 Hi everyone, this is Chicho. It looks like we're live and today is May 11th, Saturday 2019 and this is another Just drop in math tutoring session day of math. I Originally started calling these things. Let's do some math and basically I'm just here for a couple hours To help people out if they need any type of health and help in their high school math curriculum dealing with some of the Problems that they may have some of the concepts. They're having a hard time to understand basically Just a sort of a lab. I guess just drop in math during Session, okay for a couple hours And we'll most likely try to do this Every week for the next couple of months until the end of the school year for next six weeks or so anyway Because exams are coming up. We might even kick it up if they need us there If people start rolling in if they do need help reviewing a certain grade. We can just start doing Focusing on certain grades. I'm just gonna move the mic a little bit closer. Hopefully it's not Creating too much noise That way the sound volume is Is high enough do a little adjustment on this as well Again, hopefully the sound is not going to overwhelm you guys Just a little adjustments we've got our bucket of dry raised pens and Our little rag and we're gonna wipe the whiteboard with CD, how are you doing? Hey Chicho? How you doing? Doing good man. Doing fantastic. Nice chill day today We did we got the politics stream out of the way So whatever we need to talk about the politics is gone. It's fantastic. Hannah, how are you doing? Hey Chicho, I'm excited for math. Me too. Me too. And by the way on this stream We're gonna keep it strictly math now that I'm sort of basically decided to set up a Current events live stream, you know at least two or three times a month We're gonna just funnel all of that political talk into those streams and all the other streams Will be politics and economics while politics free anyway, okay? Mask, oh, how are you doing? Or mask of Raven five. Let me read that properly mask of Raven five. Hello. Hello. Hello Hello, Luca. How are you doing Chicho? How's it going doing some? economic and Econometrics myself econometrics. What are you looking at? What data are you looking at spider? Hey Chicho, hope you're having an amazing day. I am I am I woke up by the way I was just woke up. I wake up early and I check my News stream comments and all this jazz right all my analytics data with data junkie So I look at all my data right in my in my mind I see the you know, that's why math once it becomes a part of your life. You can't help but incorporate it Into your life right appreciate what the data tells you right, but I was all doing all that stuff all of a sudden on my feed Okay, we'll do two minutes of politics. I'll just tell you what I was doing this morning on my feed news feed all of a sudden I saw Twitch Chelsea fundraiser for Chelsea Manning twitch live. I was like what I've been following the Assange live streams on YouTube and other stream platforms that have been going on every Friday for last I don't know why all right and I saw this one. It was number two. I'm like what so I went on to twitch and there was a live stream of A collective right with Chelsea Manning in their live streaming on to us. I've been watching that for last I don't know. They were having technical difficulties at the beginning. I was watching that for a little while and really Gave me a boost right some people say You know talk about motivation because I tend to I go through phases. I create a lot and I don't create a lot Or whatnot, right, but I'm in general I've been this way for a while now. I'm motivated to do things, right? But people have been asking me what are some of my motivations Well, some of my motivations and I mentioned, you know, some of the ones are yesterday someone asked and I sort of said basically it's You know our lives are over in a blink of an eye So just imagine what you would do if you only had a blink of an eye left to live, right? You know, just imagine all the things you would do and all the things you wouldn't do, right? And at the same time one second may seem like a lifetime, right? So that's one of the motivations. I always keep in mind another motivation is, you know, someone asked previously is, you know, I get angry. I'm Not angry in a violent type of way, but Things piss me off. So when things piss me off, I tend to try to Compensate that with addressing the issues. Why am I? Angry or pissed off. That motivates me in a weird way, right? For example, if I'm studying in a course or something like this, if I don't understand the subject or a topic I'm like, I get a little pissed because hey, I'm not getting this So I do what I can to try to learn it, right? So that's a motivation and not a motivation is finding people that have That stand on their principles and take on power Right and manning is one of them. So The live stream this morning was pretty cool. Actually very cool on twitch I'm hooked on twitch man. Seriously fantastic Hey chicho, I hope you're having an amazing. I am I suck at not so hopefully I can learn something today Okay, okay cedar. I'll try and one thing I've been doing Properties Properties of triangle santa you want to know about properties or triangles for sure. Let's address that first looking at house prices. Whoo looking at Personally, I told people this a while ago. I wouldn't be If I had the money to spare or the inclination to take on that much amount of that wow or Stream gotta focus it if I had the inclination to take on that much amount of debt Um, I would not be investing investing in a house I think it's a bubble Hey, chicho, not right now. Anyway, but it might be inflated more. They're about to open up the money supply again slut Hey, chicho, what a day What a day I have a day jingo bells. Nicholas. How are you doing? Hey chicho? Hey chat Can't stay long moving on to scotland. Nice tonight. Just want to pop in and say how you doing necklace Hope you have a safe trip to scotland. I've never been to scotland. I'd like to go I've had scottish friends I had a scottish partner once those scots Fun time Do you get what I call the math rage happened to me a lot? Is that called math rage? Especially when the textbook leaves out proofs or long lines of algeron I can't seem to figure it out 100 luka. I've gone through texts and just Sitting in classrooms where they jump from one concept to another concept without making that link All of a sudden you're missing The core concept that connects one idea that you understood to another New idea that you need to understand if that link is gone then Yeah, I get math rage. Is that what we're calling it now math rage Hey chicho, how goes it my exam went well this morning? Awesome. Nice nice funky. You were studying for it yesterday You stayed up late. I think For the stream x how are you doing? After the triangles would it be possible to explain proofs a little further? Interested in those for sure city wrong. Let's do so we're gonna do triangles and proofs I don't know if you know what let me write this down Triangles and proofs Maybe that's what we'll start doing on these live streams Just put down what we want to cover and we'll cover it and we tick them off Right, it's just like how to study video that we put out make it to do list What do we need to do this this stream and if a lot of things start popping up I'll start announcing more math streams that we can do Okay Hello guys luka that happened to me. I think it's a A sign of poor textbook writing much at the same time. Yeah, and Richard Feynman In part of a book that he Richard Feynman was a physicist american physicist is fantastic follow his work if There's a twitter account. He died a long time ago, but there's a twitter account that shares some of those thoughts and stuff and There's a lot of lectures online and he's written a fair bit. I highly recommend them but Richard Feynman Um Basically wrote a little excerpt Regarding how textbooks are chosen at schools and stuff like this and it's basically lobbying method and whatnot. That's why they're so horrendous Okay If you if you want to know if you want if you want that little piece Put a message on discord in the general just comma. Hey teacher words that Think I'll I'll dig it up and I'll post the link to it if you want to read it Maths at university is the hardest thing I've ever done. I reckon uh luka The reason for that is is because high school education sucks, right? They're not they're not preparing People to deal with university math. I agree with you in the first wave you get hit of university mathematics It's hard like I failed my first year of calculus that I took at university I thought I could cram the last day or two to learn all of calculus I couldn't do it, right? So I wasn't mentally emotionally into intellectually maturely Prepared enough to deal with what was coming, right? And then I taught myself. I didn't really have any good teachers I took it again. I did okay One thing I've never understood are the quadratic equations I remember being taught the formula and all that but I could never comprehend What quadratics are used for or how they were created? Okay, fun. Okay. Let's put this down quadratics Those are quadratics parabolas basically your introduction to quadratics, right? I've created a lot of videos on this Um, we'll touch on it quickly triangles proofs quadratics A lot of the battle A lot of the battle with university math for me is finding a resource That's well made and doesn't assume prior knowledge. I don't have yeah, the the thing with mathematics One of the issues with mathematics is it builds on previous concepts And one of the reasons that people have a hard time at university when they get introduced to mathematics is University mathematics college mathematics post secondary Math education relies on information that you need to know in high school, right and grade 10 11 and 12 Those grades the mathematics that's taught in those grades is dependent or every grade is dependent on the grade previously, right? But what I found where a lot of people are having a hard time The reason math is not sticking Is because they don't know how to deal with fractions. They don't know how to move around an equal sign they don't know how to Break things down to prime factors, maybe functions, maybe just numbers or whatnot, right? So the main problem The main issue of high math literacy in the western world counting the United States anyway Out of europe, I think less or maybe not less. So I've had some students from europe as well that There's serious issues there as well. I was amazed that math level had deteriorated so much in europe as well Mirroring what's happened in the united states, right? But one of the main issues that people have You know, there's Math literacy is horrendous in canada united states is because math is not taught properly in elementary school and high school, right? People don't know how to add fractions how to deal with fractions or what fractions represent as soon as you don't know that You're way behind like to understand all the other concepts. You have to understand fractions and ratios, right? Do you deal with triangles? Here's triangles I'm in my third year of economics and mathematics at university and I still have high school level math gaps in my knowledge Yeah, everybody does luca. I did I got my degree in geophysics minor in mathematics worked as a geophysicist for a decade And when I started teaching mathematics, I realized I had huge gaps in my math knowledge I was amazed it blew me away. It motivated me to do what i'm doing now. It pissed me off, right? Triangles, okay The first thing you have to appreciate with triangles is we're talking about triangles that are on a flat plane We're not talking about triangles that are on spherical objects, right? So Map view of triangles is called euclidean geometry, right two dimensional, right? If you think about it One direction is considered to be One dimension A second direction is considered to be two dimensional, right? So Then when you create something like this is an area Right when you close it off this times this and we're talking about multiplication, right? There's different formulas for different shapes like this is map view as well, right if you take the um, this is r Pi r squared is the area of a circle, right gives the area that's map view Area of a rectangle or a square is the length times width, right this times this Right, so two dimensions multiply together gives you a map view Okay, there's different formulas for it, but the units of it would be basically Meter squared feet squared inches squared, right? That's what we're talking about We're talking about euclidean geometry to a certain degree, right on a flat surface You could have these on curves as well, but we're not talking about that, right? So when it comes to triangles what we have is this is Three lines that enclose an area, right? So this Would be a triangle. So it's something it's a polygon that has three angles try angle, right? Try Angle Triangle, I'm gonna show you from Triangle, right? Three angles, okay Cheers guys, enjoy the rest of the uh, see you later, Luca. Thanks for popping by Thanks for popping by got to go steady guys Cheers guys, enjoy geometry is my favorite geometry is fantastic Geometry in my opinion is the best way to make people start to get a concept. Yeah, agree 100% right? So what we got here is three lines closing An area it's a closed system, right? And these three lines where they're connected they create three angles this angle this angle Right So here are some properties that show up Based on three lines That encompass An area that has three angles in it, right? I feel like people either are more suited for algebra and logic type stuff or geometry and more creative Maybe that's the way the education system Gears it right now, but I think uh, cd I think people can go Can appreciate both Okay Can appreciate both but our current education system. Yeah, it teaches it in the way where it divides people is really weird I say to my students somewhere good better at geometry than just a straight up algebra Some are better algebra and geometry, but they're they're together You can approach of them approach them from the same problem from both sides, right? So there's some properties with this with this shape, right? One of the first properties is this All three angles added up equals 180 degrees Straight up right any triangle created on a flat surface All three angles add up to 150 degrees, right? So it doesn't make a difference. What type of triangle here's another one, right? Here's one. Here's one. Obviously, they're supposed to be straight lines, right? She showed you teach probably distribution or need or need a story. No calculus I can't have a hard time pronouncing stochastic calculus. I've I've studied it, but I'm not versed enough To teach it right now dmt by the way. I dmt. Okay Probably distribution we've talked a little bit. I will I've taken probability for sure multiple classes and I've used it doing geophysics But I need to refamiliarize myself with some of the models and some of the terminology to be able to Connect up the links. I don't want to create a whole series and have gaps in them But we can touch on certain concepts, but I'm staying away right now for from probability and calculus Just because there's so much more to talk about and I want to make sure I'm able to do it properly appropriately when the questions come up Okay, so all of these angles add up to 180 degrees this plus this plus this is 180 degrees All right, this plus this plus this is 180 degrees This plus this plus this is 180 degrees All right Now one thing we have to appreciate as well is if you draw a straight line, right? This angle is 180 degrees As well So if you take out all of these angles and break them up Put them back to back you create a straight line. It's pretty cool when you think about it, right? Why was 360 degrees chosen as the number representing a full circle? It is arbitrary or does it represent something? physical for example, why isn't the full rotation 500 degrees and a semi-circle 250 degrees first of all a better way to think of 360 degrees is 2 pi if you're talking about mathematics, right? Because 2 pi has an actual meaning behind it Okay, the 360 degrees I looked us up. This is like his math history. I'm not well versed in my math history There is a reason for it. I have to think about it. Hopefully it'll pop up We could look it up later. Okay, but a better reason better way to think about it is this. Okay, if you want to think about rotation, okay Angles if you take a full circle, right? The angle from here to here is 360 degrees. That's what we say do a 180 means turn around fly back, right? However, 360 degrees is also equal to 2 pi radians That is much more powerful to use in mathematics than this, right? 2 pi radians is a different way of measuring an angle, right? So when I say A line has 180 degrees If you add up the angle from here to there is 180 degrees This is also equal to pi radians, right? So I could say easily say that an angle on a line is pi radians I could also say that all the angles in a triangle add up to pi Which is equivalent to say 180 degrees, right? I mean if you think about it if you divide these by two 180 is equal to pi, right? What does that mean? That means and this is true for all circles the ratio right It means that if you have a radius Circle with any radius Okay The distance you travel when you travel half a circle this much Is equivalent to pi The radius pi r, okay Do you follow? So if this was five then one two Three and a bit there is one five here Two fives here three fives here and zero point one four dot dot dot pies times Five here, right? That's a ratio that exists for all circles doesn't make a difference if they're this big or huge right Now how did 360 come about? I have to think about it man. I don't want to go into the guessing game or try to remember something Right, so it's highly divisible. Yeah, I don't think it has to do with the divisibility racer kill Maybe it does To so racer kills saying the reason being sorry we deviated from this. We'll come back to this It is related to angles and we're there basically racer killer is saying 360 is two cube times three Three squared times five three squared if you break it down as prime factors, right? These are the first three prime numbers. So it's highly divisible right But it's still a very arbitrary. Thanks chichon racer kill. I see it now. Yeah, I followed chichon cool 13th on the list of highly Composite numbers is it cool Dante, how are you doing? Sorry? You didn't need to derail it. No, that's okay. That's the that's the beauty of mathematics It was it was related to angles and we're talking about angles, right the beauty of mathematics is We can it can branch off in any direction and that's one place people get lost a little bit They don't appreciate that whatever they're working on is also applied somewhere else Just give it a little twist, right? And when the twist happens if people don't see that twist they get lost in the concept It's like moving on from pre-kill to kelp right It's better than Yeah, it's better than 359. So let me kill these guys, right for now So one of the properties of a triangle is this all the angles We'll call this a b c the angle a angle a plus angle b plus Angle c is equal to 180 degrees For any triangle Yeah, pre-circular exactly what racer killers say 360 degrees is divisible by two three and five So this plus this plus this is 180 degrees. Here's another property extremely important property of triangles that people tend to forget, okay An angle controls the opposite side So this angle let me draw this in a different color pen Nicholas, right folks off off I go. Enjoy the stream folks and chicho I'll let you know what the comic book stores are like a skull off for sure. Let me know let me know Someone just gave me a friend of mine just gave me a spanish comic book. He was taking off and uh, his swiss and he just he had it And uh, you know, he knew he knows I collect comic books. So he gave me a Spanish comic book a super cool super card flip through it. I can't read it But the artwork was like old school like golden age to a certain degree. Hope you enjoy scotland necklace Don't drink too much And don't eat too much of the scottish food You're gonna actually haggis. Maybe try some haggis authentic haggis, right? So check this out angle a controls this side the opposite side Angle b controls this side angle c controls this side So by convention what we end up doing is saying if this is angle a and when we put a here We're talking about this angle, right? We say angle a is here and if it controls that side We call this little a right. That's just by convention. You could call it anything you want, right? You could draw a triangle And say this is angle w and this is a box You could call this I don't know Diamond and call this a star, right? But That really is not intuitive. It doesn't tell you that w is related to box and Diamond is related to star But if you do capital letters for angles small letters for the sides It sort of makes the processing The pattern recognition easier for the brain, right? That's one thing it's important to keep in mind with mathematics None of these are set in stone. They're not absolute but over time Mathematicians have chosen these symbols because they make it easier to process that information, right? So if we're going to call that little a An angle b controls this angle Then we're going to call this little b and capital c controls this angle So we're going to call this capital c or small c and capital c, right? That's just by convention So so far we have and by the way, we could I've done this before we could do it again, but here Let's show the logos Everything has a logo on it. So just imagine if this is a triangle, right? I'm going to decrease make this size smaller And you can see that this angle is getting smaller, right? And if I just make this side bigger this angle here gets bigger, right? So it's pretty intuitive They try 400 So we got that another property of triangles, which is good to know is There are certain properties these have that have formulas behind them, right? One of the ones is Let's do this. Here's another thing with triangles. If you have a right angle triangle, okay? If you have a right angle triangle, which which means It's 90 degrees here when you got a box when they put a box here. This is 90 degrees Right or pi over two radians If that's 90 degrees Then this plus this has to be 90 degrees, right because 180 minus 90 is 90 So that plus that has to be 90 degrees You also get the Pythagorean theorem here, which is a b c Where the formula is a squared plus Little b squared little b squared is equal to c squared, right? Like for us for this triangle We really haven't decided which side is going to be the bigger side Over here the angle across the 90 degrees because the 90 degrees controls this side We call c in general, right? and something Should become obvious right now to a certain degree For any triangle you could look at the triangle and decide if it's not a legitimate triangle Just by the numbers you can't confirm that it's legitimate, but you can confirm that it's not, right? So given any triangle if you look at the angles measure the angles or be given the angles and the sides You can decide right off the bat if this could be this is This could be an invalid triangle for example If I numbered this two I call this 55 degrees I call this 40. I should have called this 45 degrees So 45 plus 55 that's 100. So this angle here would have to be 80 degrees, right? They have to add up to 180 degrees So if I call this side two if I call this side four and I call this side seven You look at that triangle. You should be able to say if that is a legitimate triangle Or could be a legitimate triangle or right off the bat if it's illegitimate, right? And obviously what I drew Because I'm trying to prove a point. This is an illegitimate triangle because If an angle controls the opposite side then Their ratio their relationship Has to be similar to the other angles or triangle at angles and sides, right? Because if this angle is the smallest angle and if it controls this side then this side has to be the smallest side If this angle is the largest angle Then this side has to be the longest side, right? So if we look at this thing right now We have 45 degrees 55 degrees and 80 degrees Across from the 45. We have a length of four across from 55. We have a length of two How could that be this angle is bigger than that angle So this side can't be bigger than that side if anything if I had written this like this This is four and this is two right That could be a legitimate triangle because the smallest angle controls the smallest side Middle angle controls the middle length Biggest angle controls the large longest side, right? So those are some of the properties or triangles and these things have more Formulas properties that we know of right? So for example, if you don't have why should for right angle triangle as well it works There's a couple other formulas we have Which one of them is an extension of this one, right? This is specifically only for right angle triangle a squared plus b squared equals c squared. Hello, saint. Just germany. How are you doing, right? But there are some other formulas pop up one of them is an area of triangle area of a triangle Is equal to one half base times height The circumference of the triangle is easy because you just add up all the sides So length a plus b plus c is equal to perimeter The length of the sides all added up, right? Here are a couple other formulas you have related to triangles you have sign law sign a over a is equal to sign b over b is equal to sign c over c This is called the sign law basically says this sign of an angle Okay trigonometry sign of an angle of this angle divided by the length of this angle Has to be proportional to the sign of this angle divided by that link Has to be proportional to the sign of this angle divided by that link. It's just a ratio right it means For any triangle on flat two-dimensional plane Euclidean geometry This applies this ratio is consistent. It's like the relationship between radians and degrees right or the length of the Radius of a circle relative to its Circumference or whatnot another formula that we have is this the cosine law which is a squared is equal to b squared plus c squared minus two b c close Hey, it's just another formula that relates the sides and the angles of a triangle, right? That's all these are right You can get the area from the three side links as well, which is fun. You can get the area from the three side links area from the three side links From this have so musko musko musko That's cool. I think so anyway There's really a neat higher dimensional extension to pythagorean theorem, which is not well known if you have a m-dimensional Simplex in an n-dimensional space where m is greater than n. So hold on a second if m-dimensional Simplex in an n-dimensional space. Okay, so you have a uh, I don't want to call it 3d but Multidimensional objects sitting in a space that is one dimension greater than That multidimensional object if I'm reading this correctly. Anyway Then the squared m-dimensional volume of this simplex is equal to the sum of the squared m-dimensional volumes of the projections of the simplex on that on the n On the n choose m orthogonal m-dimensional substrate. Are you serious race killer? Is this topology you're into race killer? Is this from topology? What about triangles and a sphere? Are there also rules? I guess so. Yeah, there there are rules I'm not too familiar with them, but I can show you an example of one. Are you related to Tony Stark? No I don't I don't have an iron man suit I have a chicho suit Right, here's a here's triangles on a sphere, right? Here's a triangles on a sphere What should I erase? I don't want to erase that let's erase pythagorean theorem And let's erase this as well Oh, actually that one was pretty we're gonna do it on that level right it's the square It's the it's the square s s minus a s minus b s minus c etc Where s is a plus b plus c divided by two so half the perimeter Oh, wow, this is actually a straight Generalization to the pythagorean theorem think about it Two dimensions dimension two tough fiber. Well, thanks for hosting dinosaur Okay, don't tell anyone But he is Tony Stark I'm tired in your calculator and turn I was saying Oh man Koku are we doing like great six great six calculator tricks? Dude bro, I'm just How do you guys understand this? No lie. I'm just gonna allow this. It's got swearing, but Uh, right. By the way, no one is dumb. I've never really I've been teaching. I'm gonna allow this just because I want to address this In my I don't know how many years of teaching Okay 20 years whatever of working with students I have never met a legitimately dumb person and I have never met a legitimately genius person Okay, so You know just saying are you trying to find the angles of the triangle? Did he spell it wrong too so we're great for because I'm Good Yeah, for sure. It's like Everyone can speak a language Math is a language. Some languages are harder to learn than other languages Okay Most people. Yeah, there are Rick Racky Racky Like he points out most people there are There are certain segments of society very very small that have a hard time with certain languages certain Ways to process information, but that means they excel in other forms, right in general I think almost anyone could do pretty good at math Certainly high school math. Yeah, high school math for sure Right, if you get a good at math, do we get a Beard like yours, maybe maybe Is it possible to calculate the volume? It's not something you want to calculate. I've never I have it. I've had it for a long time I've never had the desire to do it, but you could do it You could take a picture take that picture into a three three there rendition program And do the 3d rendition on it. You need picture from front this way this way Scan it cap, you know take the thing and do the volume is calculus All right In my opinion a lot of teachers do not teach math very good. Yeah, they don't Okay, but here's a triangle on a curved surface, right? So if we do this just imagine a flat surface We're talking flat surface if you have two lines that are parallel, right So these two lines are parallel they're going Like this On a flat surface if these these two lines are parallel they will never cross They'll never cross right So just imagine if these two things were connected On a flat surface like a staple right Anything that's like this shape If you put them on a flat surface and let them go they'll never cross If you put these things on a curved surface they'll cross Right when they cross they'll form a triangle Right, so right now these things have 90 degrees and 90 degrees, right? But if you put these on a flat surface, so let's draw the earth let me take off this guy too and these guys So here's our earth Here's earth, right So take Go on the equator draw this line and go 90 degrees up Over here on this end go 90 degrees up You'll meet at the north pole So on a curved surface if you have Something like this it forms a triangle. So really what you got is something like this, I guess I don't know what it looks like on a flat surface that which is a meat, right is curved This plus this plus this no longer equals 180 degrees it changes Right, this would be 90. This would be 90 and this would depend how far How wide you are out here So that's how a triangle looks on non-euclidean geometry flat surface geometry You can but you need to calculate the width Of the hair every space between every hair and the Wow taco you're doing it the hardcore way, right? I would just say take the outside the Outside dimension of it and just calculate the volume there and then what you could do actually So you could do a simple a simpler way taco. You could take the whole volume, right? So you could take the volume of something let's assume this is That's what it is, right? We have a lot of Hair going down, right? So you calculate the volume of this whole thing. That's easy to do Right over area. Let's say volume. We'll talk area volume is the same thing, right? You calculate the area of that whole thing and if There's a lot of these things you don't need to calculate go through do each one What you could do is take a sample size right Take a sample size and figure out what the ratio is here And assuming the beard is consistent you can apply that ratio to the whole thing So that would save you a lot of time a lot of time, right? Which is one place that statistics and data analysis comes in useful mathematics Nice never thought of it this way teacher so cool Two parallel lines make a triangle two parallel lines make a triangle super cool when I first learned this it was it was fantastic It was fantastic. It was super cool. And this is the way I teach it. I don't think this was the way it was taught to me was Whatever it was like visualize this or you try to visualize it, right? But this is basically the way I teach it Wait the earth is not flat I believe it's possible for a 3d triangle to have 90 degrees on every Angle yeah, and larger Right if you go 90 degrees you would go here go go a quarter way around the earth Right, here's a circle Right a quarter of a way is 90 degrees, right? So you could go quarter of the way around the circle around the earth And if you go up this is 90 this is 90 and that would be 90 right You could go further And come out that way that would be 90 that would be 90 and this would be I don't know 180 to 135 whatever would be right you could make it as big as you want Within reason of course, right How can we calculate the volume and surface surface of a ball? It's it's the formulas Here let me erase this And the next one we're going to do is proofs So let me erase these guys So we got this this guy done Let's use pink And that's one thing I do whenever I get something done. I'm like to do this I cross it off Right my book list my notebooks pieces of paper. I take notes on like really makes you feel great when you cross something off, right? But we're gonna deal with we're gonna deal with something one all surface of a ball So here's a sphere, right? I forget the formulas. It's just formulas that we have right now the area of a sphere And the surface area of a sphere I think the area of the sphere is Four-thirds pi r cubed Oh, sorry the not the surface area the volume of a sphere Volume of a sphere is four-thirds pi r cubed. I believe so Right That could be wrong. We have to look it up Right and the other one is uh Okay, I'm gonna look it up. I can't put formulas on it. I'm not how to say sure Let me look this up There we go. Oh pi r Yeah volume was four-thirds pi r cubed and the surface area is four pi r squared. Okay And that's all you would do so all you need All you need to figure out the volume of surface area of a sphere Is just the radius All you need is a radius R Now we're talking about an ideal perfect sphere that it doesn't exist, right? The earth is not a perfect sphere. It's Like it not enough egg shape but sideways egg shape or whatever it is So you would have to do averages or do some form of calculus where it's changing over As the radius the position of the radius changes, right? Small triangles which can be made more exact do almost follow-up. I think right there Even on curves versus depending. Yeah, for sure. And that depends on the curvature of the Of the object that you're dealing with right if we have a sphere like this really small sphere small triangles Are gonna still show that greater than 180 degree Some angles right so it depends on the Curvature of the sphere Like on the earth you can stake out Which we have when I did geophysics you can stake out triangles, right? And you're not going to take into consideration the curvature of the earth Because you're basically Within a flat earth Right where you are If the minecraft world is infinite, how does the sun set around the earth? If the don't read that again if the minecraft world is infinite, how does the sun? Make it around the earth is the mind. I don't play minecraft is the minecraft Open world supposed to be infinitely long Then that would have to be one gigantic. Yeah, it can't be infinite first of all But it would have to be one gigantic sun one gigantic thing for you to Ask for us. I don't know thought fiver So is minecraft world soon to be everyone builds on a flat surface Is it everyone building and interacting here and they're saying that this is infinite So that means they're just adding enough servers to extend this as far as they want And they have a sun That goes around and sets So all they're doing is just Expanding this so that would make this the center of the universe The minecraft craft flat surface the center of the universe with a sun in its orbit And it would be a flatter theory, I guess I would have to look into minecraft Proofs read the proofs. Let me read some more call. Let's get caught up with the with the chat and then we're gonna do proofs right Hello, st. Martin. How are you doing? How's life? Almost in the sense of approximate. Can you explain it? So I would have to look up what that is. It consists of 20 triangles. I think also I never understood Dimaxian map By black minister for I don't know Dimaxian map either, which is based on this as I would have to look those up St. St. Germany. I don't know what those are. Let me add it on here. If we get time. I'll look it up. I so I want to sell this out. I don't know what that is. I I've caused We'll add it. I'll have to look it up Depends how much RAM you have and you can reach them and but it will take real life Days months or years. That's what taco regarding Minecraft Is an amazing shape many Viruses have really capsule the protective capsules around the virus really is that the The shape that we see of the triangles That we see like this like For viruses and stuff Yeah shapes are amazing We put out a video with math in real life if you do chat chicho search for math and art and design how to create iso deco the hedge ronds Iso deco the hedge ronds chicho iso Let me see if it pops up Bamboo I have a friend as an artist here we go Let me give you give you a link here. It's super cool. He was making Let me read the title math and art and design how to make a bucky ball and a rhombo rhomby Cozy dodecahedron with bamboo and zap straps right check this out This is a friend of mine that that's an artist that creates a lot of different shapes and stuff and put out a series of Math videos and in this one he shows us where he's using bamboo in the backyard to make a rhombo cohi dodecahedron With bamboo and zap straps super cool Nature, that's right 4 pi r squared surface area cool doesn't perfect cute Cute boys exist not as far as we know cute boy. I mean perfect cute boys In a geology mineralogy we see cute boys Are they perfect? I guess in a perfect setting you could create them you should be able to I think so the surface area is the derivative of volume with respect to r. Yeah the surface area would be this so Volume is four thirds Pi r cubed and that's a polynomial Right, and if you take the derivative of this You're going to get it's called a surface area the three comes down This is the step the three comes down in the front becomes pi r three minus one Where three kills three so this becomes four pi and that's an r squared. So the surface area becomes the derivative of the volume, right? We take the surface area again, it becomes eight pi r so I forget what that represents The earth isn't a sphere at all no snout and it's definitely not moving It's rotating It's actually a dome. Oh, you're talking about the what do you call it? The hollow earth there Oblate spheroid is a technical term for earth. Okay, cool. Oblate spheroid awesome. Thank you for that Moscow Moscow Sorry, I'm just catching up with the thing Yes, bucky bottles also named after buck minister filer who did this. Oh, really it's named after that makes sense buck minister filer bucky ball If i'm not mistaken, that's cool, Jeremy At least I know that I tried to go to india but ended up in the kribian and killed thousands of people But still people think the earth is My former camp teacher was taught by buck muscle. Oh, really? That's cool. Jacob That's super cool Uh, a perfect cuboid is a rectangular box whose sides face diagonals and space diagonals Are all integers or all integers really have to depend on The accuracy of measurement earth is not that earth Shapes in nature aren't integers So I guess not it doesn't exist You could make it with Pythagorean triples Okay, let's do proofs think about this Regarding proofs Here's a kind of proof should we do trigonometry proof 10 dreams Proof by induction. Oh, I hate induction I almost have to look this up. How about we do some simple trigonometry? For example, let's do a trigonometry proof Proof by induction. Oh my god, jaco I like I literally actually have to look look this up proof by induction. I'm what's the other one I cheat your own chat. Hello lizard. How are you doing? Same it's a big part in my math course Let's stop. Yeah, and it depends on the proof you're trying to do right Uh, hold on. Let me look this up. Let me see if I can Just read up the definition proof by induction what it requires And if something comes to me right away To do I doubt it. I haven't done a proof by induction for a long time Mathematical induction is a mathematical proof technique It is essential used to prove that a property p and holes for every natural number And so on metaphors can be informally used to understand the concept of mathematical induction such as the metaphor of falling dominoes. Oh god That's so abstract To a certain degree anyway I see the example that they're using What is it? We're trying to prove example Forming dollar amounts by coins Induction bases other than one or zero base Induction on more than one counter Prefeit Strong I want something simple induction dollar amounts revised Yeah, I would have to look these up. I haven't done these proofs by induction forever Proof by induction example prove one plus that Using a proof by induction. Okay. This one is doable Assume n equals k holes Show any you have to follow all these things you got to do Now I start with the left side No, I don't want to do that. Let's just do I can't explain that properly It's very so much Let's do it. Let's look at a different one induction proof introduction proof of finite algorithm, we do some simple just proofs that don't categorize it into one thing Prove that that's that so that's that that's that That's that that's that that's that Okay is equal to that. No, we're not going to prove by induction. My board isn't big enough for proof by induction So proofs are okay like for example prove the sum of a geometric series. Yeah, but then you get others and it's just like yeah, I agree it's just It's just a mind you know Yeah, so you prove for n equals one assume true for n equals k then prove n plus this Don't think a perfect cure has been found by one Might be out there. There might be one out there but if it's it needs to be integers or something exact It's like that whole thing about a coastline. What's the length of a coastline, right? So if you have a coastline like this You know, what's the length of this coastline and people say oh, it's this much this much And you can prove using mathematics that this is actually infinite link Because you could zoom down to smaller and smaller size and then zoom in there Then that's also gigantic and you got to add up those links And if you add do all this infinitely then the length of the coastline is infinite, right? Is it practical? I don't know Coast then because then by theory would be true for all positive integers. Yeah And then twitch stay hydrated. I guess this is a new thing. They've added that you should be drinking Reekie flow. I don't know what the reekie flow is Let's do a proof just general proof Here check this out Which proof should we do? Should we do a trick proof? Let's do this here. Let me just bring up a sample trick proof Proof trick proof question. I just want the example That we know is going to work. There we go there Prove the following identity and we'll go through it secret a plus 10 a is equal to one plus Sine a hopefully we can do it over Coast Right, okay, so let me kill this Come back to you guys ay proof Not infinitely because obviously coastlines can't be infinite, but Infinite infinitely expanding decimal notation like oh, so it's an infinitely expanding decimal notation. Okay, cool Yeah, that's the thing with me coming up with examples if I haven't looked at them for a while I forget the bunch line, right? I can give some trick proofs if you like Yeah, trick problems if you have post them we can do them, but here's one, right? So for example, take a look at this Let me put up my OBS again, right? So basically when they say prove something prove When they say prove something what they Really are saying is Prove that this side of the equation Is equal to this side of the equation, right? It on the most simplest sense, right? We're not going to do this one, but on the most simplest sense that would say prove that five is equal to Four plus one prove that this side equals that side And proving this is difficult like I don't know how to do it, right? You need like The axioms the core axioms of mathematics the five of them to be able to do and it would take you like At least this border half a board or something. I don't This doesn't interest me, right? It's not something I do because it's it's like the rudimentary structure of mathematics that you have to know The lingo to be able to prove this, right? But this one's easy Okay Four out of four people think math is hard. Yeah, that equals that So this one when they ask you to prove something What they should be really doing is putting a question mark on here, right? So I could give you a question Like this no one does but if you you are in my class This is what I would most likely do put a question mark on top of the equal sign saying hey Prove to me that this side equals that side or Is this side equal to that side? So you can't Do this x plus two is equal to five if this is a proof question You can't move anything from one side of the equation to the other side of the equation because this isn't known It's not absolute, right? So that's the first thing to keep in mind when you're trying to prove an equation You can't take anything from this side to the other side. So what you end up doing when you're doing proofs is this You take this over here and you say the left side Left side of this equation is secant a plus tan a and the right side Right side Of this equation is one plus sine a over cosf okay Then you can put a line in between these guys That they're separate And what they're asking you to do when they ask you this question is make this side look like that side or that side Look like that side or crunch these Down so they look the same So they're telling you that this Is the same thing as this you break them up And or they're asking you is this thing the same thing as this you break them up You put the left side here you put the right side here and you try to make them look the same Sometimes all it means is rewriting this Okay, crunching this out to make to make it look like that sometimes it means crunching this out to make it look like That sometimes you have to crunch both of them to make them look the same, right? But what we can do here is we'll crunch this one So whenever you got trigonometry proofs, the first thing you should do is convert everything to sine and cos Okay, and you should recognize some of the identities, but you should try to Convert things to sine and cos after you recognize identities or whatnot, right? So what we can do here is say secant here. Let's do a little side secant of an angle is one over cos of an angle, okay of theta So what you can do is convert secant to one over cos Cos theta and that's just definition identity, I guess, you know, I think a lot And then you're going to go plus and tan theta Is sine theta over cos theta So you replace that with sine theta over cos theta Okay, and then what you do you got two fractions added together And a sorry theta is a I'm replacing with theta And then you add those two guys and they're the same common denominators like ridiculously easy problem, right? So you can just add them. So this becomes one plus sine a over cos a Oh and this side equals that side. So I usually just go oh check check the works out, right? And there's way harder proofs than this I'd like to find a hard one true inometry. I'd like true proofs Yeah, yeah Jacob. Yeah, that's exactly what it is You can't prove 5 is equal to 4 plus 1 without assuming things like the existence of the natural numbers and basic Equality properties. Yeah 100 pico axioms. They're called. Is that what they're called? Uh Piano axioms. Yeah, you have to dig down if you're trying to prove that 5 is equal to 4 plus 1 like this is one of the fundamental Principles of mathematics Like how do we know that's true and that goes down to the according to Moscow, Moscow, Moscow, it's called the piano Pio, Pio, is that how you pronounce it? Pio axioms In mathematical logic, piano axioms also known as the decay we call Pio axioms. So it's referred to this you need to know this Pio axioms I'm assuming the people The mathematicians that came up with mathematics are already gone through that. You can't You know, it's reinventing the wheel in a big way. I heard of those. Don't look back to them You have to find this function equivalent. Don't mess with us Do you follow? bot mass, bet mass in that calculation in this calculation Bet mass brackets Yeah, it's a sort of bet mass, but sometimes you have to if you're solving equations You've got to go reverse bet mass if you're simplifying them you go bet mass brackets exponents I don't know what the O stands for but brackets operations. I think me Orders orders orders When you're ready, I have a couple of trick proofs. Oh, yeah for sure. Give it to us jaco How can we calculate the speed of a car that is pushed off a ramp the car that is 0.5 t Oh tongues 0.5 tons over this many pounds heavy 3,000 whatever pounds and is going down a slope 30 degrees and the runway is 55 meters long the flat Or 10 meters long and the takeoffs ramp is oh my god What distance will the car reach until it reaches the ground? So I'm assuming it's right at the takeoff. What are those values? Yeah, it threw me off. I think he's just converting him. He's saying 10 meters is 32 0.8 feet Try this one. Okay, give it to us. Let's do some trick first taco After we do the trick Post this question again, maybe maybe we'll go to What do you call it? quadratics Just learn about julia it's like 48 oh really Metrics and imperial yeah proof by induction If you want a slightly harder one try this one 10 plus cod Yeah, that was it. It should be fairly easy as well. Let's try it out Let's do one of these one more of these proofs So they're asking you to prove this right tan x conx so tan x plus cot x equals to secant cosecant secant x times oops cosecant x writing is crazy bad messy messy cosecant x right so all you do just convert either that or this I'm going to convert this one for now tan x is sine x over cos x plus cos x over sine x and then you add these up with a common denominator common denominator is cos is sine so cos x sine x and you multiply this by Sine right it's just adding fractions So this becomes sine squared x plus cos squared x and then division just means you can put them on top of each other, right? Or we don't want to do that. Sorry. We go back to that and then we have an identity sine squared plus cos squared And that's equal to one. So this becomes one over cosecant Sine x and that's equivalent to this because sine secant x is one over Cos x and sine x cos secant x is one over sine x and this just becomes one over Cos x sine x we could prove this and that's equal to that by the way This one's called proving this guy sine squared plus cos squared My name's supposed to be mask of raven mask of raven. Oh my god. I I read it I think I read it correctly once or twice and then I forgot again mask of raven Okay, I try not to forget brother Well, most mathematicians don't really care about that logicians do regarding the five plus four, I think right There's a branch of math all about mathematical logic Of course, logicians, uh, it can be called mathematicians But most mathematicians overall don't care that much about axiomatic fundamentals. Yeah It's not something I've looked at and I don't consider myself Mathematician okay regarding the physics problem. Thanks taco When we take square roots either way the value should be equal to plus or minus or minus. Yeah It should be Okay, unless there are some restrictions on the system that we're looking at Right, but in general just straight up mathematics Square root of four is plus or minus two Because Two times two is equal to four as well as negative two times negative two Usually yeah positive value, especially when dealing with things that can't be negatively Shape-sized like yeah, and then a slightly harder one prove that If you mean mathematical fantasy Oh, this is the oh the caught double angle. Okay, let's try this out The double angle stuff comes up a lot, right? So let's do it double angle one. I don't have my Trig identity sheet open up what we could do. Let's try it out Caught Caught to x I might not be able to get this one because I don't remember We could do it some of the double angles and that's supposed to equal Caught square x minus one minus one And the whole thing over to caught x I can keep these trick ones coming if you want Let's do this one and then we maybe we'll move on to the quadratics or whatnot, right? So we want to prove this equals this and I really don't know if I can do this right now by memory, but we do So cotangent is one over 10. So this becomes one over 10 of two likes and 10 is sine over cos right and if you go one over Sine x over cos x. It's not cos x. It's cos 2x. This just flips it becomes cos 2x over Sine 2x, right? So I'm going to erase these to give us more room. I'm just going to write it out like this, right? So cot 2x is really Cos 2x over Sine 2x Okay, and these are called double angle identities Or double angles and they have certain identities that they have, right? Sine 2x is easy. I'm just going to do it aside here Sine 2x is equal to 2 sine x cos x. So we're going to replace this guy with that This is a proven identity, right? So the bottom of this guy becomes 2 sine x cos x Now cos 2x has three different identities Cos 2x one of them is Sine squared x minus cos squared x Another one is 2 sine squared x minus 1 and another one is 1 minus 2 cos squared x. Okay, I gotta look this up. I don't know if that's true or not cos double angle cos Let's go trig Then image. Oh, we need is the image Trig identity sheet. What do we got? Does this have it? Oh, there it is. Okay. So this one is I got backwards already This is it's equal to this Okay, it's equal to cos squared minus sine squared cos squared x minus sine squared x It's also equal to 2 cos squared x minus 1 and 1 minus sine squared x and 1 minus sine 2 sine x 2 sine x squared Right So this is where the thing comes in with these proofs Okay. Okay, cool. We got that long correct Jacob. Thank you Taking under root of both sides So basically what we have to do now is We're going to have to decide to switch cos 2x with one of these three And what decides which one of these three we're going to choose is what we got up here And we have to ask ourselves Which one of these is if we switch up this guy with Is it going to get us closer to that? Intuitively, I would say this. I'm not 100% sure by the way, right? Now if we sub this with this we're going to have 2 let's see if this takes us anywhere 2 cos squared x minus 1 right So the question is can we convert this into that? Well, one thing we can do right now is instead of trying to make this look like that We could start crunching this guy Okay, and that guy I think I wrote out correctly. So caught squared x is cos squared x over sine squared x minus 1 and caught is 2 cos x over sine x So nothing says that you could only you only have to work on one side You can hit this and go, you know what I've converted everything to sines and cosas. I've got rid of my double angles These are cots. So let's convert the cots to sine and cosas. So at least we're in the same We're talking the same trig ratios, right? So what we do now is do the top guy first, okay? So common denominator for the top guy is going to be sine squared x sine squared x That's going to be cos squared x minus Sine squared x because if you're adding these as a fraction the common denominator sine squared x Multiply this by sine squared x multiply that by sine squared x All over this guy, but I'm going to write this sideways here What I do when I have fractions over fractions I take it sideways So we're going to divide it by this guy 2 cos And the two is up top by the way. So let me put it up top 2 cos x over sine x brackets, right? So this guy Hey, if we take a look at this We could have switched this with that That gives us a top guy, right? Cool, but let's leave this alone for now Let's deal with this dividing by a fraction means multiplying by its reciprocal. So let's multiply by its reciprocal So instead of rewriting this that you wouldn't do any erasing uh If you were doing this for school or an exam, right? You just write this down and then actually let's No, no, I need to erase because I need the space, right? So I'm going to Change this to multiplication and flip that back, right? So change it to multiplication and flip it. So I'm just going to go like this sine x over 2 cos x So you see where this came from Okay Now what you can do is go oh look Sine squared x sine x so sine x will kill one of the squares And then in the bottom we just have 2 cos x. So our final answer here would be 2 sine x cos x and at the top we have cos squared x minus sine squared x Now these two don't look the same But they are the same because this is equal to cos 2x and that is equal to cos 2x But if this is equal to that and that's equal to that that means that's equal to that and that's equal to that And that's it. This is a four-way quality, right? So all you have to do is either convert this To that or convert that to that Because I've run out of room here. I'm just going to convert this to that guy So this guy is also equal to that guy. So you can just go cos squared x minus Sine squared x over 2 sine x cos x and this side is equal to this side and we're done I think what they write is o b s or something like this All right, it's interesting The root of something squared is not equal to that thing. It's equal to the absolute value of that thing Q e d that's right Q e d Q e d I don't even know what that stands for. I think it's a latin for something. I look it up every few years and I forget It's the nature of the bees. Should we do quadratics? Let's do quadratics okay Quadratics we've talked a lot about but they keep on coming up And it's good to address it because it's something that Quad Erita Demonstrates So I guess Demonstration is demonstration Quad You can also say Just kill this How do you spell that Boom And put a happy face at the end I like that way better Thanks power Fun, let's take this down Now we've talked about quadratics multiple different ways, right? We've taken quadratics broken up into lines and stuff like this But how about this we start off with the line and then kick it up to quadratics I'm gonna start turning in proofs Like that Let's kill these guys So we got another thing done Nice, nice, nice We like it Today is a good day. We've got two out of four out of five things one is taco mentioned I don't know if we'll get to that question taco taco physics So we got two out of five done so far not bad not bad Great idea LHS and Mamma Mia I'm gonna eat papa Papa Mandarin Fun key quadratics One below quadratics is this Corsado Andron It's a shape Let me see if I can draw it Oh, yeah, this thing Cool It's basically this guy If you do buy killed it already If you have build a three-dimensional object with triangles like I can't do it. I'm not an artist You should continue this. It's a sphere that's made up of try triangles All right 20 phase 3d shape with triangle faces. Yeah It's called a perfect Sphero, perfect. Sphero is that what it's called? I forget what they're called The there's five of them five perfect Shape spheres that you can make from the things Here's quadratics Next step up from lines Quadratics are next step up from lines right away chicho. Love your liqueur. Thanks Jacob I don't know how to spell it Yeah, liqueur is weird. It's hard though. I always make a mistake. I always gotta get corrected The liqueur video is good. This summer. We're gonna make some more. I'm waiting for fresh fruit to come Okay, I need to go through my liqueur cabinet again and clean it up and consolidate some of the liqueurs And there's gonna be some space created and we're creating new liqueurs to download Gotcha Raven nice So take a look at this Quadratics are the next step up from lines So one thing we do in mathematics is this We learn a certain concept right or define a certain concept We And then we build on it, right? That's why mathematics should be the easiest course you take in high school because It then the stuff is not new, right? It's just building on previous knowledge you have so if you have a core Understanding of the rudimentary stuff from elementary school early high school Then it's easy to build up on that, right? It's just basically expanding our horizons It's like the real number set, right real number set first thing we know is natural numbers If you do a box first thing we know is natural numbers and then whole numbers and then integers and then quadratics not quadratics Rational numbers and then irrational numbers, right? And this is basically human evolution or understanding of the world around us, right? Natural numbers were one two three whole numbers included one two three and then the number zero Integers included positive and negative whole numbers, right? Rationals or integers are fractions of whole numbers and Irrationals or fractions of integers and irrationals are any numbers that we cannot write as fractions of integers, right? I'm afraid of that complex numbers and then this is real numbers complex number, we won't get into it. This is a real number set Real numbers, right? Real number set Okay So the way we've evolved our understanding of the world I think that's like a proof that's uh, has it been proven yet or not? I'm pretty sure it's been proven But uh, e is to e to the power of pi Pi i is equal to one. It's all the magic numbers put together, right? But basically mathematics evolves We know we understand something small and then we ask for a question. Hey, what happens if we layer this, right? We add something to this and then all of a sudden we come up with something new and we go. Wow, that's cool, right? That's the way quadratics Come to be if you want to think about it, right? Think about quadratics like this the beginning of quadratics being this Let's take a line and I'm going to assume we understand An equation of a line and an equation of a line is y is equal to mx plus b, right? So let's graph a line on here. Let's graph the following line y is equal to 2 x plus 1 right So we want to graph this line. We go, okay B is the y-intercept m is the slope so we go to the y-intercept which is 1 And then from there we go 2 up because that's 2 over 1 2 up 1 over And we have our line So this is line 1 Right, so we go. Okay, cool And again, we're going to assume We know what a line means what a representation of a line is That's hard So that's our first line now mathematicians play around with this or you play around with this in great 10 great 9 Maybe great 10 you play around with lines graph different types of lines going down like this going like this going like this Going like this going whichever way right you start talking about hey What if you have systems of lines, you know, what if you have this line plus this line? Or what happens when this line crosses this line and try to find solutions with these things right so you could have this you know infinite number of lines and one day someone joe blow comes along and says hey You know what we can add and subtract lines. Well, what happens if we have one line times another line, right? So let's assume we have another line. Let's graph another line Let's graph this line Line 2 with the y is equal to negative 3 x plus 4 So let's say we have this line, right? And if we're going to graph this line We're going to go to the y-intercept 4 and the slope is negative 3 so down 3 over 1 so 1 2 3 4 right And then we're going to come down 3 1 2 3 and then over So here is Our line number 2 This is line number 2 Do we even need the circles here and confuses? Let's take these guys down So there's less writing on the Whiteboard you can tell which ones what that's that line and that's that line And all of a sudden mathematicians come along or some smart ass comes along and says hey Let's create a new function and you can do this right for example Let's let's assume we have number 5 and everyone understands what the number 5 is and then you have number 6 and everyone understands what the number 6 is and then you decide to multiply 5 and 6 together And you get a new number called 30 right It's a new number created from the number 5 and number 6 and number 6 itself is created from two other numbers which are 2 times 3 prime numbers prime numbers of your prime factors to a certain degree So 2 times 3 Gives you 6 and then 6 times 5 gives you 30. So 30 is really 5 times 2 times 3 That's what the core building block of 30 is right? Well, you can do this with functions, right? So let's assume you decided to create a new number a new function And call it What should we call it? Let's call it The green pen let's call it g Right, let's say you want to create a new function called g And what we're going to do is we're going to multiply this function times this function We're going to multiply the black y times The blue y Right now when you're doing mathematics You don't want infinitely colored pens For you to be able to figure out which y is which y so the way we do it We just give them numbers. This is y1 And this is y2 That way you don't really need different colored pens. All you do is just number your y's And this is y2 So the rest of this I'm going to do with the green. Okay So what we do we say hey the new function g Is equal to y oops. That's why it's supposed to be a y1 That's supposed to be a y1 Let's put a 1 up here. We say G is equal to y1 times y2 Well y1 is equal to 2x plus 1 2x plus 1 times negative 3x plus 4 Okay, so the new function is g is this guy and we ask ourselves. Well What is the graph of that look like? Right, we've already graphed line one and line two. What's the graph of g look like? Jacob, thanks for hanging around Hope you have a fantastic day today and tomorrow Okay, so the way we're going to graph that we're going to multiply this out first. Okay So if you multiply this out this multiplies this multiplies that multiplies that multiplies that 2x times negative 3 is negative 6x 2x times 4 is plus 4 at 8x 1 times negative 3 is minus 3x 1 times 4 is 4 and then you combine your like terms and your new function is hope this is negative 6x squares 2x times negative 3x is negative 6x square this becomes negative 6x square plus 5x plus 4 This is our new function g So we multiply two separate functions that were linear that were lines And we've got a new function called g which is a quadratic So our purpose right now is to be able to graph this quadratic, right? So let's graph it and to be able to graph it. We need to do completing the square Okay, and completing the square. I have videos online for right if you do chicho completing the square You'll see it. So I'm going to go through it right now. Okay, so let's erase these guys Oh the green smudges a lot So let's do the green again. So as we start with green we stick with green Let's call the green function negative 6x square plus 5x plus Okay Let's erase this Now the way we're going to graph this That was a nice one chicho chicho. Thanks monkey. I've been doing this for a while, right? Let me bring up my stool here. Sorry about the noise again Okay, so what we need to do there's a process called completing the square here. Let me find a link for For those There's a couple of videos we have some more in a couple deleting Our videos here this this Yeah, there's all there's a few of them and there's a whole series we did of completing the square like really But this one is the ASMR version Okay, and I think it links up some of the previous versions when youtube have annotations you could have clicked on I don't think I include them in the link. So if you go to language or mathematics series three A and three B have on the stuff, okay So the way we're going to do this process is this You put this guy in brackets the x square and the x term you take the take the negative six out x square you got a compensate for this so this becomes negative five over six x Plus four right because you got to divide the five by say to your cop say for negative six times negative five over six Gives you positive five back right And then you take this guy and divide it by two negative five over six divided by two is saying is multiplied by one over two So this becomes negative five over 12. I should have picked better numbers, but it is what it is, right? You take this number add and subtract it in there. So this becomes negative six x squared minus five over six x x Let me drop this guy here x Pull um Oh, yeah, we take this guy And then we square it you get 25 over 144. Okay, you're going to circle this guy this guy You're going to add and subtract in there. So this becomes plus um Oh, sorry, this guy you're going to add and subtract in here, right? This becomes plus 25 over 144 minus 25 over 144 plus four this guy factored is x minus five over 12 squared Okay, the whole thing squared you're going to grab this guy And bring it out of the bracket and you're going to multiply it by negative six as it comes out Okay, so this guy here is becomes negative six. This guy becomes x minus five over 12 squared this becomes negative six times because this is in front of the bracket Anything that comes out of the bracket has to multiply whatever's guarding it, right? So that becomes negative 25 over 144 times negative six negative negative makes it positive Two goes into this three times two goes into that 60 not 60 72 times 72 And oh three goes into that evenly. So this goes evenly by the way 144. Let's do this 144 divided by six Six goes into 14 twice. So that's 12 to bring the four down six goes into 24 24 times, right? Oh, sorry. Yeah four times, right? So it's 24 So they kill each other this becomes 25 times 24 Okay, I'm gonna multiply this side. That seems pretty damn large Why does this seem so large? Oh because you've got a multiple of six here. It's vertical expansion by here is right So 25 times 25 is 625 Okay, and we've got one less 25. So this 600 should be anyway zero two eight 10 zero four zero four five Oh no, there's a one that comes up 500 600. So this becomes 600 Okay, so that became 600 plus 600 so it becomes plus 604. I hope I did that correctly Okay So if we're gonna graph this guy, this is what the graph g G looks like right if you multiply this line with this line The vertex is five over 12 the vertex is five over 12 and 604 I should have picked better numbers Five over 12 is less than one if that's one that's five for 12 and 604. This was four, right? 604 would be like up there, right? So we're not gonna put the dot way up there just pass my roof Right, so we're just gonna assume do this and just go. Oh, it's really high up and the X intercepts would still be this so this guy comes down Goes like this Goes like this and this point here is Let's erase this let's erase all of this This point here is five over 12 and 600 okay, so if we multiply this line which was This this point was At four and this was at one if you multiply these lines you're gonna get this gigantic thing that looks like that Which is very trippy if you multiply two lines together you end up getting a curve and the reason you get that is because You have positive and negative numbers multiplied together Which which does that for you? the infinite collection of solutions to under Underdetermined linear system Can be expressed as a single x y pair plus all Pairs that would solve the system if we eliminate the constant term So in the above case Where k is any constant How can we modify this for solutions to understand quadratic systems? Oh, wow I would have to think about that. I gotta read the thing and vice power Before before I do that, hold on a second. Let me do one more thing Kichu Let me see if I Can find the video These are really old videos so the searches are here we go Okay, take a look at this one. I call this video. I put this out In 2010 facting polynomials a graphical representation Why we factor and it's the language of mathematics video number 122 Okay, and this thing Basically does what we sort of talked about but the other way around to a certain degree Let me Many thanks fun. Like it says my pleasure man I love the thought of all Your neighbors saying this crazy tall Non-linear equation systems are far harder Yeah, non-linear much harder for sure There's no simple way to express all solutions as in the linear case. Okay, so that's regarding Dice powers question So i'm assuming that's going to be much harder than what we're doing Don't know if I want to read that again basically and I wonder if you can by if you can by Paralyzing the X and Y pairs. Yeah, I think this mathematics that On dice power and racer killer talking about there's a little bit beyond my mathematics Okay, so I'm just going to erase this and we can put another tick mark without with the quadratics. Nice Nice Without with that awesome three out of five That's good Let me erase these guys Let's see how much time we got left What else can we do? Let's see what we got. Let me have a sip of sip of tea Okay, not a sip of tea. I was thirsty It depends on a non-linear system, but it's pretty disgusting Disgusting for all of them on this you specifically choose One that's nice. Okay, so we're going to skip that one. We're going to go in our batch show Bye funky. Thanks for sticking around brother And we're almost up two hours here. Wow What was taco's question? That's fine taco's question. Where was it? That was a physics question. I don't know if we have time to go over that I have to maybe draw it out. Here we go. I don't have tacos here. I grabbed it. I'm gonna post it here look Dice power oh dice power Here's what taco posted before How can we calculate the speed of a car that is pushed off a ramp? Let's check it out Push pushed off a ramp not flying off a ramp the car that is five 1.5 tons heavy and is going down a slope 30 degree and the runway is 25 meters long the flat part of a ramp is 10 meters The runway is the slack part of a ramp is 10 meters degree and the runway Down a slope 30 degrees and the runway is 25 meters long the flat part of the ramp is 10 meters long And the takeoff ramp is 25 degrees So it's doing this So This is the car If you have a specific system in mind, there might be hope, but yeah Mask of Ravens thing specific system No dice power wanted that wanted us to deal with that question, but it sounds like Mask of Ravens math I think it's pretty powerful dice power and I think yours is as well And he's saying it's it's gonna take a fair bit to do that. I would have to understand the question itself Let's check it out can be expressed as a single x y pair the infinite collection of solutions to Undetermined linear system 4x plus 4y can be expressed as a single x y pair Plus all pairs that solve the system if we eliminate the constant terms So how do you do that? So in the above case x y equals 3 negative 2 Plus 4 negative 3 Where k is any constant Oh, I didn't even see the k in the front I didn't know how to go about that dice power Linear really is different. You can always describe the solution any easy way Nonlinear equation systems are just really hard. They are Dedicated areas of math like algebraic geometry dedicated exactly through this So I don't think I have this drawing correct Regarding what talk was it how can we calculate the speed of a car that is pushed off the ramp? The car that is 1.5 tons And is going down a slope 30 degrees And the runway is 25 meters The flat part of ramp is 10 meters and the takeoff ramp is 25 degrees What distance? The way I'm reading it is The flat part of ramp is 10 meters And the takeoff ramp is 25 degrees I don't understand. So that would have to be 25 degrees Right, but where's the 10th if flat doesn't mean just flat like this or This is 30 degrees or Well, that would be 30 degrees of permission. That would be 25 meters as well And this whole thing is 10 meters Yeah, I don't think I'm drawing a right talk Mask of Raven has got a specific system There might be Yeah, I don't know if taco's still here I don't think that's the drawing that doesn't make sense The distances don't make sense in the 25. We don't know where it starts Oh, wow Maybe next time taco we put it up. So these ones we didn't get done. We could do this one I saw the properties of an iso thing would you be? I In geometry I saw I saw hadron is a Polyhedron with 20 faces. That's what you call them polyhedrons With 20 faces the name comes from ancient greek meaning 20 and meaning seat and meaning seat Oh 20 seat the plural can be either isohedra or isohedrons There are many kinds of isohedrons isohedras isohedras, I guess With some being more Symmetrical than others the best known as the platonic convex regular Yeah, that's right, that's what they're called the platonic solids Nice Yeah, so it's one of the platonic solids The isohedron 20 faces and it's the one with the most faces as well, which is cool They're all provide the length of the platonic solids Race killer dice power says so there there's no guarantee that the null space line K K4-3.3 In the above becomes a parabola When deal when you deal with the quadratic system, for example Of course the really I'm really thinking of the x y pairs as geometric vectors Man, you guys are talking good mathematics. I wish you were doing a stream to solve this I'd watch Right Almost two hours Which was a good stream. This is a good math stream. I like that we covered some stuff, which were cool Oops, sorry about the noise game drop one of the markers Three out of five not bad Not bad Aside from that, uh, I think I'm gonna call the stream gang. Thank you for being here But dice power race color still talking about this. I'd like to follow the conversation Look your example in the linear case is just two lines Which is the same line All right, let's check it out I'll see where it takes us to show live try uh searching for quadratic equation systems If you're interested in quadratic equation systems I've done some quadratic equation systems But what you guys are talking about seems like more complicated. Oops. We're all person I sent to I meant dice power. I'll look it up to race or kill Uh systems of linear and quadratic Quadratic system algebraic solutions. No, there's a systems of quadratic equations In nothing we do systems of quadratic equations, but you guys are talking about infinite solutions with null spaces Like two parabolas either intersect once twice or they are the same. So is this what all we're talking about? Why would it sound more complicated to me? So basically what we're talking about is this just to get other people caught And myself oriented We're talking about this right Here's some quadratics Okay, you could have Two quadratics That just touch each other You could have two quadratics that only You know, they cross once they cross twice They don't cross at all They don't cross at all, right You could have quadratics that are They have the same Leading coefficient So they have the same type of opening vertical expression. So they only hit once You can have quadratics That are actually the same equations that are sitting on top of each other right You could have quadratics and lines crossing each other systems of Equations, but just a quadratic and a line and this thing there's three different basically scenarios if it touches it If it crosses it twice touches it or Doesn't touch it at all Two solutions one solution. No solution infinite solutions one solution No solution no solution two solutions one solution. So that's what they're talking about Which is cool, which is cool. We do a lot of that in high school I don't think we're doing it taking it the one step further that you take that that dice power and erase your killer talking about I count that instead of a line and it's a elliptical shape, right? That sounds right. Two ellipses can be the same not touch touch once or cross twice cool And every time you go up it gets worse and every time you go up it gets worse every degree higher It becomes more and more complicated right Welcome to mathematics. Welcome to mathematics Aside from that gang. Thanks for being here. Thank you for a cool conversations gang I like following the stuff math discussions and how they relate to high school math specifically I learned terminology and stuff, right? Um Thank you for my pleasure dice power my pleasure dice power. Thank you for being here. Super fun Um, and what we'll do next week again, we're going to do math I'll do politics and the odds are we're going to do more of the 10 by 10 puzzle and cooking stream and stuff like that I'm going to start incorporating those in there as well Okay, either way. We're going to have to Have trouble to get a Uh description to Paralyze the solution set which works for every problem Okay gang Hope you have a fantastic fantastic Okay, and I'll see you guys in a few days with more live streams and Um, I'm going to do some stuff in the background as well. So you might might seem tweeting Posting links on discord or whatnot. Okay That's it for now. Bye everyone Bye