College Algebra | Section 3 | Introduction to Complex Numbers | Victor Rodriguez | Skillshare

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College Algebra | Section 3 | Introduction to Complex Numbers

teacher avatar Victor Rodriguez, Teaching Math and Science

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Lessons in This Class

6 Lessons (36m)
    • 1. 3.1 Understand Imaginary Numbers

    • 2. 3.2 Understand Complex Numbers

    • 3. 3.3 Add & Subtract Complex Numbers

    • 4. 3.5 Dividing Complex Numbers

    • 5. Complex numbers Review

    • 6. 3.4 Multiplying Complex Numbers

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About This Class

This course will introduce you to the concept of complex and imaginary numbers. I will go through how to derive complex and imaginary numbers, and how to add, subtract, multiply and divide them. Check out the project documents for review martial. 

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Victor Rodriguez

Teaching Math and Science


Hello, I'm Victor. I am an astrobiology researcher and a math and science enthusiast. I love sharing my knowledge with the world. 

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1. 3.1 Understand Imaginary Numbers : Hello, students. And welcome to course, number three in my college algebra courses, you can check out my past college algebra courses Introduction to polynomial in introduction to radicals to get a background information for what I'm going to talk about today, which is complex numbers. And this is the introduction to complex numbers. Course, you'll see documents linked in this course that you can follow along with and use for, um, for study. Okay, introduction to complex numbers. First assed part of complex numbers. We're gonna talk about imaginary numbers, so real number is a rational number or an irrational number. And of course, an example of a rational number would be the square root of four, which is to, Or you could take your irrational number, which is a square root of two. Now, we're going to talk about imaginary numbers now, an imaginary number, if you If you If you go back to the, um radicals course if you take, let's say the square root of two, you get some number. If you take the square root of four, you got to however, what if you get the square root of negative for now? If you Ah don't think about it. If you think about it too quickly, you might say, Oh, it's to The thing is, there's no, there's no positive number. There's no number out there that would get you negative to when you square, because any posit, any number, whether it's negative or positive if you square it, will give you, um, positive. So negative. Two times negative, too, is four. So you might say that this is impossible. However, there is a work around it. It's called the imaginary number in. What I'll Do Here is, of course, if we take, let's say, radical eight. Right? If you take radical eight, you can, uh, factor this so that negative or radical eight equals radical. Four times two. So now you can do you can do that. You can take this and then you can take the radical four or the radical four portion becomes to, and then it becomes to radical, too. We did this by by taking out the factors, and then now this one, because it's a perfect square, can come out and become negative are become too in. This stays as radical, too. However, you can do this with negative numbers. So let's say you have radical four radical negative four. What you can do is you can actually take out four times negative one, because four times negative one is negative four. Then you can take this out. Take the four out in equals two, and now you have to radical negative one. Now, of course you could say Well, what do we do with this? Well, this is where imaginary numbers comes in. If you have X squared equals negative one and then X equals radical negative one equals I. Now what this means is it's a It takes a little thought to kind of put this together. We've assigned the value of radical negative one, which is right up here to be I. And then if you take I and you square it, you get negative one. So instead of writing to radical negative one, you can take this. You write it as to I and that's the imaginary number. So that concludes, Uh, this portion of introduction to imaginary numbers, my next class will be on, uh, understanding complex numbers 2. 3.2 Understand Complex Numbers: however one now that we have that introduced to the imaginary number. Now we're going to get introduced, introduced to the complex number. Now, for a small bit of recap, you had the imaginary I, which equals radical negative one I squared with then equal radical negative one times radical negative one, which equals negative one. So I squared equals one. And that's the intuitive, the intuition that we've derived from I equals negative, radical negative one. Now we can use this in more complex functions or, in other words, complex numbers. Now, first, let's do a little pure, rational numbers or pure Jerash. Imaginary numbers. For example, Negative 36 or radical negative 36 equals 36 or radical 36 times negative one. Well, if we square root, this ID equals six radical negative one, and then that can then be written as six. I now for complex numbers. So a complex number is a combination of riel in imaginary numbers. Any number can be written as a complex number. Now, what I will do is all right up here. The components of riel or of a complex number you have a plus. Be I okay now the A component is the rial number. The B is a coefficient. You know the words that coefficient. In this case, it's six, the coefficient of I. And of course, this is the imaginary, and this together is the imaginary number. So that concludes thea complex numbers section of this course. Next, we will be doing operations with complex numbers. 3. 3.3 Add & Subtract Complex Numbers : Hello, everyone. Now that you have an introduction into complex numbers and imaginary numbers, now we're going to start doing operations with complex numbers. And in this video, we're gonna be doing adding and subtracting complex numbers. So adding and subtracting. Okay, adding and subtracting complex numbers now first is right in a plus B i format. As we learned in the last video second step. The second step is to remove parentheses, and then the third step is combined real numbers, plus imaginary numbers. Now first, I'm going to write down a equation, and then we we will solve it in operated based based on the rules of imaginary numbers and complex numbers. So we have negative negative 18 plus radical negative for minus six minus negative 25 1st we need to write an A plus B. I form we negative 18. We can bring down negative 18 plus. We will keep that their plus and radical radical negative four equals to I. Because if we take out the negative one from the four from the negative four, it's four times negative. One weaken, then square root that into two and then the neck. The radical negative one equals I minus. And then we can bring down to six minus five I because 20 negative 25 divided. If you factor. Or if you separate it to 25 negative one, the 25 could become far can be square rooted to five. And then the radical negative one becomes I. So now you have negative 18 plus to I minus six, minus five I. So now we can remove the parentheses. So now we have negative 18 plus to I minus six plus five. I now it's very important that you remember to distribute the negative sign throughout the entire parentheses. Uh, polynomial. So now we can take this and combine like terms. Negative 18 minus six is negative. 27 then positive two. Or are two I plus five. I equals seven I. So this is the answer, and we can see that it The end result is an A plus B I form. We take negative 27. That's the A. And then you have plus in this plus could be minus. It doesn't have to be plus, but you have seven I, which is the B I component. Seven is the B and of course, Eyes I in 27 negative 27 is the A. Now the thing to remember is to always distribute that negative sign throughout the parentheses. Otherwise, you can find yourself making some pretty huge mistakes in when it comes to adding and subtracting really any pilot. No meals, but especially complex numbers. So that's the adding and subtracting lectures in this series on introduction to complex numbers. My next video will be on multiplying complex numbers. Thank you. 4. 3.5 Dividing Complex Numbers : how everyone Welcome to the dividing complex numbers portion of the introduction toe complex numbers video Siri's. And right now what I will be talking about is how to divide complex numbers. And now what you want to do is you want to have the denominator be rational in what? What I will do is I'll give you an example. Now, in this case, we have three divided by five I. And of course, if you remember from my radicals course said I taught you want this the denominator to be rational in the best way to do this. Of course I squared is negative one, which is a rational number. So what we can do is we can multiply by I over I and what this does is equals. Three I over five. I squared. Which five I squared is negative five. So we have three I over negative five, and then we have now we have. What we have to do is because you always want to write complex numbers in a plus B I form. What we can do is we can actually take that I right out of there and it and make it three or negative. Three fifths by. And that's your final answer right there. Negative. 3/5 over. I now we can do another one, which is a little more complex. 3/7 plus two. I now, of course, um, we want to in the because there's a polynomial in the denominator. We need the, um, congregate. Okay, which is so what we do is we multiply by seven. Negative to I because this is positive. We want toe congregate. We want to turn into a negative seven. Negative to I over seven. Negative to I. And now we can multiply this out in. What we get is we get, um, 21. Oops, 21 7 times three and then three times too negative, too. I is negative. Six I over and then we do seven times seven is 49 and then seven times negative to I is negative. 14 i and then two I times seven is 14 I and then two I times negative to I is negative four . I squared. Now, if you did the congregation right, you should get to terms that are equal or opposite negative 14 I and 14 eyes. So what? We'll do is, we'll take this and will reword it or re arrange it here. Down here, the 21 minus six I over. And of course, you have 49 and then this negative 14 I plus 14. I equals zero. So we can ignore that. And the negative four times I squared is the same as negative. Four times negative one, which is four. We can then simplify this further as 21 minus six I over 53 which then we can write it in a plus B I formed, which what we do is we can separate this too. 21/53 minus 6/53 i. And now this is the A. This is the B. And this is, of course, the I. And that's how you divide, um, complex numbers. And my next lecture will be a review on the worksheet that you can download below. Thank you 5. Complex numbers Review: Hello, everyone. And welcome to the review portion of my, um, introduction to complex numbers. Course. Now, I attached a worksheet that you can work with throughout the course, and I will actually take a couple for to review with. So first I'm going to do adding complex numbers. I'm gonna do number three. Start off. Simple three I plus I. Now, of course, this one is actually pretty simple because three I plus I is four I now we can come. Uh, change it up a little by doing number five, which is negative. One minus eight. I minus four minus I. Now, in this case, we have to combine like terms which negative one minus four is negative. Five and then negative ai minus. I is negative. Nine I. So that's number three or number number five. Now we can do, um, number nine for multiplying complex numbers in next four I times. Negative two minus eight I. And what we have to do is we have to distribute, and the way we do this is we take four i times negative two, which is negative. Eight I And then you take negative four or you take four I times negative eight I which is 32. I squared because you have I times I is I squared. So then we want to get this into a plus B. I form Well, the I squared is negative one which makes that in And of course, negative. One times negative 32 is just 32 and then ai is course negative. Eight i and then now we write this in a plus B I form, which is 32 minus eight I. And that's how you do multiplication through distribution. Now, this one isn't on the worksheet, But I'm gonna work through this one for a division. Now, I don't have the answer written down for this, So we're actually gonna work on work through this as it happened. So you're gonna see in a real time. So of course we need a rational denominator. So what we do is in this case because we have a polynomial in the denominator. We have to do the congregate, which the congregate is 12 minus six. I over 12 minus six. Hi. And what I'm gonna do is I'm gonna write it all down here, so I have more room So what you have is three times 12 which is 36 negative ai or three times six I which is or negative? Six I, which is 18 I. And then you have eight times 12 which is negative. 96 I. And then now, of course, this one I just randomly made up. So it's gonna be really messy. But then you have negative eight I times negative six I, which is positive, is negative times negative. Positive 48. Hi. Then we go down to the bottom, which we get 144. You're 12 times 12 and in 12 times six. I negative six I, which is 72. Hi. Negative. 72 I. Then you get six I times 12 which is positive. 72 I. And this is where you know you've done your conduct correctly. You should have a negative and the positive of the same value. And then six I times six or negative. Six. I time six. I is negative. 36. I squared now. I missed something over here. Um, the I I I missed the I squared. It's important not to miss the I squared for up here a negative. Eight I times negative six I. So now this is kind of messy and now we need to consolidate things, simplify things. So what we can do is we can actually go right through. And because these negate each other, we can cross those right out. And then there's a like terms which what we'll do is we'll take 36 and then these two we can combine, which is, let's see a that's four 1 10 11 144. So that's negative 144 I and then plus Now you have 48 I squared, which is negative 48 over 144 minus and any which would be minus 36 I squared is actually plus 36 and then we can ah simplify this even more. We can combine these 36 minus 48. This to is negative 12 and what we want to do is and then what we and then we have negative 1 44 i mover and then you have zero. The eight 180 are negative. 12 minus 144 i over 1 80 which then we have to write in a plus B a plus B I form which will bring up here. Negative 12/1 80. Minus 1 44/1 80 i. And now that's an A plus. B I for now. This concludes the review portion of our introduction to complex numbers. Thank you so much for tuning into this lecture series. And make sure to check out all my other lectures in on college algebra. Thank you. 6. 3.4 Multiplying Complex Numbers: Hello, everyone. This is the multiplying, complex numbers portion of the introduction to complex numbers. Series of lectures today will be doing multiplying complex numbers. Now when multiplying radicals understand that the, um, previous rules on multiplying radicals on Lee um, govern rial numbers, not imaginary numbers. So it's important to understand that when multiplying complex numbers, you do not multiply. When the radicals are negative. Instead, you write them in a plus B I form. And this will make more sense when I show you on the board late soon. So let's say we have radical negative three times radical negative six. Now we can't just multiply this instead, What we have to do is we first have to write it in a plus B I form, which we can then write it as radical three i times radical six I. And now we can multiply them. Now, of course, radical Three times radical six is radical 18 and then I times I is I squared. And if you remember from my previous videos in this lecture series, um, I squared is equals negative one, so you can then take this and you can and you can write it as radical 18 times. Negative one. And of course, you can take a square out of here, so let's do that. All right, up here. 18 radical 18 can be broken up to nine times to radical radical, nine times two. And then, of course, this is negative. One because of the I squared equals negative one. Now we can take out three radical too times, negative one which we can then disturb, or we can then multiply this through. And now we have negative three radical, too, who we were able to keep track of the imaginary number throughout. This entire equation now is part of multiplying complex numbers. I'm gonna show you how to distribute complex numbers throughout a polynomial. So what we do is we first take five times three, which is 15 I, and we treat this like a variable. So the rules off, uh, operations with polynomial supply. So you have so five I times three equals 15 I It is important to keep track of your eyes and then five I times six I equal or negative six. I equals negative. 30. I squared because you have I and I I times eyes. I squared. Now we can take this and we can simplify it as 15 I minus 30 negative one, which then you can further simplify as 15 i a negative times a negative positive. So you get plus 30 and then we can do this even further to now. We can do it as a plus B I form, in which case we will switch it around and we'll get the 30 plus 15. All right in what I'll do. And now that's in a A plus b. I format. Now what I'll do is I'll do another distribution to kind of solidify our understanding on how to distribute with complex numbers. Now we have seven negative or seven minus two. I time six plus I. Now, even though there's no be shown here, we can assume that it's one just like with any other variables. When you don't see a coefficient in front of it, you assume it's one. Uh, so now let's foil is through. You get seven times six is 42 and then you have seven times I, which is seven I. Then you have negative I times six, which is negative. 12 I. Then you have negative to i times I which is negative to I squared. Now we can simplify this a little and take you get the 40 to bring that down and then you can add these together. Ah, positive. Seven. I minus 12. I equals negative five I And then now we can take this I squared or negative toe I squared in It equals of course, this this, uh, this equals negative one. So negative one times negative two equals to Now we can write this in a plus B I form, which means we have to combine these two like terms, which is 40 42 plus 42 equals 44 minus five I And that's your answer. So this concludes the multiplying complex numbers portion off our introduction to complex numbers Lecture. The next video will be on dividing complex numbers. Thank you.