Understanding Complex Numbers and Operations

Okay, today we're going to do section 3.1 on complex numbers. So the topics that we're going to cover include plotting complex numbers, just very simple ones, adding, subtracting, multiplying, dividing complex numbers, and also how to find the conjugate of a complex number. We're also going to learn how to express square roots of negative numbers as multiples of i. So the definition of the imaginary number i is add the imaginary number i is defined as the number such that i squared equals negative one. This means that if you take the square root of both sides of i squared equals minus one, you get i equals negative one. If I was to write this by hand, I would say, write it here, I would say i equals the square root of negative one. And so if I then go the other way and square both sides, I have i squared equals negative one. Next operations with complex numbers, and we begin with what is called the standard form of a complex number. So if we have a complex number a plus b i, right, the variables a and b would be real numbers. So a is called the real part and b is called the imaginary part. If we have, say, b equals 0, so that means the 0 times i leaves us only with a, then that means that a plus 0i is a real number. Real numbers are part of the complex numbers. If, on the other hand, a equals 0, then a plus bi would be 0 plus bi, so the real part would be 0, and bi would be an imaginary number. The imaginary numbers are not. part of the real numbers so it doesn't go the other way. So just to put some numbers out if I put the number oh say 2 plus 3i this 2 would be the real part and 3i would be the imaginary part. Okay, next we can look at how to graph a complex number. So we can graph complex number on what is called the complex plane. So here to the left I have what is called the complex plane. What is normally considered the x-axis is going to be where we would plot the real part of a complex number and what is normally the y-axis is where we plot the imaginary part of a complex number. So for example, here on the right, we have graphed the complex number minus two plus three i. That means it has a real part of minus two. And so that means when we go from the point zero, zero on the complex plane left two, we've got the real part down. And then the three i, so that's the three up along the imaginary axis, that would mean three up. So one, two, three. And then we can plot there. point and that point in the complex plane represents the complex number minus two plus three i so that's two to the left and up three all right next we're going to perform the indicated operations and we're going to simplify it says make sure your answer is in standard form so that means make sure that our final answer is in the form a plus bi so the real part first you and then the complex part. All right, so first, we'll do number one. So four minus seven i plus negative five minus two i. So first of all, I'm going to remove the parentheses and group like terms. So that means the real part plus the real part. So four minus five, those are the real parts. And then I'm going to add up the complex part. So we have a minus 7i from the first one, and we have a minus 2i from the second part. All right, now I can simplify. So 4 minus 5 is negative 1. I should technically have equal signs, those are important. So negative 1, which is 4 minus 5, and then the minus 7i minus 2i is minus 9, and that is in standard. So we're done number one. All right, next we're going to go to number two. So number two, notice there's a negative sign in between the two complex numbers. So I'm going to first of all expand by multiplying that the second two numbers by the negative sign. The first numbers can stay the same minus four plus two i plus well actually it's going to be sorry negative because it's going to be negative times six negative six and then negative times negative seven i, so the negative times negative turns into plus seven i. Now I'm going to group like terms that would be negative four minus six, because that's the real part, and then the complex part plus two i plus seven i. All right, and then we can simplify that so minus four minus six is equal to minus ten and then 2i plus 7i adds up to 9i. And our solution is in standard form. All right, next we're going to look at multiplication. So number one was addition, number two we had subtraction, number three is multiplication. All right, so when you have complex numbers, you can go ahead and multiply through as you would expect. What you need to keep in mind is that when you have an i times i, that turns into i squared, which is negative one. All right, so the first thing I do is I'm going to multiply the minus 6 times 2, and that turns into minus 12, and then the i times i is i squared. So in many ways, or many people think of this as the i as a variable when you're multiplying, it'd be like minus 6x times 2x. You multiply first the minus 6 times the 2, and then the x times the x. So in that way, it's like that. So you have i squared. Alright, so now that you have this, you can have minus 12. And you can replace the i squared with negative one, because i squared equals negative one. And lastly, you can multiply negative one times minus 12, which results in a positive 12. Alright, let's look at like a little harder one. So this one has a complex number with just the imaginary part and a complex number with both a real and imaginary part, but otherwise exactly the same as one would expect. So we multiply through by expanding. So the 3i times 4, that is 12i because 3 times 4 is 12. And then the i just stays attached like a variable. And then the 3i times the minus 5i, so that's going to be minus 15i squared because 3 times 5 is 15. You have the negative from there and then i times i is i squared. Now we can simplify a little by replacing the i squared with minus 1. That's 12i minus 15 times negative 1. And lastly, well maybe we can do one more intermediate step. You can do the minus 1 times minus, that's positive, so you have 12i plus 15. And to finalize the result we want to write it in standard form, that means the real part first. And then we have 15 plus 12i and then we're done that problem. The next two, five and six, perform the indicated operations and simplify and make sure your answer is in standard form. All right, so this gets a little harder than the previous one. The previous one we had only one piece on the left and two on the right. So now we have two complex numbers both with the real and an imaginary part. What's nice is we can multiply out just like we would with a number that had no variable attached and a number that did. So all that is useful. So we would do 1 times 5 first, so that is 5, and then 1 times 2i, so that is 2i. And then I like to do the next arrows on the bottom. So minus 3i times 5 is minus 15i. And the minus 3i times 2i, that's minus 6i squared. All right, so we can simplify this a little bit. So let's see, let's the 5 stands alone, 2i minus 15i is minus 13i. And then the minus six, and the I squared can be replaced with negative one. All right, and then simplifying a little further, we have five. plus 6 because the minus 6 times minus 1 turns into a plus 6, so 6 plus 5 is 11, and then the minus 13i can stand on its own as the complex part. So now it is in standard form and we're done number 5. All right number 6. So number 6 you could rewrite that as 7 plus 2i times 7 plus 2i and apply the same technique. Or you can just use your standard formula for a plus b all squared, which is a squared plus 2ab plus b squared. So I'm going to do that. So a is 7 and b is 2i. So my first number squared is 49. 7 squared is 49. 2, so that's plus 2 times a times b, but b is 2i. And then plus. b squared. So the 2i, all of it squared. Don't make the mistake of only squaring the i. All right, so what does this equal? We have 7 squared is 49. 2 times 7 is 14, times 2 is 28. So we have 49 plus 28i. And then lastly, we have the 2i squared. 2i times 2i, so that's 2 times 2 is 4. And then the i squared is... negative one. And so then to simplify a little bit, we have the 49 minus four because minus one times four is negative four. So 49 minus four is equal to 45. And then plus the imaginary part. So plus 28i. And then we're done that problem because it's now in standard form. Okay, next is a useful techniques. So complex conjugates can be used for multiple simplifying procedures. So first, let's introduce what a complex conjugate is. So if you have a complex number a plus bi, that's in standard form, we we can denote the complex conjugate by putting a bar above there's various notations that you'll find in various books, but this is one of them. And so we say the complex conjugate of a plus bi is equal to a minus bi. So all the complex conjugate really is, is that complex number with the sign that's in between the real and the imaginary part replaced by its opposite sign. So likewise, for the complex number A minus B I, its complex conjugate is A plus B I. All right. So what's nice about complex conjugates is that if you multiply them together, you end up with a real number. And I want to show you this the long way. ideally you can do practice on your own with some real numbers like as in like two plus three i times two minus three i something like that but i'll show you in general if you were to multiply these out the long way let me show you you would say a times a is a squared and then a times minus b i is minus a b i and then you would go from the bottom you So b i times a, you'd have plus a b i, or the same as b a times i, and then b i times negative b i. And so what that would be is minus b squared times i squared, and then i squared is negative one. So what happens here, the minus a b i crosses out the positive a b i. And so what you're left with is only the a squared and that b squared because minus b squared times minus one is plus b squared. So that is what the formula here says. A complex number times its conjugate is equal to a squared plus b squared. So let's try this out. So perform the indicated operation and simplify and make sure your answer is in standard form. Alright, so according to the formula, you don't have to do the long multiplication, you can just say, okay, 5 plus 3i is the complex conjugate of 5 minus 3i and so that equals to that multiplication equals to 5 squared plus whoops plus 3 squared which is equal to 25 because 5 squared is 25 plus 9 because 3 squared is 9 and then when you add those up you get 34. All right let's try it on number 8. So you have 4 minus i times four plus i. So these numbers are conjugates of each other. And so that is equal to four squared plus one squared. And since 4 squared is 16, we have 16 plus 1. And so that multiplication is all equal to 17. And then we're done. Okay, so when we have division with complex numbers, these are the following steps you can do to in order to simplify. For the first step, write both the numerator and denominator in standard form. So the real part plus the net. the imaginary part. Okay, then we're going to multiply the numerator and the denominator by a complex conjugate of the denominator. So that's really important to notice the denominator. So for example over here we have the capital C over the complex number a plus bi the complex conjugate of that is a minus bi and notice we're multiplying the numerator and the denominator by that complex conjugate. And we want to simplify and write the result in standard form. So what that means is just multiply the C times the A minus bi, and that's just the number that stays on top. And the bottom, we can use the theorem that we just were using, right, that the complex number times its conjugate is just the A squared plus B squared. And then that way you simplify the denominator. And now the denominator is not complex, it's just a real number. Okay, let's try that out on number nine. So we're going to do this division. So the first thing we want to do is identify the conjugate of the denominator that is 4 plus 2i. And we want to multiply the top and bottom by that conjugate 4 plus 2i over 4 plus 2i, and then expand. So the numerator, we have 3 times 4 is 12, and 3 times 2i. is 6i. That's what we get in the numerator. And then on the bottom, on the denominator, we have 4 squared plus 2 squared. And so how does that simplify? We have 12 plus 6i remaining on the numerator. The denominator, 4 squared, 16 plus 2 squared, which is 4. And lastly, we have 12 plus 6i over... 20 because 16 plus 4 adds up to 20. And then we're done. I just wanted to go back for a moment to number 9 because I forgot to reduce. And I noticed that the 12, the 6, and the 20 have common factors. So let's see, what can we take out of the 20? You can take out a 2. Out of the 6, you can take out a 2. and out of the 12 you can take out a 2. So if we do that, the 12 divided by 2 turns into a 6, the 6 divided by 2 turns into a 3, and the 20 divided by 2 turns into a 10. So we can simplify this a little bit further as 6 plus 3i over 10. Okay, let's try another one. So we have 6 over minus 3 minus 4i. Okay, so the conjugate of the denominator this time is minus 3 plus 4i. So we multiply the top and bottom by minus 3 plus 4i and then we do that expanding. So we have 6 times minus 3 is minus 18 and then this and then 6 times 4 is 24i. And now for the denominator, we just apply the formula. So we have negative 3 squared and then plus 4 squared. And so then simplify one more time. So we have minus 18 plus 24i all divided by 9 plus 16. And so lastly, we can simplify that as negative 18 plus 24. I all divided by 25, because 16 plus nine is 25. With that we're done. And now we have number 11, which has a numerator that has a real part and a complex part and the denominator has a real and complex part. But otherwise same procedure. So we're going to identify the conjugate of that denominator. So the conjugate of the denominator is 3 minus 4i. and multiply the numerator and the denominator by that and then expand. And so we have, let's see, 2 times 3 is 6 and then 2 times minus 4i is minus 8i and then i times 3 is 3i and i times minus 4i is minus 4i squared. That's the numerator. And then the denominator we apply the formula. So we have 3 squared plus 4 squared. All right, so we can simplify that a little bit. So combining the 6 and the well let's see the minus 4 i squared recall that is equal to 4 because i squared equals negative 1 negative 1 to negative 4 is positive 4. So that part 4 plus 6 adds up to 10. And then we can combine the minus 8i and the 3i on the top that's here so that's going to be minus 5i and then for the denominator you have 3 squared is 9 and 4 squared is 16 so 9 plus 16 is once again 25. I'm about to have done the problem. Actually with this one we could simplify it a little bit further I just realized before pressing pause we have a five in the numerator in all the terms as well as in the denominator. So we can cancel one five, so 10 divided into five, right, that turns into a two, the five divided five is a one, so that leaves us minus i, and in the denominator 25 divided by five is five, and that is more simple. Alright, next we have a value evaluating the square root of a negative number. So just with numbers that aren't just the square root of negative one. So the square root of a negative number. So if B is a positive real number, then the square root of negative B is equal to I times root B. And more often than not, we actually write it as root B I. So if I was to do it in handwriting, root B times I. So note the square root of A times square root of B is equal to the square root of A times B is only valid when both a and b are greater than or equal to zero. So rewrite the square root of negative one as i first and then simplify. So here's some examples. Simplify. Express your answer as a complex number. The square root of negative 25. So first of all the square root of the negative is going to be i and then that's times the square root of 25. And so you have i times five and Otherwise, you can just say 5i. Okay, let's look at the next one. All right, so we have a negative out front. And then I look at this and look at the square root of negative 24. So first, I'll do the square root of the negative. So that's an i. And then the square root of 24. So that's 6 times 4. Okay. And then what can we do with that? Well, the square root of 4 is 2, so I can pull out a 2. The square root of 6, there's nothing I can do about that. So I have negative 2 square root of 6i, and that's that problem. Okay, lastly, let's look at the square root of negative 7. All right, so I'm going to first look at the square root of the negative, so that's an i, and then the square root of 7, so 7 is a... prime. I can't reduce that any further. And so in standard form, that would be square root of 7 times i. All right, next, perform the indicated operation and simplify express your answer as a complex number. All right, let's start with a. So square root of negative 27. So let's see, I'd like think of that as a square root of the negative is the i and then the 27 under the radical can be thought of as 9 times 3 and then let's look at the other radical so the square root of negative 12 i think of the square root of negative as the i and square root of 12 you can think of that as 4 times 3 okay then i can simplify so square root of 9 is 3 you Then we're left with the other three under the radical. Can't do much with that because that's a prime. So we have three root three I and then plus. And then I can do something with the four. Square root of four is two. So two and then square root of three. can't reduce that part. And then I now I look at both of these and I noticed I have like terms because both of my terms have a root three I in them. So now I can just combine those and add up the coefficients three and two. So three plus two is five, getting little equal signs here. So three plus two is five, root three, I. All right, and that's it. Alright, so next we have b square root of negative 18 times square root of negative 8. And so that is equal to well, let's see. So first, I'm going to think of that as the square root of negative times the square root of negative, what do we have i squared, and then we have the square root of 18 times eight. Okay, and so then we have i squared is negative 1, and square root of 18 times 8, that is the square root of 144. And you can recognize that as 12. So you have equals minus 12. And with that, we're done. So let's finish off with the last problem. We have... 5 plus the square root of negative 18 times negative 2 minus the root of negative 50. So we have to multiply out. I'll do that one step at a time. So first I'll multiply 5 times negative 2. And so we have negative 10, then 5 times negative root negative 50. That gives me minus 5. square root of negative 50. And from the bottom, I'll do square root of negative 18 times negative 2. That is negative 2 square root of negative 18. And then the last multiplication, we have square root of negative. Sorry, negative. Yeah, negative from the middle term here. So negative. square root of negative 18 times square root of negative 50. So just to repeat, I pulled out the negative and the negative square root of negative 50. And I just put it out front. All right, so now I'm just going to go step by step for each of the terms. So the first term is just negative 10, nothing to reduce there. The next term, I can actually break up well, two things, one, the square root of negative one or The square root of the negative is i, so I can pull that out. We have minus 5i. And the other thing I can do is break up the 50 into two factors, one that is going to be a perfect square, the other one, which is a prime. So 50 is 25 times 2. All right, so next term, I have a negative 2. The square root of the negative, or the square root of the negative 1, is going to be an i. So I can pull out an i. And then the 18 I can also break down into 9 times 2, so the 9 is a perfect square and the 2 I can't reduce. And then next term, we're still going to have a multiplication but I'm going to deal with them separately. So the square root of the negative in the 18, the negative 18 is going to come out as an i, and then the 18 I can break down into a 9 times 2. And then lastly, for the square root of negative 50, I'm going to bring down. So the square root of the negative is going to come out as pi. And then the 50 I can break down into 25 times 2. All right. And then off to the next line and see if I can reduce it a little bit more. So we have negative 10. All right. So from the second term, from the square root of 25 times 2, I'm going to pull out a 5 because the square root of 25 is 5. Now that 5 I can multiply with the 5 out front and that is going to turn into a minus 25. The root 2 I can't reduce further so I just put that next minus 25 root 2 and then let's not forget the i. Okay next term we have a minus 2i and then the square root of 9 times 2. The square root of 9 is 3 so I'm going to pull out that 3 and multiply by that 2 out front. So 2 times 3 is 6. We have a negative 6. Nothing I can do about the square root of 2. and then we have i. So negative 6 root 2 and i. And then let's see what we can do for the next multiplication. So the i here and the i here, those multiply out to i squared. i squared is negative 1, but negative 1 times the negative out front turns it into a positive. All right, so I've worked out already the i part. From the square root of 9 times 2, I can pull out a 3 because square root of 9 is 3. So I have three root two. And then for the this is still a multiplication and from the square root of 25 times two, I can pull it by because the square root of 25 is five, and the root two, we have to leave it on its own like that. Alright, so let's see what more we can do to simplify here. So equals negative 10. Let's see, we can combine these I terms we have here. the minus 25 root 2i minus 6 root 2i. So that's minus 25 minus 6, that's minus 31 root 2i, those are like terms. And then lastly, we can do this multiplication here. So 3 times 5 is 15. Okay, 15. And then root 2 times root 2. So that means that's the square root of 2 squared. So the square cancels out the root and we're just left with 2. I repeat square root 2 times square root 2 is just 2. And so then that gives us 30. So then we have negative 10 minus 31 root 2 i plus 30. And then we can finish up the problem combining the 30 and the negative 10 so 30 minus 10 is 20 20 and then the imaginary part we can leave so minus 31 root 2 i and with that we're done uh the section in complex numbers thank you for listening

If I was to write this by hand, I would say, write it here, I would say i equals the square root of negative one. And so if I then go the other way and square both sides, I have i squared equals negative one. Next operations with complex numbers, and we begin with what is called the standard form of a complex number.

So if we have a complex number a plus b i, right, the variables a and b would be real numbers. So a is called the real part and b is called the imaginary part. If we have, say, b equals 0, so that means the 0 times i leaves us only with a, then that means that a plus 0i is a real number. Real numbers are part of the complex numbers. If, on the other hand, a equals 0, then a plus bi would be 0 plus bi, so the real part would be 0, and bi would be an imaginary number.

The imaginary numbers are not. part of the real numbers so it doesn't go the other way. So just to put some numbers out if I put the number oh say 2 plus 3i this 2 would be the real part and 3i would be the imaginary part.

Okay, next we can look at how to graph a complex number. So we can graph complex number on what is called the complex plane. So here to the left I have what is called the complex plane.

What is normally considered the x-axis is going to be where we would plot the real part of a complex number and what is normally the y-axis is where we plot the imaginary part of a complex number. So for example, here on the right, we have graphed the complex number minus two plus three i. That means it has a real part of minus two.

And so that means when we go from the point zero, zero on the complex plane left two, we've got the real part down. And then the three i, so that's the three up along the imaginary axis, that would mean three up. So one, two, three. And then we can plot there. point and that point in the complex plane represents the complex number minus two plus three i so that's two to the left and up three all right next we're going to perform the indicated operations and we're going to simplify it says make sure your answer is in standard form so that means make sure that our final answer is in the form a plus bi so the real part first you and then the complex part.

All right, so first, we'll do number one. So four minus seven i plus negative five minus two i. So first of all, I'm going to remove the parentheses and group like terms.

So that means the real part plus the real part. So four minus five, those are the real parts. And then I'm going to add up the complex part.

So we have a minus 7i from the first one, and we have a minus 2i from the second part. All right, now I can simplify. So 4 minus 5 is negative 1. I should technically have equal signs, those are important. So negative 1, which is 4 minus 5, and then the minus 7i minus 2i is minus 9, and that is in standard.

So we're done number one. All right, next we're going to go to number two. So number two, notice there's a negative sign in between the two complex numbers.

So I'm going to first of all expand by multiplying that the second two numbers by the negative sign. The first numbers can stay the same minus four plus two i plus well actually it's going to be sorry negative because it's going to be negative times six negative six and then negative times negative seven i, so the negative times negative turns into plus seven i. Now I'm going to group like terms that would be negative four minus six, because that's the real part, and then the complex part plus two i plus seven i.

All right, and then we can simplify that so minus four minus six is equal to minus ten and then 2i plus 7i adds up to 9i. And our solution is in standard form. All right, next we're going to look at multiplication.

So number one was addition, number two we had subtraction, number three is multiplication. All right, so when you have complex numbers, you can go ahead and multiply through as you would expect. What you need to keep in mind is that when you have an i times i, that turns into i squared, which is negative one.

All right, so the first thing I do is I'm going to multiply the minus 6 times 2, and that turns into minus 12, and then the i times i is i squared. So in many ways, or many people think of this as the i as a variable when you're multiplying, it'd be like minus 6x times 2x. You multiply first the minus 6 times the 2, and then the x times the x. So in that way, it's like that.

So you have i squared. Alright, so now that you have this, you can have minus 12. And you can replace the i squared with negative one, because i squared equals negative one. And lastly, you can multiply negative one times minus 12, which results in a positive 12. Alright, let's look at like a little harder one. So this one has a complex number with just the imaginary part and a complex number with both a real and imaginary part, but otherwise exactly the same as one would expect.

So we multiply through by expanding. So the 3i times 4, that is 12i because 3 times 4 is 12. And then the i just stays attached like a variable. And then the 3i times the minus 5i, so that's going to be minus 15i squared because 3 times 5 is 15. You have the negative from there and then i times i is i squared.

Now we can simplify a little by replacing the i squared with minus 1. That's 12i minus 15 times negative 1. And lastly, well maybe we can do one more intermediate step. You can do the minus 1 times minus, that's positive, so you have 12i plus 15. And to finalize the result we want to write it in standard form, that means the real part first. And then we have 15 plus 12i and then we're done that problem.

The next two, five and six, perform the indicated operations and simplify and make sure your answer is in standard form. All right, so this gets a little harder than the previous one. The previous one we had only one piece on the left and two on the right. So now we have two complex numbers both with the real and an imaginary part.

What's nice is we can multiply out just like we would with a number that had no variable attached and a number that did. So all that is useful. So we would do 1 times 5 first, so that is 5, and then 1 times 2i, so that is 2i. And then I like to do the next arrows on the bottom. So minus 3i times 5 is minus 15i.

And the minus 3i times 2i, that's minus 6i squared. All right, so we can simplify this a little bit. So let's see, let's the 5 stands alone, 2i minus 15i is minus 13i.

And then the minus six, and the I squared can be replaced with negative one. All right, and then simplifying a little further, we have five. plus 6 because the minus 6 times minus 1 turns into a plus 6, so 6 plus 5 is 11, and then the minus 13i can stand on its own as the complex part.

So now it is in standard form and we're done number 5. All right number 6. So number 6 you could rewrite that as 7 plus 2i times 7 plus 2i and apply the same technique. Or you can just use your standard formula for a plus b all squared, which is a squared plus 2ab plus b squared. So I'm going to do that. So a is 7 and b is 2i. So my first number squared is 49. 7 squared is 49. 2, so that's plus 2 times a times b, but b is 2i.

And then plus. b squared. So the 2i, all of it squared.

Don't make the mistake of only squaring the i. All right, so what does this equal? We have 7 squared is 49. 2 times 7 is 14, times 2 is 28. So we have 49 plus 28i.

And then lastly, we have the 2i squared. 2i times 2i, so that's 2 times 2 is 4. And then the i squared is... negative one.

And so then to simplify a little bit, we have the 49 minus four because minus one times four is negative four. So 49 minus four is equal to 45. And then plus the imaginary part. So plus 28i.

And then we're done that problem because it's now in standard form. Okay, next is a useful techniques. So complex conjugates can be used for multiple simplifying procedures.

So first, let's introduce what a complex conjugate is. So if you have a complex number a plus bi, that's in standard form, we we can denote the complex conjugate by putting a bar above there's various notations that you'll find in various books, but this is one of them. And so we say the complex conjugate of a plus bi is equal to a minus bi.

So all the complex conjugate really is, is that complex number with the sign that's in between the real and the imaginary part replaced by its opposite sign. So likewise, for the complex number A minus B I, its complex conjugate is A plus B I. All right. So what's nice about complex conjugates is that if you multiply them together, you end up with a real number. And I want to show you this the long way.

ideally you can do practice on your own with some real numbers like as in like two plus three i times two minus three i something like that but i'll show you in general if you were to multiply these out the long way let me show you you would say a times a is a squared and then a times minus b i is minus a b i and then you would go from the bottom you So b i times a, you'd have plus a b i, or the same as b a times i, and then b i times negative b i. And so what that would be is minus b squared times i squared, and then i squared is negative one. So what happens here, the minus a b i crosses out the positive a b i.

And so what you're left with is only the a squared and that b squared because minus b squared times minus one is plus b squared. So that is what the formula here says. A complex number times its conjugate is equal to a squared plus b squared.

So let's try this out. So perform the indicated operation and simplify and make sure your answer is in standard form. Alright, so according to the formula, you don't have to do the long multiplication, you can just say, okay, 5 plus 3i is the complex conjugate of 5 minus 3i and so that equals to that multiplication equals to 5 squared plus whoops plus 3 squared which is equal to 25 because 5 squared is 25 plus 9 because 3 squared is 9 and then when you add those up you get 34. All right let's try it on number 8. So you have 4 minus i times four plus i.

So these numbers are conjugates of each other. And so that is equal to four squared plus one squared. And since 4 squared is 16, we have 16 plus 1. And so that multiplication is all equal to 17. And then we're done.

Okay, so when we have division with complex numbers, these are the following steps you can do to in order to simplify. For the first step, write both the numerator and denominator in standard form. So the real part plus the net. the imaginary part. Okay, then we're going to multiply the numerator and the denominator by a complex conjugate of the denominator.

So that's really important to notice the denominator. So for example over here we have the capital C over the complex number a plus bi the complex conjugate of that is a minus bi and notice we're multiplying the numerator and the denominator by that complex conjugate. And we want to simplify and write the result in standard form.

So what that means is just multiply the C times the A minus bi, and that's just the number that stays on top. And the bottom, we can use the theorem that we just were using, right, that the complex number times its conjugate is just the A squared plus B squared. And then that way you simplify the denominator. And now the denominator is not complex, it's just a real number. Okay, let's try that out on number nine.

So we're going to do this division. So the first thing we want to do is identify the conjugate of the denominator that is 4 plus 2i. And we want to multiply the top and bottom by that conjugate 4 plus 2i over 4 plus 2i, and then expand. So the numerator, we have 3 times 4 is 12, and 3 times 2i. is 6i.

That's what we get in the numerator. And then on the bottom, on the denominator, we have 4 squared plus 2 squared. And so how does that simplify? We have 12 plus 6i remaining on the numerator.

The denominator, 4 squared, 16 plus 2 squared, which is 4. And lastly, we have 12 plus 6i over... 20 because 16 plus 4 adds up to 20. And then we're done. I just wanted to go back for a moment to number 9 because I forgot to reduce. And I noticed that the 12, the 6, and the 20 have common factors.

So let's see, what can we take out of the 20? You can take out a 2. Out of the 6, you can take out a 2. and out of the 12 you can take out a 2. So if we do that, the 12 divided by 2 turns into a 6, the 6 divided by 2 turns into a 3, and the 20 divided by 2 turns into a 10. So we can simplify this a little bit further as 6 plus 3i over 10. Okay, let's try another one. So we have 6 over minus 3 minus 4i.

Okay, so the conjugate of the denominator this time is minus 3 plus 4i. So we multiply the top and bottom by minus 3 plus 4i and then we do that expanding. So we have 6 times minus 3 is minus 18 and then this and then 6 times 4 is 24i. And now for the denominator, we just apply the formula.

So we have negative 3 squared and then plus 4 squared. And so then simplify one more time. So we have minus 18 plus 24i all divided by 9 plus 16. And so lastly, we can simplify that as negative 18 plus 24. I all divided by 25, because 16 plus nine is 25. With that we're done.

And now we have number 11, which has a numerator that has a real part and a complex part and the denominator has a real and complex part. But otherwise same procedure. So we're going to identify the conjugate of that denominator.

So the conjugate of the denominator is 3 minus 4i. and multiply the numerator and the denominator by that and then expand. And so we have, let's see, 2 times 3 is 6 and then 2 times minus 4i is minus 8i and then i times 3 is 3i and i times minus 4i is minus 4i squared. That's the numerator. And then the denominator we apply the formula.

So we have 3 squared plus 4 squared. All right, so we can simplify that a little bit. So combining the 6 and the well let's see the minus 4 i squared recall that is equal to 4 because i squared equals negative 1 negative 1 to negative 4 is positive 4. So that part 4 plus 6 adds up to 10. And then we can combine the minus 8i and the 3i on the top that's here so that's going to be minus 5i and then for the denominator you have 3 squared is 9 and 4 squared is 16 so 9 plus 16 is once again 25. I'm about to have done the problem.

Actually with this one we could simplify it a little bit further I just realized before pressing pause we have a five in the numerator in all the terms as well as in the denominator. So we can cancel one five, so 10 divided into five, right, that turns into a two, the five divided five is a one, so that leaves us minus i, and in the denominator 25 divided by five is five, and that is more simple. Alright, next we have a value evaluating the square root of a negative number.

So just with numbers that aren't just the square root of negative one. So the square root of a negative number. So if B is a positive real number, then the square root of negative B is equal to I times root B. And more often than not, we actually write it as root B I. So if I was to do it in handwriting, root B times I.

So note the square root of A times square root of B is equal to the square root of A times B is only valid when both a and b are greater than or equal to zero. So rewrite the square root of negative one as i first and then simplify. So here's some examples. Simplify.

Express your answer as a complex number. The square root of negative 25. So first of all the square root of the negative is going to be i and then that's times the square root of 25. And so you have i times five and Otherwise, you can just say 5i. Okay, let's look at the next one.

All right, so we have a negative out front. And then I look at this and look at the square root of negative 24. So first, I'll do the square root of the negative. So that's an i. And then the square root of 24. So that's 6 times 4. Okay.

And then what can we do with that? Well, the square root of 4 is 2, so I can pull out a 2. The square root of 6, there's nothing I can do about that. So I have negative 2 square root of 6i, and that's that problem.

Okay, lastly, let's look at the square root of negative 7. All right, so I'm going to first look at the square root of the negative, so that's an i, and then the square root of 7, so 7 is a... prime. I can't reduce that any further. And so in standard form, that would be square root of 7 times i. All right, next, perform the indicated operation and simplify express your answer as a complex number.

All right, let's start with a. So square root of negative 27. So let's see, I'd like think of that as a square root of the negative is the i and then the 27 under the radical can be thought of as 9 times 3 and then let's look at the other radical so the square root of negative 12 i think of the square root of negative as the i and square root of 12 you can think of that as 4 times 3 okay then i can simplify so square root of 9 is 3 you Then we're left with the other three under the radical. Can't do much with that because that's a prime. So we have three root three I and then plus.

And then I can do something with the four. Square root of four is two. So two and then square root of three. can't reduce that part.

And then I now I look at both of these and I noticed I have like terms because both of my terms have a root three I in them. So now I can just combine those and add up the coefficients three and two. So three plus two is five, getting little equal signs here. So three plus two is five, root three, I. All right, and that's it.

Alright, so next we have b square root of negative 18 times square root of negative 8. And so that is equal to well, let's see. So first, I'm going to think of that as the square root of negative times the square root of negative, what do we have i squared, and then we have the square root of 18 times eight. Okay, and so then we have i squared is negative 1, and square root of 18 times 8, that is the square root of 144. And you can recognize that as 12. So you have equals minus 12. And with that, we're done.

So let's finish off with the last problem. We have... 5 plus the square root of negative 18 times negative 2 minus the root of negative 50. So we have to multiply out. I'll do that one step at a time.

So first I'll multiply 5 times negative 2. And so we have negative 10, then 5 times negative root negative 50. That gives me minus 5. square root of negative 50. And from the bottom, I'll do square root of negative 18 times negative 2. That is negative 2 square root of negative 18. And then the last multiplication, we have square root of negative. Sorry, negative. Yeah, negative from the middle term here. So negative.

square root of negative 18 times square root of negative 50. So just to repeat, I pulled out the negative and the negative square root of negative 50. And I just put it out front. All right, so now I'm just going to go step by step for each of the terms. So the first term is just negative 10, nothing to reduce there. The next term, I can actually break up well, two things, one, the square root of negative one or The square root of the negative is i, so I can pull that out. We have minus 5i.

And the other thing I can do is break up the 50 into two factors, one that is going to be a perfect square, the other one, which is a prime. So 50 is 25 times 2. All right, so next term, I have a negative 2. The square root of the negative, or the square root of the negative 1, is going to be an i. So I can pull out an i.

And then the 18 I can also break down into 9 times 2, so the 9 is a perfect square and the 2 I can't reduce. And then next term, we're still going to have a multiplication but I'm going to deal with them separately. So the square root of the negative in the 18, the negative 18 is going to come out as an i, and then the 18 I can break down into a 9 times 2. And then lastly, for the square root of negative 50, I'm going to bring down.

So the square root of the negative is going to come out as pi. And then the 50 I can break down into 25 times 2. All right. And then off to the next line and see if I can reduce it a little bit more.

So we have negative 10. All right. So from the second term, from the square root of 25 times 2, I'm going to pull out a 5 because the square root of 25 is 5. Now that 5 I can multiply with the 5 out front and that is going to turn into a minus 25. The root 2 I can't reduce further so I just put that next minus 25 root 2 and then let's not forget the i. Okay next term we have a minus 2i and then the square root of 9 times 2. The square root of 9 is 3 so I'm going to pull out that 3 and multiply by that 2 out front.

So 2 times 3 is 6. We have a negative 6. Nothing I can do about the square root of 2. and then we have i. So negative 6 root 2 and i. And then let's see what we can do for the next multiplication. So the i here and the i here, those multiply out to i squared.

i squared is negative 1, but negative 1 times the negative out front turns it into a positive. All right, so I've worked out already the i part. From the square root of 9 times 2, I can pull out a 3 because square root of 9 is 3. So I have three root two. And then for the this is still a multiplication and from the square root of 25 times two, I can pull it by because the square root of 25 is five, and the root two, we have to leave it on its own like that.

Alright, so let's see what more we can do to simplify here. So equals negative 10. Let's see, we can combine these I terms we have here. the minus 25 root 2i minus 6 root 2i.

So that's minus 25 minus 6, that's minus 31 root 2i, those are like terms. And then lastly, we can do this multiplication here. So 3 times 5 is 15. Okay, 15. And then root 2 times root 2. So that means that's the square root of 2 squared. So the square cancels out the root and we're just left with 2. I repeat square root 2 times square root 2 is just 2. And so then that gives us 30. So then we have negative 10 minus 31 root 2 i plus 30. And then we can finish up the problem combining the 30 and the negative 10 so 30 minus 10 is 20 20 and then the imaginary part we can leave so minus 31 root 2 i and with that we're done uh the section in complex numbers thank you for listening

Transcript for:Understanding Complex Numbers and Operations

Transcript for:
Understanding Complex Numbers and Operations