Who cares about complex numbers??

why why but most people just think about this the vast majority of people go through their lives even all of their compulsory mathematics education and even the non-compulsory mouth management education and they never find out that there's such a thing as complex numbers and you can do plenty of mattes in there like all the way up into extension one that's not easy stuff and they don't even know that these thing exists right so why do we care about them and why are we going through invest so much and them because trust me it's gonna get hard ok that's why it's for you well I'm gonna try and answer that question for you first with a simple example my favorite you do consider so this faithful old quadratus too many times before okay and being that it's equal to zero this has a solution actually it's a pair solution okay and by now I heard you know what that pair of solutions is however what I'd like to point out to you and I'd like us all to rehearse it together it is normally you would solve this by factoring because it's easy to factorize right but we go through factorizing then we go through completing the square and we use completing the square to arrive at the quadratic formula okay the quadratic form is great because you know you can use it on anything even when it's hard to factorize so I want to take this and I want to just use that skill that we know about the quadratic form I just want apply it to here we know what the conclusion movie but let us see what happens as we go through so fever mean we know the quadratic form is what numbers and I'm going to put in minus 5 would just be plus or minus square root of 5 squared which is 25 minus 24 AC which is 4 times 1 times 6 which is 24 all over in this case just two now what I want to point out to you before we go any further is that according to the quadratic formula like the quadratic formula has baked into it right it has certs like it's part of the woman you have to say minus B plus or minus the square root all you don't say the square root of then you won't get the quadratic form in and won't get the answers now just keep that in the back of your mind that's on let's continue 25 minus 24 of course is why are you squared 1 is 1 okay yes thank you sir I've got that in there their problems the reason why I need to worry about the case anyway right uh and from here we're gonna get well the negative case will be minus 5 minus 1 which is minus 6 so you get negative 3 and the positive case will be minus 5 plus 1 which is negative 4 so we get this solution now this is weird this is weird the foreigner itself has a cert in it right but then the Sun disappears sorry I said that again okay now what you're talking about is getting a multiplicity of answers like how many S's do I get but what I'm interested in is what kind of answers am I getting and the answer is I seem to start off with a sir the formula tells me to expect a sir but then the sir just kind of doesn't become a certain just disappears now this is strange right I have an answer that's completely rational right whole numbers even in this case but to get to these rational numbers I have to appeal to irrational numbers but and they just seem to vanish away now here's the important thing and this is what starts to answer this question for us okay this is a quadratic that if you do know how to quadratic but if you have a cubic formula right if I'm a cubic there is a cubic formula that exists right it's kind of you know it's analogous to this kind of thing except it's own Boston plus it's much longer like takes a whole page to write the whole thing across but more importantly the cubic formula it includes it's got baked into it negative values underneath the square root signs that appear in it right there are square root symbols expressions in the cubic formula just like they are in the quadratic formula but by necessity the cubic formula has negative values under there they have to be negative okay now this is weird though because if I give you a cubit right just considering he's a cubic okay now this cubic has completely real roots right what are the roots of this thing can you help me with them out how would I commence I can factorize right excuse me this yes and then I can factorize one more time to agree with that which gives me X minus one so this is a cubic you expect to tap three roots and there they are and they're all real roots but in order to get to those leaves your roots if you want to use the formula negative values non-real numbers appear in the formula even than what your area is completely real does that make sense in the same way that irrational numbers appear in here even if what you get out the other end is completely rational now this is a very strange idea and it gets at why numbers have the names that they do I want you to recall what were the families are numbers that are introduced to you before you remember what their go we'll start with the natural numbers and what was next after that we talk about the integers so we in the claim you think it is right what happened after that we talked about rational numbers right there we went to real numbers and then we've landed where we are which is the complex world okay now the important thing which should have noticed right is that we know what rational numbers are and we know what real numbers are but outside of these right we have numbers in here that seems be not rational and seem to be not real and they've got these strange names we call these irrational numbers and imaginary okay if they are not in those sets okay just ponder this for a second right if you are not a math student and you sort of work irrational what would that mean to you it would be like stop being irrational it means stop being crazy stop like not making sense that's what rationality or irrationality is and the whole point is that the reason why that word means that is because in here it's like this number doesn't really make sense it's just stupid it just disappears anyway but we kind of have to put it in kind of as a hack right it just makes the formula work okay it's not really there it'll disappear but you've got to put in at some point okay by the way this is Seidner a book is alone with me because um it's a really cool example you guys name the Fibonacci numbers right the Fibonacci sequence we have a definition a recursive definition for it right it needs F 1 and F 2 and then we can say F n can you help me up what's that one one that's the first term F 2 also what okay once you know that you can write F n what's the end Fibonacci number it's the previous one plus the one before that okay now we've been working with series and sequences long enough to know if these are recursive definition there should be just like a formula definition as well right we don't need to worry about it because outside the scope of the course but it exists it's this disastrous mess it's possible good question the question for why it's Route five has something to do with something important called the border ratio but I'm not gonna dig into it okay now here's the here's the really really we think as you can see just like with the quadratic formula you can kind of understand why surd's are in the quadratic formula because you know what we know some quadratics they do have irrational roots we know that right but there's not a single Fibonacci number that has any irrational part to it right but you have to have irrational you've gotta have the square root of five in order to make the formula work for any value of n any I should say any integer value of n actually any natural value in all of these square roots of 5 they all come out in the wash every single one they all disappear and that's why you get this sequence right no square roots of anything to be seen but in order to make the formula work we have to appeal to numbers that seem to be there like a hack and then they disappear now that's why these numbers here when you get negatives under me that's why they called imaginary in fact the people who really pioneer in this field they call them not imaginary numbers but they called them fictitious numbers they're numbers that another feel they're not actually there but you kind of need to stick it into your formula so you formula works question who is interested in such a kid who is interested in coming up with all this that made numbers that didn't exist made them appear and the answer is and this is so interesting you've got to go and look it up yourself okay the short answer is a whole bunch of Italian mathematician now the one you want to particularly look for if you once you go and search this top because it's a hilarious story as you discover not 10 seconds then what you want to look for is karma ca RDA and oh very important Italian mathematician around the 1,500 1,600 1,500 sixteenth century right now the reason this is such a big deal to them is that in the Italian mathematical world if you had like an office as the associate professor of mathematics was aware that the way you applied for a job which you didn't put in a CV and then you had an interview process if you thought you were good enough as a mathematician to take that job you challenged the person who had the job to a duel right now not just like any job like the French people they were all about bang bang tools right but these are Italian mathematicians they had mathematical duels now like I think I'm a better mathematician than you and this is where they're doing maybe I look I'm gonna come with ten questions right and I can answer them I know the answers to them I'll give them to you you give me your ten questions I reckon I can answer your questions faster and better than you can answer mine well as in the point is I will give you questions that I know it can be solved cuz I know their solutions right so you can see in order to win such a duel there's literally like this mathematical arms race right now lots of people believe for a long time that quadratics could be solved but cubics could not right so it's like this holy grail of like if I'm in television right and I can come up with a cubic formula it's like it's like the secret weapon that can shoot down anyone because I'll give you a cubic and you won't be able to solve it but I can write so they were intensely interested in coming up with different kinds of cubics that can be solved here for instance monic quadratics it easier so normally ones are harder to solve and it's exactly the same with cubics right so that's why they pushed really hard into this area and they forced themselves to deal with these numbers because that's how they could get someone else's job right and there were very very prestigious positions so you should look up Connor and his duels and all the people he fought with and on the secrets that were around like you know he was one of them who did one of the tricky things what's really hilarious is that you know how we talked about all Pythagoras's theorem and like theorems that a name not two people right one of the reasons why a lot of the results that we have are named dr. people is because we don't know who came up with them because they founders all and instead of publishing kept it secret because that's your secret weapon why would you publish yeah exactly so people would discover it they would use it there with all these jewels and then when they died they would take this secret with them to the grave and then someone else someone else decades later would rediscover it me like oh this is cool and then you know it was a lot of detective attack you do so well it just says something about you okay so so so so so right can we just come back this is the question I was trying to add someone right why did people go to why do they want to go to like this field of numbers were just so hard to work with they are hard to work with as you just go back um the answer is they didn't want to they didn't want to go there but in order to develop the weapons they needed they were forced to go there because imaginary numbers they arise naturally very quickly out of the regular old counting numbers and the ways you can combine them algebraically right you've just got to deal with them and so they're there and so mathematicians with it

Channel: Eddie Woo

Views: 958,384

Rating: 4.9306254 out of 5

Keywords: math, maths, mathematics

Id: 19c4c3SwtS8

Length: 13min 52sec (832 seconds)

Published: Wed Nov 04 2015