okay so can you do this problem and if you cannot then you're going to have a difficult time passing algebra 2. but no need to panic we don't need to get into this state and be like oh my goodness i'm going to fail algebra 2 no that doesn't have to be the case at all okay but we do need to uh fix this real quick if you're at the algebra 2 level college algebra level anything beyond algebra 1 then you should you know definitely be able to at a minimum identify what's going on here and obviously even better be able to actually do this problem okay so what is the topic that we're going to be talking about here okay do well let me just ask you that do you even know what we're talking about well hopefully most of you out there is saying i think it has to do with this imaginary little little i thing and you would be on the right track the topic here is really complex numbers okay complex numbers and complex numbers um is a pretty big topic in and of itself and we're trying to do something here we're trying to multiply two complex numbers that's really the problem and this is something that you probably were introduced to at um in algebra one at least the concept of complex numbers uh but in algebra two you definitely need to know how to work with complex numbers and this is not that difficult so if you are you know upset that you didn't know what this is no need to panic we'll quickly review this and by the time we finish with this video you will know exactly how to do this problem okay but before we get going let me quickly introduce myself my name is john i'm the founder of tablet class math i'm also a middle and high school math teacher and over many years i've constructed what i like to believe is one of the best online math help programs there is of course i'll let you be the judge of that if you're interested you can find a link to my math help program in the description of this video but if you need to take a full online math course i can help you i have several courses or if you need assistance in a course that you're taking i can help you okay i have full lessons and i teach you how to solve the most uh common problems you're going to encounter as a middle school high school and college basic entry level college level mathematics so i literally solved thousands of problems so if you're interested you can follow uh find my math up program follow the link in the description of this video now if you are a student uh one thing i must stress to you is importance of note-taking okay my golden rule of math is i've just seen it through decades of teaching math is those students who take the best math notes almost always have the best math grades and the reverse is true okay some of you unfortunately might say i don't need to take math notes it's not that important what's more important is for me to check out my cell phone and talk to my best buddy in my class and of course that's the way i was and there was no cell phones back in my day but if there was i'm sure i would be on it not paying attention to the teacher taking good math notes okay you got to take great math notes it's an absolute 100 percent essential um skill that you must master okay if you don't uh take note-taking seriously you will pay the price unfortunately okay so take a look at your notes if you haven't been um you know doing the right things then start improving there's always a room for improvement but if you start improving your math notes your your math grade everything will start to follow but in the meantime you need something to study from so i actually offer excellent detailed uh comprehensive math notes those would include pre-algebra algebra 1 geometry algebra 2 and trigonometry so if you want to check those out you can find them by following the links in the description of this video all right so complex numbers that's a topic and um let's just do a quick uh rehash of why complex numbers are important this is going to be a very very brief um rehash because this is a big topic okay so in algebra one let's say and uh previous okay to all your elementary schools and we'll put algebra 2 over here because that's kind of what we're talking about we were dealing with what kind of numbers well primarily you're dealing with the number line right we have like 1 here 2 here negative 1. negative 2 we have like pi square root of 2 negative 3 two-thirds what kind of numbers are these yes you guessed it this is the real number system okay real number system and that's what we've been using and we still use it obviously in all mathematics but uh exclu exclusively so in algebra one and uh prior to that you've been using these type of numbers the real number system but somewhere along the line you encounter the need to to learn another number system so let's just quickly see where that kind of happened so let's take these two equations here x squared equals 16 and x squared equals negative 16. so this equation x squared equals 16 i'm saying what number times itself is positive 16. so the way we solve that we just take the square root of both sides and we get x is equal to plus and minus 4 because a positive 4 times a positive 4 is positive 16 and negative 4 times negative 4 is also a positive 16. so what number times itself is positive sixteen well uh positive four and negative four so this is a basic quadratic equation and there's always two solutions to our basic uh two all quadratic equations and here they are positive four and a negative four now let's take a look at this quadratic equation x squared equals negative 16. so if i gave this question i would say 50 of the students would be like oh x is equal to negative 4. all right then i would have to give you a big smiley face take a bunch of points off and then be you know crying and whatnot i'm like no no no no don't need to stress what we need to do is learn say a different number system here because look if you're saying x is equal to negative 4 what you're telling me is negative 4 times negative 4 is negative 16. no that's positive 16. so what number times itself is going to get me back to a negative 16. well we're like yeah that's a good question so let's go ahead and take the square root of both sides okay like we did over here and go into your calculator and take the square root of negative 16. okay your calculator might start to smoke or shake or say something to you it might come out with a little error message or something like i don't understand because your calculator primarily is operating with real numbers now if you have a more sophisticated uh scientific calculator you know you could plug this in and it will give you the right answer but for the most part any basic calculator you know even like your cell phone is it's not going to understand it it's going to like i don't understand this question i don't understand how to get the answer that's because you know we're they're it's thinking in terms of only the real numbers we need another number system we cannot uh solve this equation x squared equals negative 16 with these numbers over here we need more numbers okay we have to go get ourselves another number system and that would be the complex number system all right so it's a very very um you know important step up in your understanding of mathematics because uh the complex number system is over here the real number system actually lives inside the complex number system you've got to understand this now if you're saying well this is a lot of information i don't remember all this i just want to know how to do the problems if you're thinking of those terms like just show me how to do the problems guess what you're not going to be able to really master and comprehend this material and then math is going to be a struggle because you're not just going to be able to memorize and and just do problems you have to understand okay and put all the pieces of the puzzle together okay now uh complex numbers all right it's a big topic i'm not going to go into uh all of complex numbers but with complex numbers they're in the form of a plus b i there's an imaginary component and a real number component now with any number all right you can do various operations with complex numbers we can add them we can subtract them we can multiply them and there's kind of a basic concept of division and there's these things called complex conjugates etc we can graph these things but primarily uh when we're solving more sophisticated equations now beyond algebra one okay into algebra two and uh college algebra pre-calculus et cetera you know we need uh to really understand complex numbers okay so that's uh you know my uh sales pitch to you why complex numbers are important so what are we trying to do right here well we're trying to multiply two complex numbers you got to know you have to know how to do do this um do these type of basic problems now if you look at this this looks very much like doing a problem like oh x plus three times x plus five if you you know are thinking this this looks like an algebra binomial problem like this times this that's exactly right that's good okay this is exactly how we want to start this problem we just use that foil method okay first outer inner last okay then we're going to go ahead and combine like terms uh more or less so let's go ahead and do this problem all right so again we're going to treat this just like a binomial so we're going to go 3 times 2 that is 6. and we'll go 3 times this negative 4i that is going to be negative 12 i will put that right there and then 5i times 2 that is a 10 i and then we get 5i times this negative 4i all right so let's just be careful here 5i times a negative 4i so what's that going to be well we're going to multiply the numbers 5 and 4. so we're going to get 20 and then i times an i is an i squared and we got this negative sign so that's negative 20 i squared right so let's just make sure we understand this 3 times 2 is 6. 3 times this negative 4i negative 12i 5i times positive 2 10i and then 5i times a negative 4i is negative 20i squared okay so now we have to combine like terms and here we have a number six it's by itself so we'll just keep this here and um here these guys are exactly the same they both have an i so a negative 12i plus a 10i would be a negative 2i okay so that's good so now we just have to figure out what to do with this thing negative 20 uh i squared all right so uh the first thing we need to understand uh to be able to simplify this is what is uh the definition of i okay so i equals the square root of negative one all right again i don't want to turn this into a complete lesson on complex numbers but hopefully you understand this right so i um this imaginary part is uh equal to square root of negative 1 so i squared would be what well we just square both sides and this square root goes away right so in other words here if i is equal to square root of negative one if i square this and i square this i get i squared is equal to negative one okay so i squared is equal to negative 1. so this right here is negative 1. okay so now i can simplify this because this is really negative 20 times negative 1. so negative 20 times negative 1 is a positive 20. all right so this is plus positive 20. so that's what this really is right here okay plus positive 20. now i could combine like terms i have 6 and 20 and that's going to be 26 minus 2 i okay remember we want to write complex numbers in terms of the form a plus b i our real number first so that's 26 and then our imaginary part in this case that's 2i over here and that is the answer okay we have something called closure um in mathematics meaning that if we are multiplying two numbers a complex number times a complex number the answer itself will be a complex number okay and so we want to write that uh you know as such right so 26 minus 2i and that is the answer now if you were able to do this problem i would say that that is pretty awesome i would definitely give you an a plus and 1 000 percent so i know that makes your day but again you know you don't want to get overly confident because this is a fairly easy problem all right there's a lot to know about complex numbers uh but this is a good start but the most important thing is to know why we need them okay i mean that's the whole essence of mathematics it's not able to do these um um you know processes you know this is like following a recipe it's like oh yeah i can do this problem but what does this mean or why do we even have to do this well that's the real essence again of understanding math and if you um are taking notes paying attention to your teacher then you know you'll be like oh yes i know exactly why we're doing this and it's important because this stuff will you know um come up again uh in the future especially if we continue to take more math and i hope that you do continue to take more math beyond algebra 2 because math is so cool trust me you'll love uh taking more advanced math especially calculus and beyond okay so uh if you enjoyed this video some way found it helpful okay please consider smashing that like button that would be helpful to me and if you're new to my youtube channel please consider subscribing i've been on youtube for a good 10 years at the time of this video and i have hundreds and hundreds of videos um on my channel there for you from basic to advanced obviously i love teaching math but if you want my best uh work uh follow those links in the description of this video but with that being said i definitely wish you all the best in your mathematics adventures thank you for your time and have a great day