• A. $50$
• B. $14$
• C. $49$
• D. $1$

Dividing Complex Numbers

Now that we know how to add, subtract, and multiply complex numbers, let's learn how to divide them. Whenever I divide by a complex number, I end up with a fraction that has an i in the bottom. So, I'll have something in the numerator divided by a complex number that has a term with i in it. Looking at this fraction, I’m not really sure how to simplify it and come to a solution that makes any sense. This i in the denominator is problematic, I don't know how to deal with it, and I want to get rid of it in any way that I can. To transform this complex denominator into something real, we learned to use the complex conjugate. I'll walk you through how to use the complex conjugate to divide complex numbers and arrive at a solution. Let’s jump right into an example.

I want to find the quotient of these numbers: 31+2i. The first step is to eliminate the i in the denominator. My first step is to multiply both the numerator and the denominator by the complex conjugate of the denominator. Since I want to eliminate the i in the denominator, I will multiply by the complex conjugate, which for 1+2i is 1-2i. Multiplying the numerator and denominator by this conjugate doesn't change the value of the fraction. Expanding this, the numerator distributed into the complex conjugate gives 3-6i. With the denominator, using the foil method, the terms produce 1 from 1*1, -2i from 1*-2i, 2i from 2i*1, and -4i2 from 2i*-2i. Simplifying the i2 term to -1, we get positive 4. Collapsing terms, I am left with 5 in the denominator. I now have 3-6i5.

Next, I expand the fraction into real and imaginary parts, yielding 35 minus 65i. This fraction is already in its lowest terms, so this is the solution. That’s all there is to dividing complex numbers. Let me know if you have any questions.

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. $22+16i$

Find the quotient. Express your answer in standard form. 

$\frac{-5+3i}{-7-4i}$

• A. $\frac35+\frac45i$
• B. $18i$
• C. $23-41i$
• D. $\frac{23}{65}-\frac{41}{65}i$

Here’s what students ask on this topic:

What is a complex number and how is it represented?

A complex number is a number that combines a real part and an imaginary part. It is represented in the form $a+bi$, where $a$ is the real part and $b$ is the imaginary part. The imaginary unit $i$ is defined as $i^2=-1$. For example, in the complex number $3+2i$, 3 is the real part and 2 is the imaginary part.

How do you add and subtract complex numbers?

To add or subtract complex numbers, you combine their real parts and their imaginary parts separately. For addition, if you have $a+bi$ and $c+di$, the sum is $\left(a+c\right)+\left(b+d\right)i$. For subtraction, the difference is $\left(a-c\right)+\left(b-d\right)i$. For example, $4+3i+2+5i=6+8i$ and $4+3i-2+5i=2-2i$.

How do you multiply complex numbers?

To multiply complex numbers, you use the distributive property or the FOIL method. For example, to multiply $\left(2+3i\right)\left(4+5i\right)$, you calculate: $2\times 4+2\times 5i+3i\times 4+3i\times 5i$. This simplifies to $8+10i+12i+15i^2$. Since $i^2=-1$, the expression becomes $8+22i-15$, which simplifies to $-7+22i$.

What is the complex conjugate and how is it used in division?

The complex conjugate of a complex number $a+bi$ is $a-bi$. It is used in division to eliminate the imaginary unit from the denominator. For example, to divide $3$ by $1+2i$, multiply both the numerator and the denominator by the conjugate of the denominator: $3\times 1-2i$ and $\left(1+2i\right)\left(1-2i\right)$. This results in $3-6i$ over $1+4$, which simplifies to $3-6i$ over $5$, or $3/5-6/5i$.

How do you divide complex numbers?

To divide complex numbers, multiply the numerator and the denominator by the conjugate of the denominator. For example, to divide $3$ by $1+2i$, multiply both by the conjugate $1-2i$. This gives $3\times 1-2i$ over $\left(1+2i\right)\left(1-2i\right)$. Simplify to get $3-6i$ over $1+4$, which is $3-6i$ over $5$. Finally, split into real and imaginary parts: $3/5-6/5i$.