Let's make sure we're live. There we are. Awesome, awesome, awesome. Hi everyone, this is Chichou. Welcome to another live stream. Today we're doing drop-in math tutoring session for the 2020 year and it's number 12 since January. We're doing one of these things. And it's basically drop-in math tutoring session, open discussion. I'm making myself available for people if they have math questions about high school mathematics. We can deal with that, right? Brett Kelly, how are you doing? Good evening. Good afternoon from my end anyway and good evening to you. Hope you're doing well. Aside from that, our little intro, I'm on Patreon. Patreon.com backslash, Chichou, CHY, CHO. If you want to follow this work, Patreon is a great way to follow this work. If you want to support this work, Patreon is a fantastic way to support this work. We're live streaming on Twitch, twitch.com backslash, Chichou live. I had a Chichou live for Twitch. And basically this is where we live stream and then we upload the videos to YouTube and the chute and maybe in the future to other platforms as well. Welcome to a live stream. Nice to have you here. I do announce these live streams 30 minutes before we go live on Twitter, Gavs, Mayans, VK and other and all the links will be in the description of this video. For a lot of live streams, we're going to be uploading the audio to SoundCloud, right? When I'm recording with a lapel mic, these math moves we're not uploading yet. We might at some point when I do a little upgrade to the standing mic and stuff like this or get some wireless lapel mics happening. How are you doing? You're from the US. I'm in Canada. Sam Hitt, my man. How are you doing? Welcome, welcome, Grand Prix. Hey, Chichou. Hope you're having a smooth day, having a fantastic day so far. Nice day. Had good math discussions with students and we're doing more mathematics now. So life is pretty sweet and it's sunny out here. And Grand Prix, thank you for hosting, by the way. Awesome. And it's sunny here, so actually set off a beach umbrella inside the house. Let me show you. It's the end of it. So that way the sun is not going to totally glare out the whiteboard, right? Maas, how are you doing? Hey, hey, keep them eyes dry. I know. Crazy, crazy Sam Hitt. I did not expect what happened last stream. I'm right now supposed to be bad luck, but it's pretty good luck so far. As for the videos, we're going to be uploading these to YouTube and BitShoot. Everything goes to BitShoot and as much as possible goes to YouTube, but as far as we, you know, as long as the sensors don't deplatform us, right? Heat index days, 105 degrees Fahrenheit in Oklahoma. Hot, hot. How's the humidity there, by the way? Witch please, how are you doing? I just want to thank you after talking to me about my this Calia. I can't even pronounce that. I was able to do my first mental math. Thank you. Oh, you're very welcome. You're very welcome. I'm glad it helped out. And there's a few people on chat giving advice as well. So that's fantastic. That's awesome. Good to hear. Good to hear. And thank you for coming here to mention it as well. I'm glad we're on the right path anyway. I know it's helped out my students. So that's fantastic. Awesome. I'm going to take these things down over here. That way we've got a clear board or clear screen. And if there's any math questions anyone ever has, we can definitely deal with it or open discussion. We can go wherever you guys want, almost wherever you guys want. So I wake up this morning at 9 and start doing English and French work until 1300 hours. So 1pm. And I relax to math Twitch stream. I really miss school. Nice. Yeah, there's some people that missing it. Some people are not. Learning is fantastic. Learning is a good thing. It makes you feel good from what I, you know, the way I feel about it. It's hard work, by the way. Learning is hard work. You know, you might have to put in a lot of hours to learn a certain concept. But once you learn that concept, rock and roll, right? Mille. Mille. Mille. Mille. Mille. How are you doing? Hope you're well, Chicho. And the same for everyone else watching. And you as well. Thank you for popping onto the stream. Graham, how are you doing? I need to keep walking, but it's just so hot. It's unpleasant. Oh, is it? Here is really nice. But tell you the truth, gang, I miss the hot. I really do miss the extreme hot. 68% humidity. Actually, that's not bad. 68% is on the humid side, but that's not bad. Ripper, how are you doing? Greetings, Chicho. How are you, brother? Chat, how are you? I'm doing good. Two days ago, you made me cry. In a good way. So much love. Wow, crazy. Oh, no, no. Oh, no, not Joe. Oh, no, not Joe. That's what it is. Oh, no, not Joe. Hope everyone is doing well. Eduardo, how are we doing? Welcome, welcome. Monty Ray. Hi, hi, hi. How are you? How hot does it get there? It doesn't get that hot here. It gets in where I am. It gets about hot as 30 Celsius. Again, that's pushing hot, right? 30, 32, seldom does it go above 30. 30 degrees Celsius or whatever that is in Fahrenheit. But I grew up until I was 10 years old in the desert in Southern Iran. And it used to get so hot there that the asphalt on the ground used to become malleable. So as kids, we could walk around and put sticks into the asphalt, into the road, and it would stick up. It was good. I lived in a place where there was tarantulas and scorpions, and I lived in the desert. I think it was sort of attached to the Arabian desert or something, so it was crazy hot. I loved it. I want to move to the Pacific Northwest. Texas is brutal. Pacific Northwest, though, man, living in the temperate rainforest is a pretty sweet thing. I can honestly tell you, that's absolutely amazing. Have you ever seen a moose? Yeah, I've seen a moose. I've seen elk. I've seen bear. I haven't seen a cougar. I'm pretty sure a cougar has seen me, because when you see a cougar, the odds are you're in trouble, okay? So he's gotten close, or he or she has gotten close enough for you to be able to see him. So in general, if you're in the forest, we have a lot of cougars in BC. They're like cats, right? They are cats. They track, they hide, and stuff like this, so you have to be careful. So I've never seen a cougar. I've seen a grizzly in nature. Grizzlies. Grizzlies are big. Grizzlies are huge, right? Sam, I know this might be simple, but can you go over the concept of real numbers, please? For sure, we do. I know you've seen murderous raccoons. And by the way, the raccoons, the raccoon video, raccoon fighting in a tree, that's my most popular video right now, it's getting a lot of hits, right? When we moved into this new place, we had a couple of raccoons making love on a branch, on a tree. So there should have been a video coming out following that, raccoons having sex in a tree. But when we're recording the video, it was just all of a sudden, we're like, oh wow, wow, we're signing a lease, by the way. And it didn't record. We didn't get the recording. Maybe they come back and make some love. Ripper Chicho, I don't apologize, haha. I'm glad my words, along with the chat, echoing them. We're able to move you in a way that you see the impact that you have on us. After all, that's the goal of it. That is the goal. That is the goal, Ripper. And I appreciate it. It was amazing. I tried to go read the chat to calm me down. I read the chat, it was like, oh my God, this is too much. I couldn't walk away because we're live streaming. Lots of rain today. Elder God, but I don't move all day. Lots of rain. Oh, you move a lot, man. 500 push-ups, or 1500 push-ups. So we're going to do real numbers for sure. Okay. Moose are actually pretty huge. They're gigantic moose. And they could be mean. You know, you have to be careful around moose. Mooses can be, like bears are pretty chill, right? Mooses are, to a certain degree, more dangerous than bears. Okay. Really? Didn't know this until recently. Yeah, mooses are gigantic. Hey, Chicho, finally made a stream. Hello, the trash man. How are you doing? I saw you pop into the discord and you asked me about discord and you were on the, what do you call it, premier a couple of days ago. So welcome to our live stream. Ripper, Chicho, but I will absolutely take my receipt and look you body slam. Let your body slam. One day, right? Oh my God. I don't even know if I could do it. There's no way I'm not in shape to be able to do anything like that, man. You will rip me apart, twirl me and throw me out of the ring. Hey, everyone. Hope all is well. Trash man. It's time to chill with Chicho. Nice. Hey, Hannah. How are you doing? Grant, Chicho? I know I've asked, but does it snow a lot there? Not too much on the coast itself.  So we got a lot of micro climates in the Pacific Northwest. Lots of micro climates in the Pacific Northwest, in the Pacific Northwest. I'm moving into a new space in three weeks. Just have to sign my lease and pay my bills. Nice. What made you fall in love with math, Chicho? What made me fall in love with math? What really did it for me was teaching it when I got into teaching math. That's when I really began to understand mathematics. I always had an appreciation for math. I liked math. I realized how much power it gave me when I was studying in high school and university. That's why I ended up getting my math minor. I always knew it was powerful and it was a power that I would like to have. It's like playing a video game, right? Do you want that magic sword that annihilates everyone? Yeah, of course you do, right? You'd be silly not to acquire. Do the work you need to do to acquire that magic sword to be able to slave whatever creature you encounter. Math is the same level, right? But I didn't really fall in love with it the way I have until I started teaching it. So for me, it's teaching mathematics that I'm in love with, I guess you could say. Last year I was on a canoe vacation in Sweden. The double-decker bus we were in, taking us back home, hit a moose, full speed. Oh, annihilated the bus. In the middle of the night, one of the scariest experiences of my life. Luckily, nobody was seriously injured. Wow. Yeah, mooses are gigantic. They're horse-sized, right? And I've hit a deer going at 120 clicks an hour in a car. It takes out the car. Mooses are like cows on stilts. Yeah, 3,000. Congrats. Oh, lots of chas. Okay, guys, I'm going to doing math. Yeah, perfect citizen. Nice. Uh, I came to say hi. I was super busy and I will be tomorrow. Taco operator. Thank you very much for popping up. Say hi. We're going to deal with the real number set right now. I'm also from the UK. Nice. Lots of UK friends here, which is fantastic. Have yet to see a tutoring session on Twitch. Just here off curiosity. Awesome. It's I'll, I'lls one. Welcome to our math stream. VC. How are you doing? Hey, Chico. Do you tutor set theory at all? Like, uh, cardner, uh, cardin, narrow tea and proofs and stuff. Not really. We don't encounter that too much. I've done a little bit in the past because it was part of the curriculum, but over 10 years ago, they took it out. So I haven't had practice with it. Yes, Chico. Perfect timing. Starsky. How are you doing? How's life? Warwick. Love these math streams. Awesome. Starsky. UK, UK, Chico. Gina. How are you doing? How are you doing? Should we talk about the real number set? Now, real number set, by the way, I've got a couple of videos out there. If you do Chico real number set, uh, I used to have it all in one video when we did it, but that was loaded on, believe it or not. Google had a website where you could load on Google video, but they took that out, right? Chico real number set. So on YouTube, it's not on bit shoot yet, but on YouTube it's broken down into two parts. Okay. There is, uh, it's the third video I put out in mathematics, third and fourth videos. Okay. So let me, oh, that's on video. Here it is. Here's part three. Here's part one of the real number set. Okay. Now keep in mind, this is literally the third video I ever shot of me recording of me doing mathematics and me being in front of the camera. Okay. Like literally this is me at the beginning stages of creating math content, the third video I put out. It could have been even the first video because I probably made my intro video before I made this video. So this was possibly the first video I ever shot of me being in front of the camera. Maybe. Okay. There's part one. There's part two. They used to be together, but Google took out their Google video platform, right? So all that stuff got deleted. Google. Hey, Chico, Laura, how are you doing? Midlands, people from Midlands, part where they buried with work again. So I will be lurking Catholic traditionalists. No worries. Hope the work goes well. Okay. Serious life question. Assassin on we. Serious life question. Should I study well in high school, such as act and sat over summer? And do you regret studying or not studying well in high school? And would you regret and do you regret in high school? Look, high school sucks for most people, especially now. A lot of people don't like high school. It's school to prison pipeline, whatever you want to call it, right? Indoctrination. You're in a big class size.  There's a lot of violence. There's a lot of drugs. There's a lot of stupidity. The teachers, you can say all that stuff, right? Irrelevant. It is your responsibility to educate yourself. If that means when you're going to high school, there's certain powers you have to accumulate in life, okay? You need to be able to learn how to read and write. You need to be able to learn how to do mathematics. You need to learn history, but not indoctrination from centralized education system. I tell people, as soon as you graduate high school, for the first two to four years of your life, you should be re-educating yourself, right? Learning about your history, looking to English, start doing some writing, just either creative or analytical writing, look at mathematics. Just learn what school didn't teach you, right? Because you're in the learning mode, right? So after two to four years after high school, you should be educating yourself into what school didn't teach you, right? So in your question, do I regret learning? No, I don't regret learning. Was there a lot of wasted time? Yes, there was. Was there anything I could do about it? Not really. Not at the time, not with what I knew, right? Do you regret? I don't know. That's about it. That's all I can say about that, right? You look so much younger. I look so much, it's a goatee. But if I change the color of this, not really, but hey. That was put out in 2007, by the way. That's 13 years ago, right? Fellow teacher, nice. We will be creating first custom edited videos this summer as well. Good luck. Good luck. Good luck. What does the W number stand for? Yeah, we'll do Gino. Good, thank you. And how are you? Oh yeah, Gino. Chicho, your KD ratio video on graphing, did an excellent job in introducing my little brother, third grade to graphing. Awesome, awesome, Miro. I'm so happy about that. That's the kill-to-death ratio, by the way, in gaming, right? We did a little stream and we looked at ratios and stuff, and it makes sense, and you're like, oh, that's it. Awesome. I have a law from there. If you have x equals that, what is the confuse? Okay. Is that going to be a trick number, Evan? Interesting. Chicho, have you ever made a video along the lines, things that high school didn't teach you? It's probably incorporated into all of my videos, Miro, right? It's probably incorporated into all of my videos. I will be trying to channel the good math vibes from this stream, applying some fairly gnarly math in the project I am working on as I listen. Awesome Catholic traditionalists. If it's shareable, share it on our discord page, right? No, it's not. That would be a long video. Why not? If you're confused, how do you know if it's wrong? Okay, let's deal with a real number set, right? Let's deal with the real number set. The way I explain the real number set to people, by the way, a lot of these videos I'm making, they're geared towards teaching what it is we're talking about, but they're geared towards teaching how to teach what we're talking about. And this is exactly the way I teach it to my students because once the kids, my students know the angle that I'm coming from, why I'm explaining things this way, what it all means, then they learn easier, right? Because they know why, which is the first thing that everybody wants to know. Why? Why do we have to do this? Why do we learn this? Hello, Mr. Hezakaya, how are you doing? Like we say, don't go gray, go pink. So take a look at this thing, the real number set. The real number set, most people would look at it this way, right? You see this thing, you see this thing, you see this thing, you see this thing, and then they break this down into two equal parts, right? And then they put real number here, real number set, and then they put what's called the natural numbers, whole numbers, integers, rational numbers, irrational numbers here. I'm going to put them in, but we're going to explain this a little bit further, right? So we're going to do this, and we're going to go natural numbers, we're going to go whole numbers, we're going to go integers, and rational numbers they call Q, in my part of the world anyway. Some places I think they call it Z, right? It's rational numbers, we'll write them down here so you see what they are too, right? And then in mathematics, if you want to say not rational, not a certain variable, you put a line over top of the letter. So if Q means rational, a line on top of a Q means not rational, shorthand in mathematics, right? That's all it is, okay? So the way I explain this, and the way I'm going to explain this to you guys is this, right? This is the way you have to look at the real number set. The real number set is human evolution, sort of explaining to you where, how we have come to be where we are in understanding numbers, right? In understanding mathematics. We write Z, so for Q you guys write Z, so Q could be Z, and it could be not Z, rational and non-rational. Thank you very much, lots. Okay, all right, mean use, right? So take a look at this thing. So if this is an explanation of where we have come from, right? Let's break it down into something that's tangible, something that people can relate to, right? I also want to show there should be a setting in your webcam setting. Depending on the computer, it could be in the various places you can turn off the autofocus function. I'm noticing it is getting in and out of focus, which would be a pain for future editing. Yeah, it does. It does get into, it does do that, but one thing I do, I bring things close. Like for example, here's my snack today, it's sort of melted, it's sunnier, so it's hot. This is hemp hearts, right? Take a look. This is hemp hearts on top of steel cut oats, breakfast that I made with fruit inside, fruit and nuts. So this is steel cut oats with apple, cherries, dates, I got pumpkin seeds in there and walnuts in there, and I put honey in here. So what I do, I sort of mix this thing around, right? And that's the reason I haven't turned off the autofocus, by the way, because I bring things close to the camera and whatnot. Thank you very much for the bits. Oh, no, not Joe. See, take a look. It's very delicious. Hopefully you can focus. Sometimes it has a hard time. Check that out, right? Super delicious. Super delicious. Okay, look at you being healthy. It's very good.  So the odds are today I'm just eating that, right? Oats are so good, yeah. Too many carbs, health are gone.   Very tasty, very tasty, very tasty. 420 per cup hemp. Nice. Ham parts on a ham. Nice. So take a look at this thing. Consider this to be human evolution, the real number set, right? So just imagine if we were, you know, hundreds of years ago, thousands of years ago, right? If we were living in a village, right? And just imagine there was a village of all sheepherders, right? That's what I use in my explanation in those videos, right? I say, if you were living in a village hundreds of years ago, thousands of years ago, whatever years ago it was, because you really don't know human history that well, right? Well, you could mix that for inspiration. Really, human history, we continuously keep on finding out that we have been on this planet in a form of a civilization or tribal community for not thousands of years, tens of thousands of years, but hundreds of thousands of years, right? So however far back you want to go, because if you look at some of the structures that we have in the world right now, right? From the Aztecs to the Mayans to the Egyptian pyramids to some of the structures in Asia and Europe and the Americas, some of the indigenous structures that they built with the sort of lining things up with the equinoxes and stuff like this, there was mathematics involved in that, right? So take yourself back to a time where you lived in a village and in a village of sheep herders, right? So everybody in that village had sheep and if you lived in a village, there was usually a central point where, you know, tavern or whatnot and this was your village and everybody in that village is a sheep herder, right? These are my stick figures, they're not very good, right? Oh, this poor guy doesn't have legs, right? So just imagine everybody in that village was a sheep herder, right? And whenever you wanted to get together, you came into the village, right? And let's say every Saturday night, everyone went into the village and they partied, right? You got everyone's partying, light up the place and dance, right? And then you would sit around the table and you talk about life and what do sheep herders talk about? Well, they talk about sheep, right?  Oh man, one of our sheep just had a sheep, right? We've got a little baby lamb, I think sheep is lamb, little lambs are baby sheep, right? I think so anyway, right? Oh, we can now say we have five sheep, right? And then Bob over here says, oh wow, that's great, I only have three sheep and, you know, Fred over here or Jonathan or Margaret or Elizabeth or whoever, well, it sits around and says, hey, I got eight sheep, right? So through that concept of sheep herding, you would have something called natural numbers where people had one, two, three, four, five, six, seven, eight, nine, 10, as many sheep as possible, right? That would call natural numbers, right? So natural numbers would be one, two, three, dot, dot, dot. If every family member in that village was a sheep herder, everybody would have at least one sheep and that would call the natural numbers, the counting numbers we call them, right? Now, just imagine there was some job low from outside the village that made its way to the village. He was a passer by, he was going through, right? Maybe he came from a different village. He came from a different village and goes to this village and Saturday night goes to the party house. We do everyone's partying, dancing, they sit around the table, they start talking. John, Fred, Margaret, Elizabeth, they start talking about their sheep and they turn to this guy because there's one big, gigantic round table. They sit around, talk to this guy and go, hey buddy, how many sheep do you have? And the guy goes, sheep. He goes, I've got no sheep, zero sheep. Now for these village of sheep herders, if they were living in a bubble, right, and they didn't know the concept of anything outside of their village, right? They couldn't understand what having no sheep meant, right? They would go, what do you mean no sheep, right? What, if you have no sheep, then what do you do, right? So this guy would go, well, I don't have any sheep, I'm a blacksmith, right? They go blacksmith, they go, yeah, you know, those metal things you buy, you put at the bottom of your horses feet, if you have any horses or the hammers you use and stuff like that. So these guys, they would just go, what, what, what. After a while, they would begin to understand that there was something beyond, beyond the natural numbers, right? And this would have been a bigger subset than this, right? So these natural numbers, one, two, three, four, five, six, seven, eight, nine, ten sheep, whatever sheep, all of a sudden beyond that, there would be something called zero sheep. The concept of zero would be introduced, right? Now, as, because this goes beyond the village capacity, right? It's another bigger subset, right? Because maybe, let's erase this, these guys would go, oh, you know what, outside of our village, right? There's a bigger area where there's people that have no sheep. They're blacksmiths, right? They have zero sheep and all of a sudden you would go, okay, let's call that the whole numbers and we include the number zero with the whole numbers. Hopefully you can see that, right? You include the number zero with the whole numbers.  Okay, let's take this down. Let's make it a little bit bigger so it gives us a little wiggle room, right? And apologies about not reading the chat, but I want to make this thing flow properly, okay? I don't want it to be interrupted too much, so my apologies again, okay? So let's assume we got this, right? What's this called? The real number set, right? What do we got? We're going to cut it down the middle, but keep in mind, this doesn't mean to have equal number of numbers here as there is here. It's just a visualization, right? And then we have the natural numbers, natural numbers, and these are one, two, three, dot, dot, dot, goes on forever. And then we got the whole numbers, whole numbers. We got zero, one, two, dot, dot, dot. Okay. And then we got the integers, integers, and then we got rational numbers, which is also written as z, and then we got irrational numbers, which is written with a bar on top, right? Right? So the whole numbers includes the natural numbers, right? It just adds the number zero. It's a larger area. It's a larger set, right? That's the whole numbers. Now, just imagine these people finally after a certain amount of time, they wrap their heads around what it means for people to have no sheep, right? They might have different jobs, right? They might have different things they do, so there's zero sheep involved, and zero means a lot. We'll come back to this, by the way. It's huge, okay? That's why we broke it, or mathematicians, they broke it into a new subset, because it was such a gigantic step in our understanding of the world and understanding in mathematics, right? And zero existed before. It was just us defining what the number zero meant, right? Now, just imagine humans are evolving, right? Population is growing, there's travel happening, and after a few hundred years, all of a sudden some person enters the scene, which is outside of this region, outside of the set, right? And goes to the village, party house on Saturday night, right? Les, how are you doing? Party house, and they dance, they drink, and they're gonna eat food, they all sit around the table, the big table that they have, and Bob, Fred, Elizabeth, Margaret, and all these people start talking about their sheep, and they also talk to this blacksmith, or whoever else that was here, maybe there's sheep herders here too, right? There's a sheep herder from here, that one in there, they ask this sheep herder, how many sheep do you have, right? So they start talking about their sheeps, and their blacksmith, and people that don't have any sheep, right? There are people that do other things that don't have any sheep, right? They talk about the number zero, right?  He goes, oh, I don't have any sheep. Oh, they go, oh, you don't have any sheep. You must be a blacksmith. And he goes, no, not a blacksmith. I'm not interested in blacksmithing. I'm a merchant. They go, what? What's a merchant? They go, well, I don't have any sheep, but I promised some people that I would deliver them sheep, so I'm negative to sheep, right? I need to get my hands on a couple of sheep so I can deliver it to somebody because I made them a promise there was going to be a trade the deal happening, right? And they go, what? You got negative sheep. You want to buy sheep, so you want to buy somewhere our sheep. They go, yeah, that's because I owe somebody sheep. They go, oh, what does that owe mean? It goes, well, it's negative. And all of a sudden, these people get introduced to the concept of negative numbers, right? Which is what integers are. Integers are negative natural numbers, right? But not just negative natural numbers. This subset, which includes these guys now, like a bigger subset, more information, these people here introduce, include the natural numbers, include the number zero, as well as negative all numbers, right? So integers are negative two, negative one, zero, one, two, dot, dot, dot, right? And these guys go, wow, there's people that are negative numbers, negative sheep. They need sheep. They cancel out positive sheep. If I have two sheep, they need to sheep. That means together we have zero sheep. What? It includes this guy, this guy, and the negative integers, right? So it takes a few hundred years, however long it takes, for this tribe to really be able to digest negative whole numbers, right? And then they start to realize how to deal with negative whole numbers in equations in their mathematics, right? Makes sense? That's where we are right now. And keep in mind, integers includes whole numbers and natural numbers, because this is human evolution. If we're evolving, we're not releasing information as we acquire additional information on top of that. We're building it, right? This is building the information, right? Well, what happens beyond this? Well, beyond this, there's people out here, people out here that maybe live in the city or something, right? Includes all of these people out here, right? The world is getting bigger and bigger. Human evolution, the world is getting bigger and bigger. These people come along on a Saturday night, right? They see this party. They see the lamp on the, what do you call that thing where you twirl it around and everybody's going around cooking it and eating it on the spitfire? Is that what you call it? Where people are eating and they're going, wow, we've never seen this before. We don't have this. Let's buy some of this product and take it back to our village, and eat it up, right? But the whole sheep is crazy. We want to buy half a sheep, right? So these guys are sitting around the table. They talk about their sheep. They talk about the number zero. They talk about people needing sheep, whole sheep, whole negative numbers, right? But then these guys come along and say, hey, listen, that's a gigantic sheep. We guys grow a good sheep here, right? We raise good sheep, but we don't, we can't do anything with a whole sheep. It's too much. We don't have the capacity to refrigerate it, and we don't have enough people to eat it before it goes bad. So I want half a sheep. And these people go, what in the world is half a sheep? No one's ever bought half a sheep from us. But what are you going to do with half a sheep? They go, well, we're going to cook it up and won't you just cook up a whole sheep? And he explains again. They go, okay, half a sheep. I don't know. Sure, I guess we could sell you half a sheep. And that's what rational numbers are, right? What rational numbers are, they would have explained to people that rational numbers are numbers that repeat or terminate. That's what they explain to everybody in school, what rational numbers are. These numbers are numbers that repeat or terminate, right? Repeat or end. I don't like that definition. It's a working definition. It's okay. But let's take this guy out. What are rational numbers? Rational numbers are numbers that you can write as fractions of integers. So rational numbers are fractions of integers. And fractions of integers tend to end and repeat, so for example, the number two is a natural number. But when something is a natural number, it's also a whole number. It's also an integer. It's also a rational number. Why is it a rational number? Because the number two you can write as two over one. One over three, negative seven over five, 0.3. What's 0.3? 0.3 is three over 10. What's 0.3 repeating? 0.3 repeating is one over three, right? So any number that repeats or terminates is considered to be a rational number and belongs in this subset. Now rational numbers, as you can tell, if it's a number two, it can be a natural number. But two over one is also a rational number. So whenever you're trying to explain what a number is, which subset it belongs to, all you have to do is take it down to the lowest subset that it can go in the pillar. So if someone asks you what the number two is, you say it's a natural number because that implies it's also a whole number, integer, and a rational number.  It's a whole number. It's not a natural number. And when it's a whole number, it's also an integer and a rational number and a real number. So for example, the number zero, the number zero, you could just write down zero over one. Zero over one is just zero. If they say, hey, what's negative, what's the number negative three? Negative three can be written over negative three over one, which is a rational number, but you don't need the one. So negative three goes down to the integers. So you always have to take it down to the base subset that it belongs to. That's what rational numbers are. What are irrational numbers? What are these guys? Well, if these guys are numbers you can write as fractions of integers, these guys are numbers you cannot write as fractions of integers. Can not write as fractions of integers. Okay, what are numbers you cannot write as fractions of integers? Pi is one. The root of any prime number is an irrational number. It doesn't end and it doesn't terminate, right? So pi, for example, is 3.1415, I believe, dot, dot, dot. So there's no pattern beyond this, right? It doesn't repeat. Rational could be 3.1415, all repeating, which would mean 1415, 1415. That would be an irrational number. If it doesn't, there's no pattern. If it doesn't repeat, it's an irrational number. Okay, that's what the real number set is. Now, why is this important? How does this play out? Let me show you where some of these boundaries occur and what they mean. Okay, interesting, a very convenient table. I think I'm starting to remember the school curriculum. Nice. Now, I'm going to erase these guys, our sets, our worlds, our villages that we created. Okay, let's take these guys down. What you can ask yourself is this. Let me take this number out too so people don't think it's a magic number, right? I got a new, new thing. I got to wash the other one. So, so you ask yourself, what's the big deal? So what? We went from 1, 2, 3, all the way up to infinity to 0, 1, 2, 3, all the way up to infinity. What's the big deal? Why have a boundary? Why have this line here going from here to here, right? Why not just write it down as color whole numbers, right? What's the big deal? It's just one number. It's just a number zero. Well, here's the big deal. The number zero is huge, gigantic, enormous. If the number zero didn't have the properties that it does, right, there's two main properties that the number zero has, okay? If it didn't have those properties, then mathematics would not be what it is and the limitations of mathematics would not be what they are, right? Here's the thing with zero. This is why we make a distinction. As soon as we were able to define the number zero, and I believe it was someone in India that defined the number zero, okay? Here's the problem with zero. The problem with zero is the limitation of mathematics is the only, as far as I know, the only limitation of mathematics is we cannot divide by zero. So this step from here to here introduces two things. Number one, no dividing by zero. The universe explodes when we divide by zero, okay? It is an impossibility according to the mathematics that we know. In this realm that we exist in, in the realm of mathematics, the most powerful language that we've been able to come up with to try to understand the world around us, okay? Even this amazingly powerful language, right? The most used language on earth between humans and humans, humans and machines and machines and machines, right? In this language that we're able to use to send people to the moon to put satellites into orbit, to send satellites outside of our solar system, right? With this language that we've been able to come up with, we can't do one thing. We can't divide by zero, right? No dividing by zero. Limitation. That's a huge boundary, by the way. Huge boundary. Here's the other power of mathematics. And by the way, there's a video I put out called, if you do chicho, travel at the speed of light. Why we can't travel at the speed of light? Chicho, why we can't travel? All right, I'm just gonna say it's speed of light. Chicho, speed of light. Let me find it for you. That way I'll link it up. Here's the video, okay? For those of you in chat, if you want to keep track of this, let me link it up in chat. And I'll provide in the description of this video once we load this thing up, okay? In that video, I go through explaining how I finally was able to understand why we can't travel at the speed of light. It's because we get a division by zero. In Einstein's paper on the electrodynamics of moving bodies, right? If you've ever gone to school or if you've ever watched science fiction, if you haven't had your head in the sand for the last 50 years or 80 years or 100 years, you would have heard that we can't travel at the speed of light. It's a limitation that we have in physics, in this physical life that we live, what we can understand. Now, in science fiction, you've got something called warp speed where you can bend space or whatever it is, right? Okay. Now, the reason we can't travel at the speed of light is because in the equations that Einstein presented in the paper called electrodynamics of moving bodies, if we travel at the speed of light, we get a division by zero in the equations and the world universe explodes. Boom. That's it. Simple as that. Okay. Now, this is the problem with zero, right? It explains to us the limitations of mathematics. Here's the power of zero, two. The power of zero allows us to solve equations. The power of zero, zero, allows us to, let's just put it here, solve equations, right? How does it allow us to do that? Here's how. If we have simple equations, if we have just one variable equations, for example, x plus two is equal to zero or equal to eight, then you subtract two, subtract two, you get x is equal to six. That's easy, right? x is equal to six. Well, as we evolve, right, human beings evolve, as we evolve, right, we start to deal with more complex problems in our lives, right? So, we get more complex equations in our lives. So, for example, let's assume we have the following equation. x squared plus five x plus six is equal to zero. Now, for us to be able to solve this, we can't just move these guys around. If we move these guys over there, we're going to get x squared is equal to negative five x minus six. But to solve an equation means get x by itself. Right now, how are we going to get x by itself? We can't just do it by adding and subtracting and stuff like this. What we end up doing is, we do something called factoring, right? We ask ourselves, what are two numbers that multiply to give you six, add to give you five, and you can break this thing up. Here, let me show you how this works, the mindset behind it, right? You can break this thing up and go, oh, this can be broken up into x plus three times x plus two, right? So, we've got two things multiplied together to give us zero. Okay? Yeah, lost 69, 69, 69. The problem is, for sure you can do that, but you still can't be able to get x by itself. So, for example, law said we could do this with this equation, right? We could factor out an x here, and we've got x plus five plus six is equal to zero. The problem with this is, we still have to solve for x, right? This goes to there. But how do you get x by itself? You've got two x's here, it doesn't work, right? And I'll show you this. I'm going to keep this. I'll show you where the thing occurs, right? So, here's the power of zero. This, keep this in mind, okay? Power of zero. Power of zero is this. Power of zero tells us this. And by the way, we've got a video on this. So, if you do chi-cho, power of zero, let me get that for you as well. Power of zero, okay? I put out a full-on video just on this. The power of zero, because it's an extremely important concept, right? Without this power, we couldn't solve complex equations. It'd be impossible, right? Here's a link. Power of zero. Payapa. Thank you very much for hosting. Let me get myself back to where we're supposed to be. So, check this out. This is the power of zero, right? And that video goes through it. It's like a 45-minute video going through it. But we're going to do a speedy Gonzalez start, right? This is the power of zero.  How can you have a whole bunch of stuff multiplied together to give you zero? How is that possible? How could you have two things? A times B equals zero. How could you multiply two things to give you zero? Well, the only way that's possible is if at least one of them is zero. If at least, and at least is important, at least A, B, C, D, E, or F, at least one of them is zero, right? We don't know which one is zero or which one could be zero, so we set them all equal to zero. So we say, oh, in this situation, A can equal zero, B can equal zero, C can equal zero, D can equal zero, E can equal zero, and F can equal zero. That's the only way that's possible or a combination of all those, right? So it could be A and C are equal to zero, and B, D, E, and F can be whatever they want to be. It doesn't make a difference. Whatever number you have times zero is zero, right? That's the only way it's possible. This is the power of zero. Why is it the power of zero? Because zero is the only number that works with. You can't say, hey, how is it possible to have A, B, C, D, E, F times together to give you two, right? If you have that, you can't say, oh, the only way that's possible, if at least one of them is two. That's not true, right? You can't say, oh, the only way it's possible if one of them is two. That is absolutely not true, right? You could make A two, and the rest zero, and you end up getting zero, not two. You could make one two, all the other ones, one, except for F, and make F a gazillion, and you get a gazillion. Like it doesn't work, right? If you make all of them two, it's just two, two, two, two, two, four, eight, 16, 32, 64. That equals 64, so it doesn't make sense, right? The only way this is possible is if at least one of them is zero, right? If this thing is equal to zero and at least one of them is zero. Well, that's the way we end up solving this equation, right? We just took something that we couldn't break down and factored it into two things multiplied together to give us zero. That means we can split it up and set each one equal to zero. X plus two is equal to zero, so X is equal to negative three, and X is equal to negative two. So if we plug in X is equal to negative three for X in this equation, this side equals zero, same with negative two, right? Now check this out. Over here, we had someone, I forget who it was, say, oh, let's factor it like this. Let me rewrite this. I should have moved this over a little so we had a little room, but I'll try to write it so you can see it. You can factor out an X from here, and you're going to have X plus five, and then you got plus six is equal to zero, right? Here, let's draw a line here so we don't get confused about this, right? Well, what we can do is if we want to continue to solve this using the move the six over becomes negative six. So we got X times X plus five is equal to negative six. So we got two things multiplied together to give us a negative six. Is it true that either X is equal to negative six or X plus five is equal to negative six? Absolutely not. You can't say, oh, this means that either X is negative six or X plus five is equal to negative six. So this would be X is equal to negative 11 and X is equal to negative six. That's what we get if we did this, right? Well, if you plug in negative six for X here, you're not going to get this side equal to zero. It's not going to happen, is it? Negative six squared, negative six, here, watch this. Negative six squared plus five times negative six plus six. Well, negative six squared is 36. Five times negative six is negative 30. 36 plus negative 36. Six plus six is 12. It doesn't work, right? It doesn't work. Oh no, not Joe. Thank you very much for the bits. And someone else was doing some bits. So thank you very much. If I'm not noticing it, and thank you for the subs and thank you for the follows and stuff. But when I'm doing a little bit of mathematics, I like to make sure we don't lose the train of thought because if I get distracted, sort of sometimes lose my train of thought game, okay? Is that clear? So check this out. What just happened? And by the way, you can expand this to huge functions, right? If we do this, for example, here, you could do this. I'm not going to write down the polynomial thing for it, but if you could have this, x plus two, three x minus one, two x plus one. Let's put an x up front. Five x minus one is equal to zero. Well, you got one, two, three, four, five things multiplied to give you zero. So you can set each one equal to zero, right? x is equal to zero, x plus two is equal to zero, three x minus one is equal to zero, two x plus one is equal to zero, and five x minus one is equal to zero. So this is x is equal to one over five, x is equal to negative one over two, x is equal to one over three, x is equal to negative two, and x is equal to zero. So whatever function this creates, as a polynomial function, those would be the solutions for that polynomial function. Crazy! We couldn't do this if we didn't have this power of zero, right? That's why we have a boundary between natural numbers and whole numbers. It's pretty important. It's huge. It's gigantic. Really. It's humongous, right? Wow! Things we cannot do with zero. And as soon as we get introduced to something, now we can solve equations. Very cool. Very cool. The fundamental theorem of algebra is exactly that a degree n polynomial can be factored in n linear factors. Yeah, racer kill. How are you doing? Only possible if if you use complex numbers. Yeah, we need the complex numbers as well, right? So some of these things might not have a solution, right? That's the reason we have a distinct boundary between natural numbers and whole numbers. That's the reason we came up with the concept of whole numbers. Why do we have this? Whole numbers to integers. Integers. Negative whole numbers. Well, negative whole numbers introduce or negative, well, negative whole numbers introduce another problem in mathematics. That's why we have this boundary, right? There's something called, they called it, they used to call it imaginary numbers. The reason they called it imaginary numbers is because from what I understand, racer kill knows the history of mathematics better than I do. So few other people I think. But the way I understand it, the reason we called things imaginary numbers, and I'm going to show you what they are, is because mathematicians used to think that was a byproduct of the mathematics. They didn't realize what imaginary numbers meant, or if they existed in the real world. Later on, we found out that imaginary numbers were something that was part of our reality, so they started calling it complex numbers, right? And the issue here is this, okay? The boundary between whole numbers and integers. Let's call this the boundary between natural, here, let's write it out properly so it goes in the right direction. Natural numbers to whole numbers, right? Well, what's this boundary between whole numbers and integers? The boundary between whole numbers and integers, think about it this way. It introduces complex numbers, racer kill. A complex number is a number of the form a plus bi, where a and b are real, and i is the imaginary unit. So basically here, I'll explain to you guys what that is, how we write it out, right? But ask yourself this. So basically here, I'm going to say for the boundary here, it introduces the concept of complex numbers. So introduction of the concept of complex numbers, and those numbers are a plus bi, right? Where a and b are the element of, are the element of the real number set, and i is defined as a square root of negative one. Okay? Now, what does that mean? Well, check this out. Was first used for solving degree three polynomials. Is that where it was used first? That's cool. And degree three polynomials are x cube plus something, something. Degree three polynomials could be something like this. f of x is equal to x cube plus 2x squared plus 1x plus 2, right? That's a polynomial to degree three, right? Cubic polynomial, I guess you can think about it that way. But for sure, that's what it would have been. That's right, right? That would have been one of the first places you encountered. But basically, you can think about it this way. How to derive the value of i. Here's where i or complex numbers comes in. Now, ask yourself, what's the square root of four? Right? What's the square root of four? Question. What's the square root of four? Now, square root means that you're looking for two numbers that are identical. Well, I think to note is that negative one has two distinct square roots. Negative one has two, plus or minus, yeah. Two. That's half your answer, by the way, lost, right? Because in the square root symbol here, you're really asking yourself, break down four into two numbers that multiply to give you four, yeah, plus or minus two, right? So two numbers that multiply to give you four are two times two, right? One times four, negative one times negative four, and negative two times negative two, right? Let me write this bigger. Negative two times negative two. So those are the combinations you can have, right? Square root says if you find two numbers inside the square root symbol that are identical, you can bring them out as a single, right? So you can write the square root of four either as square root of two times two, or square root of one times four, or square root of negative one times negative four, or the square root of negative two times negative two. Now, these two aren't identical, so they're irrelevant. So we can just take them off the board, right? So let's take these guys off. How do you like my little eraser thing? More precise. So we can say with two numbers identical, square root of four can be, you can break it down into two times two, or negative two times negative two. And square root means if you have two things that are identical, you can bring them out as one thing, right?  So it's just two. You can grab two, negative two, bring them out as negative two. You got nothing left on the inside, so it's negative two, right? So square root of four is plus or minus two. I'm going to erase this. We need the space, right? I'm going to erase this. So square root of four is plus or minus two, right? Plus or minus two. Well, ask yourself this. Question B, what's the square root of negative four? Well, negative four, we're looking for square root, right? So we don't care about one times four or negative one times four and stuff like this. For real numbers, the square root of A is typically defined as the positive number, which squares to A. For a complex number, it's not as easy to define the square root, but there are ways to do it. Is there? Okay. Thanks, RacerKill. So check this out. Square root of negative four. So we'll forget about the one and four multiplied together to give you four, right? But we can do this. We can say square root of four can be negative two times two, right? Or it could be two times negative two. But these numbers aren't identical. So we can't just grab a negative two and a two and bring it out as positive two or negative two. It doesn't work, right? So what can we do with this? What we can do is say, okay, we want to create two identical numbers. So separate the negative from the two. You can multiply any number by one and not change it, right? So we could say square root of negative four, if you break down four, it's two times two times negative one, right? Then what you could do is say, oh, two times two, those are identical. I can bring these guys out as a two, right? And inside the square root symbol, we've got negative one. So we've got the square root of negative one, right? And there are infinite number of numbers that behave this way, right? Square root or even root of any negative numbers is the root of that number times square root of i. Now because this occurs an infinite number of times, we have a special name for the square root of i and we call it square root of negative one and we call it i, right? So we can say define i right here as a square root of negative one, so this just becomes two i, right? So why is this boundary between whole numbers and integers so important? Because it introduced the concept of complex numbers, imaginary numbers at the time, but later on we find out it's complex numbers which is crazy cool. Wow! We might as well create a new subset and call that integers because it's a huge leap in human evolution, in human thought, human understanding, right? Crazy cool, crazy cool, right? Okay. What else we got in this real number set? What else we got in here? Let's take a look at something called prime numbers, okay? An important thing with complex numbers is that solutions to polynomials with complex coefficients will always have complex solutions. Ah, really? Okay, cool. That's super cool to know as well. So basically when you had a polynomial, if you had a complex number there, it always would have a complex solution, which isn't true for real numbers, obviously, which isn't true for real numbers, obviously. But real numbers can, the polynomial can jump into the complex numbers. Is it only going to be complex numbers raised to kill? By the way, polynomial complex coefficients will always have, is it, have complex solutions only or will they also have real solutions? I'm assuming they also have real solutions, right? Is that true?  Because we're going to need the space to talk about something called prime numbers. Sure, but a real number is also a complex number, but a real number is also a complex number. As long as b is zero, right? Is that what you're talking about? When b is zero, it could be expressed as a complex number. Is that true? Is that right? Race or kill? We'll wait until he answers that. Yes, that's what I mean. b is zero. Yeah. So as, because you could set b is zero and that's part of the real number set. So the i disappears, so you have your real number, right? Which is cool. Which is cool. Let's take this down. Let's look at something called the prime numbers because prime numbers are really the driving mechanism behind the real number set. So we haven't even got to the driving mechanism of the real number set yet. Crazy cool, right? That's how important this thing is. 73, my favorite prime number, is it? Nice. What's my favorite number? Who wins the prize? What's my favorite number? Let's kill this. 69, 3, no, no. There's somebody here with that number. Death 420. You, you didn't guess my number? 420. I think it officially became my favorite number a few streams ago. I can't think traditional 420, right? Is that stream usually slightly behind the chat, whereas an issue on my end? Is it, it could be, there could be, well yeah, there is usually chat in real if you're watching on Twitch. It takes a little delay before it pops up under the screen here. Okay, you're on, it's very fast here, is it? That's not prime. 73 is not, oh, 243? No, 243 is not prime. Have you ever seen Karl Monk video about math and ancient history? I don't think so. I don't think so. Now, what's the driving force behind the real number set? We're getting sun coming in here. How far can it go? Hopefully, we won't block this out, most of it. I got my, by the way, I got my, someone said, you shouldn't do this. I got my umbrella set up here because we got a skylight and the timing is, the sun sometimes blacks out the board or whites out the board, right? He only asks his favorite number, not his favorite, yeah, my own favorite number, not my favorite prime. Favorite is 0, Sam hits. 0 is a good number, very, very powerful number, right? So, what are prime numbers? What are prime numbers? Prime numbers. Numbers. Now, prime numbers are numbers that can only be divided evenly by one and themselves, right? So, for example, just imagine you have to follow numbers from number one to 20, right? One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, right? I could ask you, how many unique numbers do we have here, right? Now, unique numbers, you could think of 20 as being a unique number if you want, but it's not. It's made up of prime numbers multiplied together to give you 20, right? So, number one is a special number. It's not a prime number. Number two breaks down into two times one. Number three breaks down into three times one. Number four can be broken down into, we talked about it, one times four, four times one, negative four times, negative negative and all that jazz, but we're talking about natural numbers for now. We're only talking about positive numbers, right? There's only one 20, but you can non-prime though. I say it's not unique. I know people are going to burn me for this. 20 must be unique. No, it's not unique. 20 is made up of prime numbers multiplied together. It's not an atom. It's a molecule, right? So, it's not an element if you want to think about it in chemistry, right? It's made up of elements, right? So, think of prime numbers as you're building blocks of nature, right? But they're building blocks of numbers, okay? So, if you look at the periodic table of elements, that's what prime numbers would be in mathematics and there's an infinite number of prime numbers, right? Number four can be broken down to two times two. That's two prime numbers multiplied together to give you four. Number five is one times five, so it's a unique number. So, what we're going to do to make sure this is not messy, we're not going to split any number that is just one times itself because that's the definition of a prime number, right? So, what we're going to do is just circle the ones that only break down into one times themselves. Prime numbers are certainly important in some fields of math, but they are also irrelevant and also irrelevant in others, for sure. There must be, right? But they're really, really important in all of mathematics in high school. There's no doubt about a racicle, right? And they're ridiculously important in relation to the real number set, right? For other number sets and stuff like this and very complex mathematics, I'm pretty sure prime numbers probably don't have a huge part to play in it. I haven't gone that level, right? One, seven, five, seven, thirteen. Yeah. One is not a prime number. Consider it just to be a unique number. I'm not sure what the name for one is. Racer killed. Do you know what what they call one? I just call it a special number. Unique. Unique. So, two is a prime number. Three is a prime number. Four is not. Five is. Six can be broken down to two times three. Seven is a prime number. Eight can be broken down to two, two, two, made up of prime numbers, right? Mirror about math and ancient civilization per flood. I.D. Imagine. Imagine. Nine is three times three. Ten is two times five. You're noticing that these numbers, if they're just multiplied together to give you, they're prime numbers multiplied together to give you that number, these numbers are called composite numbers. They're made up of prime numbers multiplied together. Identity, unity. But, yes, it's not a prime number for convenience space. Yeah, for convenience, basically. I used to include it in my prime numbers, but it's not, right? So, unity, identity. Okay. All right, so far. Cool. And then you got eleven. Twelve is two times six, and six is two times three. So, it's just prime numbers multiplied together. Thirteen is a prime number. Fourteen is two times seven. Fifteen is three times five. Sixteen is two times eight. Eight is two, two, two. So, it's just four. Two is multiplied to give you sixteen. Wait. Two is not. Is not. Is it? Yeah, two is a prime number, for sure. Two is the only even prime number, because the only way you can multiply two natural numbers to give you two is one times two, and that's the definition of prime number, right? Numbers that can only be divided evenly by one and themselves. Numbers that can only be, actually, we should call it natural numbers. Natural numbers that can only be divided, divided evenly by one and themselves. Apologies about my writing, but from saying it, hopefully. You know what it says. Seventeen is prime. Two times nine. Three times three. Oh, sorry. Yeah. Three times three. Nineteen is prime. Twenty is two times ten. Two times five. Right? So, from the number one to twenty, we have twenty numbers, but we have one, two, three, four, five, six, seven, eight, nine building blocks. Right? Nine prime numbers. Okay. Catholic traditionalist, primes are a very important number of computer science applications. Cryptography, hashing, tax scheduling, etc. Yeah. Like, for example, right now, there's no way we could have been streaming, right? If we didn't have prime numbers, because prime numbers are the key building blocks of all everyone's passwords. Okay? Prime numbers is the reason. By the way, correct me if I'm wrong. Those of you who know this level of mathematics much better than me, right? But prime numbers is the reason we're able to do banking online. Prime numbers is the reason why we have passwords, why we can have secure communication, right? Prime numbers are it. Okay. Did you post the video link in Discord? Yeah. Nice, tasty. You're talking about the same thing. You're pushing this thing, by the way. So, I would recommend you don't push it too far, because you might get timed out, because you're totally off topic talking about this stuff, right? So, if you want to share information, we have a Discord page. Go to our Discord folder, Math folder, and post it. You're very welcome to share information, right? But if you're trying to hijack a discussion, it becomes silly. Okay. Awesome. We'll do. Thank you. Awesome. There are other ways to Crip, but you're kind of right. There are other ways to Crip, yeah? Yeah. I want to learn some of that stuff, man. For some reason, I'm really interested in that stuff. There is a reason for it, by the way. Credit card numbers are prime secure. Yeah.  Quantum computing rolled out on a mass scale, then no password is secure anywhere, right? I'm pretty sure about that. But again, people can correct me if I'm wrong on that deal, right? So, prime numbers are the building blocks of all the real numbers. That's what we get from this, right? Prime numbers are the building blocks of the real number set. And why are they important? Well, they come into play here between irrational and irrational numbers. They help us reduce fractions, right? Right, but no, I can't, right? So any root of any prime number quantum computing is necessary for the true free exchange of information. It's, it will, the problem is it's gonna be rolled out and only a, it's gonna be not available to me and you. It's gonna be initially only available to the certain small sector of society. They're too slow, yeah. Quantum computing, yeah. Thank you very much, Graham. I was about to hop in there. Chicho, the trash man. Chicho, could you use quantum computing to encrypt at a more complex level? Also, I believe so. You must be able to, right? Or is it a laws of mathematics thing? I, you know what? Once quantum computing comes in, I think what's gonna happen is you can throw the rulebook away. We really don't know what's gonna happen. We really don't. Okay. Thank you very much, Catholic traditionalists. Makes sense. Using prime numbers keeps fractions and ratios simplified by their nature. Exactly Utah Jazz, right? There are algorithms which are better against quantum computer methods. Is there a race to kill? Not only Piggy, how are you doing? What's up, what's up? Talking about the real numbers and the prime numbers. Sir, how can we get a negative number from prime numbers? How can we get negative numbers from prime numbers? You multiply by negative one. Remember, one is a special number, right? This guy here is a special number. Or what did you call it? What did you call it? Racer kill? You called it unity? What did you call it? Africa? What you called it? The thing is that quantum computer algorithms are more efficient at some things. Prime factorization is one thing. Okay. So basically what happens is because there is, and by the way, the reason that prime numbers are used for encryption, okay, secure communication, is because there are infinite number of prime numbers and no one's been able to, and from what I understand, the proof is out there that there's no pattern to prime numbers, right? So prime numbers, there's no pattern to when they appear, right? So whenever you find a prime number, there's a huge number of prime numbers that have been discovered, but there are new prime numbers being discovered all the time. And those new prime numbers are unique things that we discovered. It's like finding a brand new element in the world, right? You just found a new prime number. Incredible, right? It's a very important, very important unit. So prime numbers are in units, right? Quantum convenience seems dangerous, seems dangerous, Allah God. I agree with Racer Kill. I think there are ways to tie up systems enough to secure things, but if you really want to want to be secure, you can continue to store our data on local systems and five 24 floppy desk grab unit. Would that change the game of Bitcoin farming? I believe so. I believe so, bust, bust-starten, bust-starten, or bust. I believe it's going to change the game for a Bitcoin and mining. I'm not 100% sure on it, but from what I'm understanding, it's going to change the game, right? So the question is, how is it going to change the game? It would be like inflation and zombs, if it becomes too easy, when the value yes, and that is the thing, Trashman. Is negative 2 prime? No. Geno, we don't consider negative numbers to be prime. Negative numbers, 2 is a prime. Negative 2 is the prime number times negative 1, right? So Racer Kill can clarify this, or Catholic traditionalist, but you don't have negative primes. You just have positive primes. Basically, prime numbers are natural numbers that can only be divided evenly by one of themselves, not just integers, right? So natural numbers. So we kick it down all the way to the bottom, right? It does follow a general trend. Prime numbers? It's not exact, though. It's not exact, right? There's twin primes and stuff like this. Actually, a unit is a number which has an inverse. The identity is one. I think that, oh, is that what it is? So what? No, that couldn't be it. I'm thinking something else. I like the Carl... Oh, dude. Nice. Great minds. I have to admit that the fact that prime numbers are infinite just hurt my brain. Yeah, they're infinite and unpredictable, right? There are certain patterns, I guess. There are certain things that you can sort of get in the red ballpark, but you still have to, what do you call it? Force it, right? Rigor, right? All right. Thank you. My pleasure. Kind of yes, but usually is defined as the positive number. Yeah, Gino. That's the way I understand it as well, for prime numbers. Racical. I have just started looking at things like lattice base cryptography. Oh, Catholic tradition is getting me excited. I wish I was there, man. I wish I was there. There are infinite number of prime numbers, but prime numbers are infinite. Doesn't mean anything. Oh, okay. Let's rephrase that. There are an infinite number of number of prime numbers, but prime numbers are infinite. Doesn't mean... Okay, cool. Right? So I worded it slightly wrong. Okay, you touch us. Yeah, it just seems wrong. Two is a prime number. I must write this down. And it is the only even prime number. So it makes it even super special, right? But my memory is fighting it. I know. It's because it's so nice. There's a couple. It's two. It's even. How could it be a prime number if it's even? But it is, right? But it is, which is fantastic. But the number of prime numbers below X is asymptotically to X. Oh, really? I didn't know that race to kill. That's a very famous theorem called the prime number theorem. Well, I must have read this some time ago, but I totally forgot about it. So the number of prime numbers below X is asymptotically X over log X. So it's a function. It becomes asymptote. So the number of prime numbers, as you on a Cartesian coordinate system, go up really fast. There's a lot of prime numbers initially, right? But then it becomes asymptotic. So think of it this way. For us to be able to create the number one to 20, we need nine numbers. We need one, two, three, four, five, six, seven, eight, nine. With these nine numbers, well, not even these nine numbers. With these, with one, two, three, four numbers, we can create the numbers up to 20, right? Or up to 20, there are only nine prime numbers, right? So there's a handful of prime numbers all the way up to 20. If you go to 100, there might be a couple of dozen prime numbers, not even, right? If you go to a thousand, there might be a couple dozen or a thousand, way less than 100 prime numbers, right? If you go to 10,000, maybe you're breaking 100 prime numbers. If you go to 100,000, maybe you're into a few hundred prime numbers, right? If you go to a million numbers, maybe you're into a couple of thousand prime numbers, right? So just imagine from the number one to number one million, there's a million numbers, but you could generate all those million numbers by a few thousand numbers. So right there, you don't need a million database. You can crunch all that down to a thousand database, a few thousand database. So it makes it a lot smaller, it makes it more manageable, right? Incredibly powerful. Racer kills are gods like racer kills.  You could think it in terms of probability. If you pick an integer in the interval one to X, there's a probability one over log X that it is a prime number. Ah, cool. One over so thousand. Okay, okay. Oh, I don't want to graph this. I don't want to graph this. I want to have this up. So it's like the opposite of exponential. Logarithms, yeah, in large part, right? Special type of exponential anyway. Is that why they work so well for password protection? Yeah, exactly. You have to go through so many numbers to find prime, so many numbers to find primes. Credit card security again, credit card security again, right? Super cool stuff, super cool stuff. This is actually what we just went through is actually more in depth than what I did in my real number set, the first videos I put out. But I did link this up with I think a couple of years later, I created another video or set of videos talking about the square root of prime numbers and the power of zero and irrational numbers, not irrational numbers, well, irrational numbers, but integers and imaginary numbers are complex numbers. So link that up in that way as well when we're talking about square roots and radicals and stuff like this. So it's pretty cool. It's pretty cool. Super cool stuff. I still don't know what you're with stand for. I have no idea what that means. 15 is not prime. 15 is not prime because 15 can be broken into three times five. So it's made up of two prime numbers multiplied together. Oh, whole number set. Know what your whole stands for. Whole number set is just the natural numbers plus the number zero, right? Yeah, natural numbers plus the number zero. All numbers ending in an even number, not prime then. Yes, there are no even prime numbers. The only even prime number is two because any even number can be divided by two. Right? Log X grows slowly. So prime numbers become more rare, but the rarity doesn't grow that fast. But the rarity doesn't grow that fast. Okay. What is zero prime? Zero factor is one. Zero factorial is one, isn't it? I'm pretty sure it's one. You're catching me at time when I'm thinking about this. Expect two, sure. And five. Yeah. Would X over log X be a ratio? Yeah, for sure. Any fraction is a ratio.  Yeah, any ratio is just any fraction. I think a lot of mathematicians say zero is a natural number. Do they? Did he put it in here? I've always learned it here as a whole number. Gina. Well, it's number divided by another, but it's likely not going to be rational. Oh, did you say ration? No, ratio. Yeah, ratio. Gives that. Okay. So prime numbers can't be made from two evens and any numbers times of stuff. Yeah. Prime numbers can only be made up. If you're taking two numbers to multiply to get your number, if the only two numbers that you can multiply to get that number is one and that number is considered a prime. If any other two numbers multiply to give you that number other than one and the number itself, then it's not a prime. It's called a composite number. She shows right. Whole numbers include zero. Naturals doesn't. Yeah. Yeah. Fractions and ratios are just different ways of expressing the same thing. Yeah. But their interpretation is different. Fractions are part of a whole. Ratios is a comparison between two things. Right? Race or kill. Good with numbered patterns, but not maths really. Yeah. Elder God is good with patterns. Problem solving is crazy fast. Race or kill. Zero being natural or not, or not depends. If you read a book, you're going to have to check what convention that author uses. Okay, cool. Good to know. I actually didn't know that they're mathematicians that consider zero to be part of natural numbers. I always associated with the whole numbers. It's just an introduction because it's such a huge step in human knowledge, evolution, right? But it won't be a constant, right? Because x moves. Yeah, it won't be a constant. Are you talking about the Gina, the prime numbers increasing? Proxy King. I think now has changed. Now zero comes in natural numbers. Oh, does it? Oh, I'm too old school then. I'm too old school then. Trash math. Fascinating how these things can change through interpretation. Yeah, really. Utah Jazz. Also, I got C's in high school math and never took math in college, so I'm very out of my element. Utah Jazz, the C's really don't mean too much. You might have understood the mathematics really well. It's just not the not you can perform in the tests that they were giving you, right? Real elongated musk. It is 1 AM. Oh, God. You got lost in the stream. Funny learning balance. If you include zero, then you get a monoid under addition. So it is nicer algebraic structure. Okay, cool. I feel like I keep on getting little bits of understanding, but my brain is mostly frog. What is that? A West frog. Trash one. Sometimes people write n underscore zero for including zero. But yeah, you're going to have to check for every source. Not every person uses the same notation. Yeah. I took a college course in math two years ago, and the natural numbers included zero. So cool. We see. Can you earn money finding undiscovered prime numbers? I believe so. I believe so. There is notoriety, notoriety for sure. And from what I understand, RacerKill probably knows this. There are websites where they give you money for proofs. I'm pretty sure there's going to be a website that you can generate money by finding primes. Gina, yeah, I was thinking maybe X over log X would trend to a constant value, example Fubunuchi. Yeah, it would continue to increase though. It still continues to increase, but very slowly. Right. I will argue for the side till the death. No, perhaps if you find newest largest one. Yeah, new and the RacerKill. If they found the largest one, I'm pretty sure there are prime numbers all the way up to the largest ones that haven't been discovered yet. Right. Or are their algorithms just going through just grunt, just forcing it trying to find prime numbers. So the largest prime number, no, because there would be people trying to find prime numbers jumping the algorithms. Right. So they must be unknown prime numbers that are smaller than the largest known prime number. I don't think it's bounded, but it grows slowly, it grows slowly. Hi, everyone. Hope you're all doing fine. Crazy bro, Athen. How are you doing? You've taught jazz. Discovering new prime numbers seems pretty similar to discovering new digits of Pi. X grows much faster than log X, regarding the asymptotic. Right. So log X, log X is just a Y value of it really, but X grows much faster. Oh, P versus NP. 34 to 57 viewers. Low high this stream. Ah, nice older guy. Fun. We're kicking up closer to the 75 average. Right. Funny. Twitch. RacerKill. Well, there are so many primes. There are like billions or trillions of prime numbers below the largest one we know. Okay. So it doesn't, it hasn't gone sequentially. There's billions or trillions below the largest one we know. Really. It's no use having a list of that. No, no use at all. Crazy bro, Athen. It's doing fine. Thanks for asking. Hope you're also doing fine in these weird times of changes. Yeah, doing fine. I'm doing much better than most. There are people that are in major turmoil areas. So I wish them the best. Right. There are some people that have to completely reevaluate their existence because their world is shattering. Right. So yeah, being, knowing what's going on politically economically in the world is a stabilizing force. Is a stabilizing element in your life, which is why I push the concept of politics, economics, and specifically looking, looking at our world through the realm of mathematics to make people more anti-fragile. Right. VC. I wish, I wish you could answer that question. What did VC ask? Is P equal to NP? Is P equal to NP? Well, it looks more like it's in the tens of millions of primes below the largest. The point stands though, the point stands though. So it's not trillions. It's tens of millions of primes, most likely below the largest prime known. Very cool. So tens of millions of unknown primes below the largest prime known. Right. Math, best place to rest. Fun, fun, fun stuff. Great question, by the way. We hadn't done the real numbers yet for a long time. We just went off on it. Could it, could it possible to develop a computer program that could, that could find unknown prime numbers? People have been trying to work on that forever and there are programs running, I believe, trying to find new prime numbers, but it's more a rigor. There's no pattern to it. Right. So it's trial and error and zooming into an area and trying to find the primes. Right. Did you see what Spain did? They start to give fixed income to 2 million poorest people of Spain and they want to cover whole populations so that everyone get a fixed income. That would be cool, actually, even though I'm against universal basic income, by the way, because right now it's a solution, right? Long term, anyway. I'm not for it for the long term, because right now is a band-aid solution to a lot of the problems of the world because of centralization of power and theft through Wall Street. But in the long term, it's enslavement, right? Have you ever taught the Colts conjecture to anyone? No, Gina. I don't, I don't even know what the Colts conjecture is. Maybe there's a question to Racer Kill. VC prime factorization is in the NP class of computer problems, but unclear if it's NP hard. I have no idea what that means. Racer Kill. Actually, all big primes have been of the form 2 to the power of P minus 1, 2 to the power of P and then minus 1, because it's easier to prime check such numbers. So these are some methods that people have come up with to check to see if a number is prime, once the number could be nominated as candidate of a prime, right? Is that it? Also, it can be shown that if a number 2 to the power of N minus 1 is prime, then N must be prime. So it's twin primes. They're twin primes. Is that correct? Isn't there approximately below largest MNC's prime? Learning balance. Colts conjecture also sometimes referred to as 3N plus 1 problem. I gotta look into that, Catholic tradition must. I was talking about the number of primes below, not numbers. The tens of millions. Have you ever seen the movie Contact? Yeah, for sure. I liked it.  There were some Hollywood aspects to it that I didn't like, but it was fun concept. A pot point in it is that extraterrestrial communicate with us using prime numbers. And I think that's one thing that has been part of science fiction for a long time, Utah Jazz, right? That a lot of people say, oh, there's science fiction out there that says if we encounter aliens, it'll be through music that we'll be able to communicate and stuff. But it's not really music. It's prime numbers. I think what do you call it? Close encounters of third kind also had prime numbers, but with notes or whatnot, right? But it's pretty much, I don't know, accepted or believed fact, believed theory that when, if we encounter alien species, that the only way we're going to communicate with them is through prime numbers, right? Through mathematics. Thanks for the informative lesson. Looking forward to the next set of streams. Nice. Enjoy it. Enjoy this very much. I did. So my, so learning balance over that is what I computed Wolfram. I think you're right. Thanks. How many zeros is that Wolfram? I'd like that website, by the way. Fun, fun conversations gang. So what do we say? Should we call the stream? That was fantastic discussion for Thursday afternoon or Thursday evening, depending on where you are, or Friday morning, depending on where you are. By the way, gang, thank you for the subs. Thank you for the follows. Apologies if I didn't catch them all, and I know I didn't catch them all. Utah Jazz, of course, wouldn't primes be different if we didn't use a base 10 number system? And what are the chances that space aliens would use that? Maybe I'm overthinking it. I don't think base 10 matters, really. Does it? Racer kill would know.  That's all these zeros after this. Are there streams scheduled coming up? I haven't scheduled anything, trash man, right now. I've probably announced the next set of streams in the next couple of days, but I'm thinking of, thank you very much for the bits, Jive's Zen. I'm thinking of maybe doing a cooking stream on Sunday, but I don't know yet because I have to go shopping to get the supplies we need. I want to make some dolmans, right? But I need to go get ground beef and some other surprise and stuff, supplies. And I don't know if I'll have the time to do it by Sunday, but I'm going to try to shoot for Sunday, maybe. But if I announce it, it's going to be tomorrow evening that I'll say. So the best I can do is maybe get people a one day warning or maximum two days warning. Okay. It's completely unreasonable to have some list of them. Wolfram was the only reason I survived GCSE math. Yeah. I was born the prime year. Nice, other God. No base. Doesn't matter. Yeah, base shouldn't matter for prime numbers. I was born on the even year. Yeah, me too. I'm very common. Me and you are a very common racer killed. I was born on the odd year. You could probably guess which one. You thought 1987 hilarious. Is that a prime number? Is it a prime number? Prime. Nice. Is it prime? Nice. Well, we take apart Furman theorem, Furman theorem. Okay, gang. Thank you for being here. By the way, fantastic conversation. Thank you for the help, racer kill, Catholic traditionalists and everyone else that was helping in the mathematics. Thank you for your questions. Thank you for the conversations. Thank you for the subs. Thank you for the bits. Thank you for the follows. If you want to follow this work, I'm on Patreon, patreon.com, backslash, CHO, CHYCHO. If you want to support this work, Patreon is a fantastic way to support this work. I don't put anything behind paywall. I will continue to share as much as I can for as long as I can. So if you want to follow this work, you can just follow through Patreon and get notifications as to what we're sharing. And if you do like what you see and if you do have the funds, Patreon is a fantastic way to support this project. Goodbye. I'm going to see you next time. Great stream. Great stream, trash man. Okay. We are live streaming this on Twitch, twitch.tv and .com, backslash, CHO live, CHYCHO, L-I-V-E. If you want to participate in these live streams, twitch is where you want to be at. Okay. You too, Catholic traditionalist. Have a great rest of your day, folks. Be blessed. Be blessed. I do announce these streams on Twitter, Gavs, Mines, VK and LO. And we share additional information there. And all the links will be in the description of this video. So if you want to follow this work, you can follow us through those platforms. Those are the ones that I'm active in right now. A lot of the streams, we're going to upload the audio to SoundCloud as podcasts. Right now, I'm not uploading the math streams or anything that involves visuals to SoundCloud. We might, at some point, once I get caught up with all the other streams that we want to load on SoundCloud, those streams and videos that we've shared over the last 14 years on YouTube and over the last few years on Bitshoot. Right? So there's a lot of content to be loaded on SoundCloud. If you want to listen to these things, what we're doing in audio format, SoundCloud is where you want to be at. Okay. And I do upload these streams, everything to Bitshoot, technical difficulties allowing. Okay. And because of the censorship on YouTube, we're holding off on uploading certain things to YouTube because we don't want to get deep platform and get flagged by the algorithms that are trying to censor information. Okay. So if you want to follow everything, you want to subscribe to the Bitshoot channel. And I'm totally okay if you want to watch these things on Bitshoot instead of YouTube, even though if YouTube is monetized for us, right? Not everything, YouTube keeps on demonetizing certain videos. But if you are on YouTube, you want to contribute, support this work, you can also join YouTube membership. And that is also a fantastic way to support this project, as well as following, subscribing through Twitch and joining in our Patreon page. Okay. Aside from that gang, thank you very much for being here. Thank you for everything. Mods, thank you for taking care of business. And I'll try to announce the next set of streams in the next couple of days. Maybe by Saturday, I should have something on by hope. And if we do, and if I do end up going shopping, I'll try to do a cooking live stream on Sunday, but no promises. Okay. Aside from that, I hope you have a fantastic day, morning, evening, afternoon, and I'll see you guys in the next video. Bye for now.


think people make the mistake of not actually thinking about why they do things so for example you see people going after they say they want a million dollars and you ask them why I'm like you know what I was raised in a poor family my mom my dad always fought and eventually they got divorced over money and you're like okay I get it but then you ask them why again and they're like you know I want the safety and the security of having this money and you ask why again and they're like I just want to have freedom and I want to have options and then you go why why why and then you realize that they didn't really need a million dollars they just needed like if they made six figures doing something they enjoyed they would have plenty of money to do all the things they wanted to do and I think the trap is we do all these things and we put ourselves on like this hamster treadmill like so deep in the matrix for our entire lives and then we don't even know why we did any of it like for me I was super frustrated by the fact that I had a full-time job and I was working 40 hours to just pay my rent to exist in a place that I didn't even particularly like that much and just so I could exist and repeat the same day over and over with the four hours free I had every day and to me this was like absolute insanity like I had no idea why I just didn't understand why humans would do that for 40 years their whole lives and so for me that was one of the big it was an impetus for me doing something and creating my own business so I could regain those things that's a great classic reason that people change is pain 90% of the time I mean I can only think about my own story where I don't know that many people who think that they want to acquire a lot of money so they don't have to struggle without the struggle or people I hear all the time they're like I want to just find one person marry them and stay with them you almost never hear that from people that had like happily married parents like that deep emotional energies from people that are like very traumatized by something in their past or have been the child of a person who's gone through multiple divorces or the person who had never had any opportunities in life or they envied all these people that could travel the world but they couldn't because of whatever economic circumstances I think for most people it's interesting when I talk to people who even hired me as a coach it's like 90% of the time maybe 90s in exaggeration maybe more than 50% of the time they've recently had a death in the family or if they're hiring me for their health one of their friends either their parent has died or a friend their age just died and that's put in them like this that feeling a sense of mortality like you know we all intellectually know we're gonna die but you know it's like in Tuesdays with Maury where he says you know everyone intellectually knows they're gonna die but we don't really believe it because if we did then we would actually change and I think whenever people either realize their own mortality which is easy to see when you see a parent a few decades older going through a health crisis or when they pass away or when it's just that day-to-day malaise where you just are dying slowly like the thorough most men live lives of quiet desperation just repeating the same day and that's supposedly a lifetime I think for most people it's frustration or desperation that almost always leads to change and it's very rarely like these happy-go-lucky everything's unicorns and rain both people that have the kind of fire the hunger that actually results in doing something impressive people don't change mostly just because it's easy right like even it seems weird because you could think okay how can a person who's so in debt or how can a person in so much physical pain or how can a person so overweight that they hate themselves how can they not change it seems logical but I think for a lot of people you know I had a woman who hired me as a coach and she said I am when I dug deeper and I asked her all these questions and I was like why do you really want to lose weight like why now what's going on she's like when I was younger I was she's like not to be egotistical I was pretty sexy and she goes I had guys flirting with me all the time and now that I've gained some weight I feel like if I lost weight again I would get attention from men and cheat on my husband so like it could be so simplistic for us to think that she just needed to lose weight and there was this deep logical reason that she just had to do the work and eat right but the story was way deeper and way more compelling and so for a lot of people we don't change because it's like that the devil you know versus the devil you don't right like she knew I would not like myself as much or I didn't like how I looked or felt but at least I don't know if I open pent doors box I don't know what would happen so I think for a lot of people it's the fear of well this kind of sucks but at least I know what's gonna happen versus if I step into the unknown I don't know I might be the ridicule of my friends and family I don't even know what to do I don't know how it's gonna work out and I think oftentimes it's just that the pain has not gotten big enough and deep enough that people are willing to just short-circuit themselves and their habits I mean I guess in the interest of timely advice the best advice recently was from my friend Jackie who said to radically follow your intuition above all else and this is coming from someone who had a very successful practice in New York City bought her first place when she was young in her 20s burned herself out and her brother died of drugs and she realized that she really wasn't living the life that she really wanted to be living and certainly she was killing herself physically and her advice to just radically follow your intuition it's very simple but again it's like dying no one really believes it can work like no one really believes they're gonna die no one really believes if you just follow your intuition implicitly just 100% trust it that you could actually do something as concrete as make more money like we get it with love and friends and like traveling but like really concrete things I think it really surprised me because when I was young interesting thing about my life because I was so into spirituality my life was filled with synchronicity there were coincidences every day to the point where I would have these coincidences that were so freaky that you just like a laugh they're so perfect it's like you know there's an intelligence and that it's hearing you and as I got older I didn't even realize how infrequently that that was occurring it just stopped for like years maybe five or ten years and it wasn't till last year or really the beginning of this year that I started doing this again and all these synchronicities have started happening again and so when you live that way it makes you realize that there is like this this quote infinite intelligence and that it it hears things and it when you follow the things that excite you and that intuitively resonate it there is this kind of synergistic effect and just for lack of better word magic happens.


You got more followers than Beyoncé. We're in Times Square, New York City. Got the red mic. Let's go interview him. What's the best thing to take photos of in New York? The best photo is in Tan Square. This, my camera is a R5 Canon. How come you didn't get a red one? The best photo is Macho with my costume. I am super hero Latino, Chapulín colorado. Ha, ha, ha, ha. Sígamelos buenos. You got more followers than Beyoncé. How do you get 752 million? I think mil in Spanish means thousand. Yeah, right, right. How much do you charge for a photo? It depends. For one day maybe 200, 300, 400, 500.


Tied up in Tinsel, by Nioh Marsh. The book's dedication reads, for my godson, Nicholas Dacre's Mannings, when he grows up. CHAPTER ONE HALBURDS In my sire, said Hillary Bill Tasman, joining the tips of his fingers, was flung into penury by the great slump he commenced to scrap-merchant—you don't mind my talking—not at all. Thank you. When I so describe his activities, I do not indulge in fachetsie. He went into partnership in a rag-and-bone way with my uncle Burt Smith, who was already equipped with a horse-and-cart and the experience of a short lifetime—uncle, by the way, is a courtesy title. Yes. You will meet him to-morrow. My sire, who was newly widowed, paid for his partnership by enlarging the business and bringing into it such items of family property as he had contrived to hide from his ravenous creditors. They included a mison-bowl of considerable monetary, though in my opinion little aesthetic value. My uncle Burt, lacking expertise in the higher reaches of his profession, would no doubt have knocked off this and other heirlooms to the nearest fence. My father, however, provided him with such written authority as to clear him of any suspicion of chicanery and sent him to Bond Street, where he drove a bargain that made him blink. Splendid! Could you keep your hands as they are? I think so. They prospered. By the time I was five they had two carts and two horses and a tidy account in the bank. I congratulate you, by the way, upon making no allusion to Steptoe and Son. I rather judge my new acquaintances under that heading. My father developed an unsuspected flair for trade, and taking advantage of the depression bought in a low market and after a period of acute anxiety sold in a high one. There came a day when, wearing his best suit and the tie to which he had every right, he sold the last of his family possessions at an exorbitant price to King Farouk, with whom he was tolerably acquainted. It was a Venetian chandelier of unparalleled vulgarity. Fancy! This transaction led to most rewarding sequels, terminated only by His Majesty's death, at which time my father had established a shop in South Malton Street, while Uncle Bird presided over a fleet of carts and horses, maintaining his hold on the milieu that best suited him, but greatly increasing his expertise. And you? I, until I was seven years old, I lodged with my father and adopted Uncle in a two-room apartment in Smalls Yard, Cheapjack Lane, EC-4. Sample complete. Ready to continue?