hello again everyone Welcome to our discussion of section 3.5 where we will talk about complex numbers so complex numbers uh is a vast area of mathematics but in this course we will have this is the only section where we will talk about it and in the future those of you who will be taking uh higher level mathematics especially if there is any math Majors among you for sure you will see uh courses that will you will have to take courses that are specifically focusing on complex uh numbers okay such as complex analysis is one is one very cool course uh for mathematicians those of you who are electrical engineering or computer science Majors will again take courses that have uh that deal with complex numbers uh more than other majors and even mechanical engineers will deal with this okay so uh before we delve into this topic I wanted to remind everyone to please skim through the textbook section 3.5 in this case before you watch the video or you attend the class and take careful notes about things that you feel are important to you as you read uh the the textbook through that section uh or write down your questions as you watch the video or read through the book make a note of your questions and then bring me your questions and then uh review the lecture notes and solve all the problems either in the book or in my ex worksheets without looking at my Solutions first and then once you have your Solutions already completed then watch the video or then uh look at the Solutions in the book and complete all the homework assignments by the due date um and then uh review review review throughout the semester okay this is reviewing the previous sections as well as this section okay so the complex numbers do plot in a in on a plain like that so the this complex number has a real part the real part is a and an imaginary part which is B so how do we know B is the imaginary part well it is associated with I with this I which we will talk about what this I represents but it's associated with the imaginary piece of of this whole number so on a plane the real number is plotted against what we know as the x axis that is the real axis that axis becomes the real Axis or real number axis and the vertical line or the vertical uh or the y axis becomes the imaginary um the the axis for the imaginary part of this complex number which would be B so B would plot along the what is known as the y axis which is now the imaginary axis and a which is the real part of this complex number would plot against the x-axis formerly known as the x-axis and currently known as the real number axis so let me make it clear because this keeps as a matter of fact I have it as a note here uh I have it as a note and I talk about it in class but for for some reason when I ask questions like this on an exam people a lot of people uh still make an error so given this complex number A plus b i given that a is the real part and I'm saying it here a represents the real part and B just B not I the B only represents the imaginary part okay so a is the real part and B just B is the imaginary part okay so if if somebody ask well as a matter of fact I have examples here and before we get to it let's talk about the letter i that you saw up here so what does the letter I represent the I is equal to aare root of1 how is that possible well this has been this is possible because of an agreement between all the mathematicians so it's a definition thing just like the regular dictionary that defines various words because those words we all agree mean those things like chair a table and and so on this in the mathematics world this was defined square root of1 was defined as the letter i as the imaginary as the as what that letter or I represents so if that's the case then I squared if you square both sides you get1 because this sare root of1 squar will cancel the radical expression and we end up with just a NE one okay this is I've already box this but this is very very important in when we deal with this definition is very very important to remember and apply when we deal with complex numbers okay so now having said that answer these questions before you see me write the answers down in other words please pause the video and use a pencil to answer these these two questions and then come back to the video and see what I how I have answered these okay so I'm assuming you've done that and you already have answered these great so the real given this complex number the real part is three the imaginary part is just8 that's what I was mentioning here up here a lot of people write down um let me use this piece of a scratch paper if you don't mind a lot of people write the imaginary part as8 I that is not correct the8 because it's associated with the letter i is the imaginary piece so the imaginary Part is8 not8 I okay so this one again I'm assuming you've already answered this question which is great and now you're here to see how I'm going to answer this well if you break this up into the real right now the real and imaginary pieces are mixed together we need to separate that so by breaking it up in two different piece so - 2 7 - 4 4 7 I right if you break this up into pieces you end up with that so now it's clear what is the real part and what is the imaginary part so the real part is -2 over 7 the imaginary part is -4 over 7 okay all right so number three please pause the video answer these questions on your own given given what we've learned in this part of the page in these few lines uh before we get to that so remember all that and then answer these two questions once you've done that with your pencil come back to the video and see take a look at my solution okay so I'm assuming you've already done that and you're now back with your Solutions ready fantastic okay so this plus is a neutral algebraic sign so I'm simply going to remove it and I'm going to write down -5 + 5 I + 7 - I okay now we can just like if I was an X we can combine the like terms right - 5 + 7 that would be 2 5 i - I would be plus 4 I okay so that's the final answer to that they just say carry out the operation so what is the real part here two what is the imaginary part four okay we know that so again I'm assuming you've already answered this question so the minus I have to be careful is not neutral like the plus algebraic sign is right so I need to distribute this into to that and then combine it with these terms so let me do that -1 + i - 6 minus * minus positive 9 I okay so let's combine the like terms -1 + 9 POS 8 I plus uh well let's see no what did I do no I I have an error here guys let me create break that with my thick pen here that's not the constant so that's the imaginary part so -1 - 6 it is - 7 sorry about that my apologies so that's minus 7 and then I + 9 I is + 10 I so what is the real part of this complex number minus 7 what is the imaginary part 10 okay okay all right so here we have a multiplication going on okay please answer that pause the video answer this and then come back to the video for best I'm saying this for best learning results okay I'm assuming you've done that and now you're back so we're going to foil this just like we would do it with even if we had real numbers so 6 * 1 plus 6 * * I right plus 6 * I so I just distributed the 6 into these two terms then plus - 4 I * 1 plus - 4 I * I okay so let's simplify 6 * 1 6 6 * I 6 i - 4 I * 1 - 4 I plus -4 I * I I got to be careful here because we have -4 here time I * I let's bring this back I * I is I 2 right I 2 is I * I it's equal to -1 so I * I which is I 2 I'm going to do this in pieces so you can follow everything so now that is equal to minus1 right we just know that I is equal to < TK of1 I 2 = -1 so let's take this one step further we have 6 plus uh 6 I and - 4 I add up to 2 I right postive 2 i - 4 * -1 because i^ 2 is -1 = 6 + 2 I -4 * -1 pos4 so what do we have we have 10 + 2 I okay what is the real value of this complex number 10 what is the imaginary part of it two okay okay so the complex conjugate remember that when we had radical expression like this let me bring a piece of scratch paper here so let me use this as a scratch paper remember when we had uh the side is better so remember when we had say say 1 over um < TK 3 u - 2 we would have to rationalize this right how would we rationalize the denominator we would multiply the numerator or we would multiply the denominator by square < TK of 3 + 2 and also we could to rationalize the denominator we couldn't just multiply the denominator and not do the same thing to the numerator right so we have have to say < TK of 3 + 2 also multiplied by the numerator so what we ended up with we would get < TK of 3 + 2 over 3 - 4 and then we'd get squ < TK 3 + 2/ -1 or um minus minus in parentheses squ < TK 3 + 2 but we'd have to this is remember how we used to rationalize the denominator when we had the radical expressions remember that we're going to have to do that here because we do not want a complex number in the denominator of a fraction so we're going to have to multiply this by what is known as the complex conjugate see complex conjugate right there we got in order to rationalize the denominator of a complex number right we right now we have a complex number in the denominator we cannot have that we want to simplify this how we're going to multiply uh the denominator and the numerator just like what we did with the radical expressions by the complex conjugate of the denominator so what is a complex conjugate then okay so that that's what this this segment explains so suppose we have a complex number 7 + 2 I right then the complex conjugate of this complex number is 7 minus 2 I so just change the middle algebraic sign to its opposite if this was 7 - 2 I the complex conjugate would be 7 + 2 I now it is 7 + 2 I so the complex conjugate of this would be 7 - 2 I here suppose we have 3 - 5 I then the complex conjugate of this complex number is 3 plus 5 I okay another example suppose we have 1 - I then the complex conjugate of this complex number is 1 + I okay so that's the complex conjugate and the importance of it is that we're going to use it to rationalize the denominator of a fraction that has complex numbers in it okay we don't want to have just like we didn't want to have radical expressions in the denominator of a fraction we do not want to have a complex number in the denominator of a fraction okay okay so now here one of the exercises we want to uh rationalize the denominator of this complex fraction so that we do not have any complex numbers like that so the complex conjugate of the denominator here is 1 + 3 I right so so complex conjugate of the denominator is equal to 1 + 3 I okay so what do we do with that we use to multiply the numerator and the denominator by that in order to get get rid of the complex number in the denominator so let's do that we have uh we start with 3 - 4 I / 1 - 3 I multiplied by 1 + 3 I / 1 + 3 I so what do we get we get in the numerator we get 3 - 4 I * 1 + 3 I / 1 - 3 I * 1 + 3 I okay so now we're going to foil this like we learned we did with regular numbers right so 3 * 1 three times PL three times or or so plus 3 * 3 I okay plus - 4 I * 1 plus - 4 I * 3 I okay divided by here here we need to use the special product formula remember that we we talked about it early in the semester so remember x - y * x + y what we get is X if if these two terms are identical and the only difference is the sign algebraic sign in between them minus and plus we get x2 x * x^2 - y sare minus these two term multiplied by each other we can only do that if this algebric sign is different we could not do that with plus and plus or minus and minus only if they have opposing algebraic sign in the Middle with identical terms on both sides we can do that so this is equal to that so let's uh put that property apply that here so that'll be 1 squ right these are identical terms and the only thing different is the middle algebraic sign so 1 2 - 3 I 2 Okay so let's simplify I'm going to come here so all the way to this side 3 * 1 is 3 3 * 3 I 9 i - 4 I * 1 - 4 i - 4 I * 3 I -4 * 3 is -12 right - 12 I * I is -1 I * * I is I 2 and I 2 is1 remember that I 2 is -1 I * I is i^ 2 so -12 * -1 so this - * 3 - 4 * 3 is -12 I * I is I sared which is1 okay divided by this is 1 1 2 is 1 minus 3 2ar is 9 I 2 is -1 okay so I distributed the two in into the denominator 1 2 is 1 minus minus is minus 3 2ar is 9 I 2 is -1 okay so let's simplify this a step further so in the numerator what do we have we have three 9 i - 4 I is POS 5 I -2 * -1 is pos2 divided by what do we have here 1 - 9 * -1 is- positive 9 right 1 + 9 right that gives us that so let's simplify again 12 + uh 12 + 3 is 15+ 5 I / 10 right so we've been able to rationalize the denominator the complex number is gone now we have uh 10 can can we simplify this absolutely so let's do that so we have 5 * 3 + I over um 10 right so this this goes away we have one this goes away we have two so we [Music] have 3 + I over two can we simplify this absolutely let's write it down we have three halves plus 12 I and that's how we are able to end up with that expression and that we've done that because it says simplify the result we've simplified the result and write the real and imaginary Parts separately separately that we also did right if you leave it like that the real and imaginary part are are still mixed but when you write it like this then the real part is three halves the imaginary part is one2 and the two pieces are written separately okay so that's the final answer for that okay so I hope for this one if you didn't already for the next one please pause the video and write down your own solution and then come back to the video and see how I work this problem out okay so I'm assuming you've done that and now you're back great so the complex conjugate of the denominator is equal to 2 + I okay 2 plus I is the complex conjugate of the denominator we need that in order to rationalize the denominator of this complex number okay so let's do that so we have 10 + 5 I over 2 - I right multiplied by this fraction multiply by the complex conjugate of the denominator so 2 + i/ 2 2 + I okay then we're going to multiply 10 + 5 I * 2 + I / 2 - I * 2 + I okay so let's foil this 10 * 2 10 * I right plus 5 I * 2 plus 5 I * I so let's foil this using that special product formula which which I talked about earlier x - y * x + y is x^2 - y^ 2 right so we have that scenario here so we have 2^ 2us I 2 okay so we're going to uh simplify it down here I'm going to start on this side so um 10 * 2 20 10 * I + 10 I 5 I * 2 + 10 I plus 5 I * I right that will be plus 5 * I 2 or1 right I * I is I 2 I 2 is equal to1 so that1 you see here is I * i i 2 is -1 okay ID 2^ 2 4 minus I 2 is1 okay so let's simplify this a little more 20 this is - 5 5 * -1 is - 5 + 20 that's 15 10 I + 10 I that's 20 I / 4 - -1 that's 4 + 1 which is 5 okay can we simplify this more yes absolutely and we should because they're asking us to simplify this as much as possible and write the real and imaginary pieces separately or separately so Factor the numerator 5 * 3 + 4 I right if you distribute F excuse me if you factoring is the opposite of the distributive property so if you redistri rute the five into this parenthesis you should get 15 + 4 I plus 20 I so that's how quickly you can check your own work when you're factoring these Expressions that over five and of course the five and the five cancel out and we end up with uh just 3+ 4 I and we we don't have to do anything more because the real part is three the imaginary part is four okay okay so here in all of these please pause the video write down your own solution and then come back to the video and see what I done I hope you'll do that okay I'm assuming you've done that and you're back to see this solution so this one has a negative power which we don't like right so the first thing we should do is rewrite this expression such that it has a positive power or the negative power is gone so take the reciprocal of it you get one 1/ 2 - 3 I and that gets rid of the uh Nega power now the complex conjugate of this now let me bring this down so the complex conjugate of the denominator and is 2 + 3 I so then we're going to multiply the numerator and the denominator of this fraction by 2 + 3 I okay let's do that looks like I don't I have very little space to work with here I don't know if I can fit all the solution here to I hope so I hope I can so 2 + 3 I over 2 + 3 I right equal 2 + 3 I * 1 is itself right 2 + 3 I / by now we once again we in the denominator we will apply this rule if we have identical entities being multiplied with an algebraic sign in the middle being the only the difference between them we end up with x² minus Y 2 okay so here it we end up with 2^ 2 minus 3 I squared right like that okay so in the numerator we have 2 + 3 I in the denominator we have 4 minus 3 2 is 9 9 i s is1 right if you distribute this outside power into three and into I you get 9 and i^ 2 which is minus1 and this minus multiplies Itself by both of these so what do we get we get 2 + 3 I / 4 - 9 * -1 POS 9 = 2 + 3 I y = 13 should we separate this yes they ask us to separate this so 2 over 13 + 3 over 13 I and that's where we stop like that okay okay so this next one is much easier than that and again I hope you've answered these questions and you're just um watching the video to see how I'm answering to compare Your solution with this so that is we can rewrite this as Square < TK of -1 Time Square < TK of 81 okay we can do that we can separate the squ root of 1 from that so if I bring back you know that we defined squ < TK of1 as I and I S as1 right so we know we know root of1 is equal to I so that is I okay so that is equal to I times sare root of 81 is 9 right time 9 or 9 I okay so that's the answer for that part H again let's separate theseare root of 1 from this so we have < TK of -7 is < TK -1 * < TK 7 < TK of -63 can be written as < TK of -1 * < TK of 63 so we have < TK of -1 timesquare < TK of1 right this is I so a square root of1 is I so we have that is I let's so I * < TK 7 time another I now square root of 663 is uh Square < TK of 7 * 99 right 6 square root of 63 can be written Asun of 7 * 9 so we can separate these so this square root of seven can multiply Itself by that okay so we can write this as let's I * I is1 because it's I 2 right I * I is I 2 and i^ 2 is equal to1 so -1 time < TK 7 time let's separate the these two < TK 7 * < TK 9 okay now < TK 7 * < TK of 7 is is equal to 7 so we have- 7 time < TK of 9 is 3 right -1 * < TK 7 * > 7 is just 7 < TK of 9 is three so we end up with 21 like that okay all right find all solutions of the given equation and express the final answer in the form a plus b a plus b i so the assumption is we know that that this this quadratic equation has complex Roots okay so let's find them so first of all do you see any way of factoring this no because 17 no two numbers can be multiplied and give us 17 and add up to get to give us8 right um okay so we're going to have to use the quadratic formula uh so let me write that here a is = to 1 B is = to8 C is equal to 17 okay all right so let's plug it into the U uh quadratic formula minus B is - - 8 plus or minus Square Ro TK of B b^ 2 or - A 8^ 2 um minus 4 * a * C which is 17 all of that over 2 a 2 * 1 okay so let's simplify this what do we get we get x = - -8 is POS 8 plus or minus that's - 8^ 2 is 64- uh 4 * 17 is 68 so - 4 * that uh 68 yes I'll double check I'm pretty sure it's minus 68 but uh why not double check right so right there 4 * 17 16 68 okay so POS 64 minus 68 will give us Min - 4 so plus or minus Square < TK of -4 so divid 2 right okay so that is clearly a complex number so x = 8 plus or minus uh Square < TK of1 1 * < TK of 4 / 2 then X is equal to 8 plus or minus I squ < TK of 4 is 2 right or 2 I divided by 2 okay and then uh let me CU I have enough room here let me come up here and so like that and then so we have simplifying this a little more let's factor out the two in the numerator we end up from with 2 * 4 uh plus or minus I right over two so so this two and this two cancel out we get x = 4 + orus I so you can leave the answer either like that or you can say that the answers are uh x = uh 4 + I and x = 4 - I like that on an exam this would be okay if you left it like that I but uh it depends on what the online homework or how the online work uh online homework um management system accepts your uh response so if they accept it that like that you can enter it like that if not you can separate the two answers and ENT enter them separately okay usually in computer based exams or homework Management Systems they would want you to do it this way so on a piece of paper I'd be okay with that but on in the computer it would you would probably have to do that okay separating the answers okay so now the next one x2+ + 64 that's easier to solve for x right simply move the 64 to the other side that's - 64 then take the square root of both sides so square root of x^2 equals plus or minus right when we take the square root if it's the even root we're calculating we get plus or minus plus or minus of minus of - 64 4 that's clearly a complex number on the left we have X on the right we have plus or minus um s < TK of -1 * sare < TK of 64 and then we have x = uh plus or minus I uh time 8 right s root of 64 which is 8 so let's write this in proper format proper form X = plus and minus 8 I like that okay okay so here we have another quadratic equation uh can we Factor this uh I don't see it how because what two numbers would multiply to 10 and add up to positive 10 and add up to six nothing right 10 and one they multiply to 10 but do they add up in any way to 6 10 - 1 is 9 10 + 1 is 11 so there's no way 10 and one can add up to six how about two and five well two and five add up to 10 multiply to 10 but can we add or subtract those two numbers in any way and get six no okay so it's not possible to factor this so in that case we have to use the quadratic formula a is equal to 1 B is equal to 6 and C is equal to 10 Okay so using the quadratic formula we have minus b or - 6 plus or minus Square < TK of b^ 2 or 6^ 2 - 4 * a * C which is 10 all of that over 2 a okay so if you haven't done this already please pause the video complete this solution and then come back to the video okay I'm assuming you've done that which is fantastic xal - 6 plus or minus we have 36 - 40 over 2 2 A by the way is 2 * 1 right so let me correct that that was 2 * 1 that's one a was one so which is two so simplifying this a little more we get - 6 plus or minus this is again 36 - 40 is -4 < TK of -4 / 2 right so then we have uh like that so so that is we can rewrite that as plus or minus < TK of1 time < TK of 4 we know < TK of1 is equal to I right we learned that on that first page right there root of1 is equal to I and i^2 = -1 over two so let's simplify this a little more we get - 6 plus or minus < TK of 4 is 2 and root of netive 1 is I so plus or minus 2 I over 2 okay so now let me put the rest of this here so what do we get we can factor out the two from the numerator we get x equal uh 2 * or let's factor out well it makes no difference factor out two and then we get min -3 plus or minus I / by 2 right then the the twos cancel out we get x = - 3 + or - I so you can either keep the answer like that on a piece of paper or you can write it separately on a computer as x = -3 + [Music] I and x = -3 - I like that okay all right so so then one last note before we close this uh before we wrap up the lecture on this section and we kind of did this let me bring back we did this this is an identity I want to speak about uh but we already applied it when we were working on when we were working through this part on page three of this worksheet part H < TK of -7 * < TK of -63 see I we couldn't say negative * negative positive when you have a complex number and this is clearly complex number because you're taking the even root of a negative number so you have a complex number you cannot multiply the negative numbers in in the roots and say positive 7 7 * 63 and then that is equal to posi 21 you would end up with a different answer right if you did that we cannot do do that and that's what this property is saying it's saying that square root of a or some complex number time square root of some negative number which is complex is not equal to is the left side is not equal to if you were multiplying two negative numbers under the same radical expression it's very important to remember okay if you had them separate it's different than if you had them under the same radical expression because so here we know that the left side is a complex number each one is a complex number which would reduce to that then the left side the right side would be po you because both negatives are under the same radical expression you could multiply them and get a positive value You' get square root of ab on the left you have I's from the complex number so soqu root of I * I is -11 * < TK of a so the result on the left and the right are different so just remember when you have complex numbers separately you cannot in a radical expression you cannot multiply the negatives and under the different radical expressions and say I have a positive number no only if the negative numbers are under the same under the same radical expression which happens a lot actually using the quadratic formula it happens quite a lot when you end up with a negative number and then you add a larger positive number to it and you eventually you end up with a positive number not a complex number so if the negative numbers are under the same radical expression you can multiply them and say they're positive but not if the negatives are under different radical expressions okay all right with that thank you for persisting through this video uh for now have a nice day and I'll see you again next time