Math Storytelling: Part One

Okay. Hello there. Well, let's see. Today we're going to try doing something a bit different. Today somebody had the idea that this should be math storytelling day. So I will try and tell some stories about math. And I guess let me try and tell a little bit of the story of math, starting at the very beginning. So we might start off with kind of, what is math? Math is kind of a system of abstraction about the world. It's a little bit like back in the day before human language was invented. Maybe one could imagine people were just, like, pointing at things to say. That's what I mean, and so on. And then eventually this idea emerged that there could be a word like it's a rock, it's a mammoth, it's whatever, an abstract word that could describe a mammoth, any mammoth. That's kind of a system of abstraction. That is our version of human language. And there are other systems of abstraction like logic and so on. And a really big one that is the one I've spent my life working on is computation and the idea of following arbitrary kind of computational rules. But we're here to talk about math. So let's talk about the kind of abstraction of the world that is math. And specifically, math tends to deal with abstractions that involve things like numbers and shapes and space and so on. Okay, so let's start off with kind of, where did things like numbers come from? Things that we have today, they all had to get invented somewhere. You might say, aren't numbers just totally obvious? How could you ever be in a world without numbers? People say that there are indigenous tribes in the Amazon jungle and places like that who don't have words in their languages for numbers larger than essentially three. How can that be? Well, if you're hanging out with a group of people, and then you would typically say, well, you know, I'm hanging out with John and Jane and Mary and Peter and so on. Each person has a name. You don't necessarily say, I'm hanging out with seven people. You don't need to collectivize. You don't need to abstract the idea of number. Each person has a specific name. Okay, let's say you're keeping goats. You give each goat a name. Every goat is different. There are no. The goats aren't identical. But you might decide one day, perhaps because you start doing commerce and you're trading, and you say, I'm going to exchange, you know, five goats for seven sheep or something like this. And then you've got to be talking collectively about the goats, collectively about the Sheep. And for that you start needing the idea of numbers and the notion that you should just count the sheep and not just talk about every sheep as an independent individual sheep. That's kind of the key idea that leads to the notion of numbers. And so we don't know when that was first invented, but certainly from tens of thousands of years ago. There are examples of tally sticks where people would mark off, kind of make a mark, a series of marks. And maybe they were counting something, maybe they were counting the number of mammoths they saw that day or some such other thing. So then the question comes about, so how do you start off? You've got this idea of numbers. How do you represent numbers? And I suppose, and now let's actually go modern and computational here. Let's say we're going to represent a number. And our sort of first approach to doing that is to just say we're going to have a. We're going to say we're going to represent the number, I don't know, 13 or something here. And the. And so we might do that by just having a bunch of the. A bunch of things marked off on our stick. I happen to write them as ones here, but I could as well have just written them as little marks on a stick. Well, you know, if we've got, let's say, you know, 5 marks, most of us can. Can immediately say, oh, we kind of get the idea that's five. We can compare it with this, there's more there, and so on. But one of the things we might want to do is to find sort of a more organized way to represent numbers than just having to mark off one mark for every we representing a number just by that number of marks, so to speak. So in terms of sort of the language of mathematics, one could say that writing a number this way is writing it in unary, in essentially base one. But so a typical thing people do when they mark things off, you know, when they're checking things off on some piece of paper or something, as they might say. And again, I'm just using. I'm just doing this to. As a way, okay, they might mark off every time they get to five tally marks. They might sort of group those together and put a slash through them or something like this. And so then we can. There. It's a little bit easier for us to see, okay, there are two groups of five and then there's three left over. You know, we have 10 fingers. So we have a habit of doing things like saying, of grouping things in groups of 10. So we might say, here we go, we've got something where we've got our number. It's what we would call 43. But we've got 44 groups of 10, and then three ones left over. Okay, so then the question is, how does one start to represent numbers? This is sort of the beginning of an approach to making numbers a little bit easier to deal with is not to have to take sort of each one on its own, so to speak. Well, historically, there were a bunch of different methods developed, I think. Let's see if we can pull one up here. Let's see. Let's try this. Let's try something like 53 in. Let's try something like Babylonian. This was an early number system. See if this works. Wake up. Wake up. Okay, maybe I should make this a little bigger here. Okay, so that was. That's the Babylonian for 53. And I guess we can see five little marks of this kind and three little marks of that kind. If we go and say, let's do 78. Okay, we're seeing slightly different markings because that mark there. I don't know, I think I'm guessing that. Oh, yeah, that mark there represented 60. That. That mark there represented 10. And then we got eight left over. And in the Babylonian system, I think if we try and go to 128, for example, we'll probably get two marks for 60 and eight marks for one there. So this was kind of a notation where you have something that means, oh, no, actually, this is just showing the 60 is off in this corner and this thing here and I guess is a 10. I'm guessing if we go back down here. Well, we can. Let's just use some technology here. Let's say. Let's just make a list of Babylonian numbers from 1 to, let's say 25. And. Okay, there we go. So 1, 2, 3, 4, 5, 6. Okay. And this, I guess, is the mark for 10. That's the mark for 11 and so on. So here we're just. We're essentially just doing the same kind of thing that we did up here, except that we're using this mark to indicate a whole block of size 10 here. Well, there were other systems people used. So, for example, another one that was used was the Greek number system. Let's see what that one looks like. Okay, let's look for some smaller numbers there. Let's do the same thing for Greek. Let's say here. Okay, so here's the Greek number system. And the Greeks kind of had an interesting idea, but it ended up kind of being a goofy idea in the end, which was they said, well, we already have something which is in sequence, which is the letters of our Alphabet. Alpha, beta, gamma, delta, epsilon. Well, that's funny because that's not a letter that I think is a digamma, which is not a letter that's in the standard Greek Alphabet. And there's eta, theta, iota. And then here, I guess they're using for that iota is kind of the symbol for 10. And then they're using these tens again. But one of the things they did here was, was they said, let's just use the letters of our Alphabet to represent successive numbers. Which might have seemed a good idea at the time, but when letters of the Alphabet kind of went out of circulation, that probably made it really confusing. Well, then the next kind of another number system is the Roman number system, Roman numerals. And if we look at that, it had another scheme for representing things where we've got 1, 2, 3, and then 5 is an important thing. So we represent a 5 with a V. And then the 4 is represented by saying, well, it's one less than the 5 and so on. And so this is, if we keep going here up to 100 or something, we can see this whole system for dealing with Roman numerals, which no doubt was quite convenient when you just sort of glanced at a number. It was certainly a lot easier to tell what was going on where you saw XXVI than if you saw a bunch of tally marks or something like this. But when it came to doing things like manipulating numbers, adding them together, this was not an easy system to do that in. But so the big idea that. So the Roman numeral system was the one that was used for for many years actually basically in the Western world until the 1200s. But there was this sort of alternative idea, the Hindu Arabic system of numbers, which is the one that got brought to the west around 1200 by folks like Fibonacci. And that had kind of a different idea. And its idea was to say, well, let's just take these things where we have these blocks and let's just say we're going to have. We're going to use the position of a digit to represent things. So here we're saying, well, there are three separated ones. We'll just put a three digit there. And here there are four blocks of 10. So we'll put a four digit to the left of that 10 to indicate that there are four blocks of 10 here. And if we were to, let's say we were to do something like this. We were to say. Not sure I can quite. Let's say we do 143 here. Then the idea would be we would. Then let's try and do this. We could partition this again in groups of up to 10, see if this works. Never tried this before as a way to explain numbers. Okay, so now it's a little hard to see what's going on here. Maybe if we put. Let's do this. Let's map frame onto this at framed onto. So I'm using, obviously, computational kinds of things to do this. Okay, here we go. Maybe we want to go down to level three there. Okay, so here we've got. Now we've got. We're beginning to have sort of a different number system where we've got. We can see these sort of big mega blocks. I should have probably gone even further. Let me just try doing this. I wonder whether this will work. Let's see whether this works. Now we got the same thing. Okay, so what we're seeing is the individual ones down here. Then we're seeing these blocks of 10. And then we're seeing every block of 10. Every 10 blocks of 10 is being separated off in another big block here. And so this is a way of thinking about kind of our positional number system where we say there's one block of 10 tens or one block of 100 there, we put down a one there. There's then four blocks of 10, and then there's three individual ones here. And so that's how we get kind of this positional notation for this number 143. And so that's kind of the. And this is then a way of representing numbers where we can represent even pretty big numbers. I mean, if we try and use. For example, let's try and use the Babylonian number system. Let's see how far it can go. Let's try representing the number. Let's try just representing the number a million in the Babylonian number system. I don't know whether it'll have a symbol for that. Okay, we managed to make it to a million. All right, let's. Let's live even more dangerously. Let's try and go to 10 million. Okay, well, we got some funky things showing up there. Let's go to 100 million. Still thinks it can do it. Maybe this one can be extended. That's interesting. I bet if we try and do this in the Greek number system, it will just give up. Ah, it only supports numbers up to 10,000. So we can go down Here, let's look at the number 9,000, the Greek number system. Okay, I think what has happened here is that to represent. Let's look at what happens at 900. Okay? That's the symbol for 900. I'll tell you what, let's go look at all the numbers from. Let's say we want to go up in hundreds, so let's just go 100 there. I goes from 100 to 1,000 in steps of 100. Okay, well, so they're just using other letters of their Alphabet. Rho, sigma, tau, epsilon, phi, chi, psi, omega. I don't know exactly what that is. That might be a. I think that's just an image here. But now for. This is for 1000, they're going to this weird kind of stacked representation. Let's say we go up to. Let's go up to 2,000 here. They're kind of having a hard time, I think. I guess that's the symbol for 1000. Is that a stacked up like this? But it's getting funkier as they try and go higher. One feature of our positional number system is you can represent as big a number as you want. So I can just type in a huge number here. And this number, we can say it's, you know, six ones, five tens, four hundreds, three thousands, et cetera, et cetera, et cetera. We can get as big a number as we want this way. And of course, if we want to say, you know, that number, we can give it a name in English and we can kind of. We typically have this convention that we call out every thousand. So it's a thousand, there's millions, billions, trillions, quadrillions, quintillions. And at each stage here, we're kind of blocking this. Just as you can see, as this number is displayed, it's got little spaces between every block of three numbers. And each block of three numbers we're then sort of calling out with a name. That's the thousands block, that's the millions block, the billions block, etc. And usually we just use these names as a way to talk about these numbers. We don't try and do computation with numbers that have been written out with their names like this. But in a sense, we're doing something pretty similar to what folks did in the past. But we've got this nice, clean notation that doesn't have any of these words lying around in it. So kind of the. If we're the thing that. The. So, okay, so we've got this idea of numbers and the idea of Numbers, counting numbers, you know, 1, 2, 3, 4, 5. That idea certainly existed in the earliest recorded history. We have the idea of zero as a number, and the idea of negative numbers came much later. I mean, the idea of zero as a number maybe was 2,000 years ago. Maybe the idea of negative numbers is pretty much 1500 and came actually at the same time as imaginary and complex numbers came. Maybe we'll talk about those in a little bit. But let's, you know, why did nobody need a representation for zero? Well, what is zero? Zero is nothing. Why would we need a representation for zero? You only start needing a representation for zero when you start doing operations with numbers and you try and do that in a systematic way. So let's talk about operations with numbers. So the first thing we think about doing operations with numbers is just look at the successive numbers. Just say the successor operation, which is basically a thing that adds one to a number. It's interesting that in Wolfram language, for example, we don't end up having a separate function, you know, successor, that just adds one to something. But let's just, for our purposes here, just because it's going to be useful, I think let's just invent a successor function. I'll just write it here. It's an extremely simple function, just adds one, and so we can say, what's the successor of 10? And we'll get 11 and so on. Now, you know, an interesting question is, if we've got this successor operation, what happens if we just keep doing it? So we have this very convenient function nest. Actually, we'll have nest list, which will go ahead and progressively do something. So, for example, let's just do Nest FX3, for example. It will say I'm going to do F and then I'm going to do F on the result of that. F on the result of that. I'm going to do that three times. So what do we get? If we take the successor of 0 and we take the successor of 0 three times, rather unsurprisingly, we get the number three. Let's say, however, that we took the successor. We took the successor of six three times, what are we getting? Well, we're getting six plus one plus one plus one. In other words, we're getting six plus. We can think of that as getting six plus three. So in a sense, getting. When we add three, it's like three times do the successor operation. So in a sense, we can think about addition as being successive successoring, so to speak. If we want to add three, we do successor three. Times on the thing we already got. And then that adds three to what we have. So that's a. So this operation, by the way, the notation of a plus sign to represent addition, that didn't come in until the 1400s. People were just describing things in words up until that time. And then the idea of, well, let's systematically represent this addition operation with a symbol that was separate from our numbers. So, okay, so the question then would be this idea of taking an operation like the successor operation and just applying it many times. What happens if we take the addition operation, for example, and apply that many times? So what we have there, we'll say every time we want to say x goes to x +2, for example, and let's start with zero, and let's do this seven times. And so what happens? Well, and I could have written, by the way, for the successor. Well, let's just do it this way. We're saying we're going to be adding two. So we could have written this. That. We could have written this X+2. We really wanted to be pedantic. We could write the X +2 as a Nest of a successor of. See, what am I doing here? I could say, oh yeah, of X2 times. So that's our way of representing this thing here means add to. We could write out the successor there if we really wanted to. We could write that out as in terms of another thing where we say this goes to thing plus one. And that will be. So that's saying we want to take the result of successoring two times and say five times here, if we want to the result of successoring five times on our original number. And then we want to take that successoring, and we wanted to take that result of doing that successoring, which is addition. And we want to do that seven times. So what is this iteration of. Of addition? Well, the iteration of addition is multiplication. So in this particular case, we'd be saying, let's say I had done the example here with three. For instance, this is saying do the addition three times. And doing the addition three times is equivalent to multiplying by three. So, okay, so let's see what happens if we keep going here. Let's say we say, so we've got this multiplication by. So let's say we have multiplication by two. And let's say we've got the multiplication by two. And we start off well, I'm going to start here with a one. And we're saying let's successively multiply by two every time. Okay, so if we Multiply by two every time three times. What do we get? We get what we can represent as 2 to the power 3. Just like this operation of doing successoring, of doing addition three times, we represent like that. Or the operation of doing just the pure successor three times, we represent like this. So the plus is kind of the. We don't have a standard notation for the just do successor, but we have a standard notation that says do the successor operation three times. Do the nested successor operation three times. That's a. Times. Do the nested times three times. That's a power. So obviously, given this, we might say, well, what happens if we go on with this? We could say instead of this, we could say, how do we keep going? We could say we've got two to the power three. And we could say, well, we could do this in a couple of different ways here. We could say, let's just do the squares each time. That's not very interesting. What we could do here is to say we've got. Just like we've got a. We could say we make 2 to the 3 by saying 2 times 2 times 2. We could say the next level up, we'd say 2 to the 2 to the 2, and that's 16. And if we go 2 to the 2 to the 2 to the 2nd, it's going to be that. And if we go 2-to the 2 to the 2 to the two to the 2, how big is that going to be? Oh, wow. That's a very big number. So this operation of successively taking powers, it's usually called tetration or power towers, and numbers get very big very quickly. So, for example, we could represent this if we wanted to. We don't have a notation for this that's built in to Wolfram language because it's not an operation that one does very often. But we could say, for example, two up arrow five to represent this thing of taking two to the two to the two five times. And that number gets very big. So, for example, this number, I'm sure that number is way bigger than the number of particles in our universe. So, yeah, it's 10 to the power 19,000. So that's the 10 to the 80th Protons in our universe. So that's vastly bigger than that. So we get to a very big number very quickly. By the way, one of the things that's kind of a feature of what's going on here is you might say, how do you interpret that thing that's written down there? Two to the two to the Two to the two to the two. What is that? Because it matters whether if we take two to the two to the two, we get. Well, actually, this. Let me do one more level, because otherwise we're not going to see this happen. Two to the two to the two. Like that, we get 256. If we take two to the two to the. Oops. To the two to the. How many twos do I need? Two. Whoops. To the two to the two, we're going to get that giant answer there. Did I have the wrong number of twos? I had the wrong number of twos there. Let me get rid of one of the twos. Okay, we get a different answer. So it matters how we put the parentheses in here. It matters whether we're assuming that wedge groups to the left or groups to the right. Do the same thing with plus or something. We say two plus three plus four plus five plus six. Okay? We get that result. We can put parentheses in any way we want. It doesn't make any difference. And that's a feature of the plus operation. We talk about it. It's this property of associativity that's true for plus and isn't true for power. For multiplication, same thing we can do 2 times 3 times 5 times 6 and it doesn't matter. That's the same as 2 times 3 times 4 times 5 times 6. So this, this. So one question is, when we invent a notation like plus, we might. Or when we invent a notation like wedge for power, we potentially have to make a decision. What does it actually mean? In the case of plus, we got away with not having to make that decision of does it group to the left or to the right? For Wedge, we do have to make that decision. So when we were designing Wolfram Language, we made the decision wedge is going to group to the right, which means that if we just type 2 to the 2 to the 2 like this, it's going to assume that we group those things to the right and it's going to compute this rather than computing this. These things are pretty tricky. For example, let me show you another example. If I say, let's see, 2 over 3 over 4 over 5, what does that mean? Because it matters which way we group that. If we said 2 over 3, let me just put in some arbitrary grouping here, 2 over 3 over 4 over 5, we'll get a different answer. So it matters what 2, 3 over 4 over 5, how that's grouped. And in that particular case, we choose to group to the left. So if we say two, let's see, two to get the parentheses right, two over three, over four. Oops. And that whole thing over five, and that will give us one over 30. If we were instead to make the convention that we grouped to the right, we get a different answer. Let's type it in and see what we get. Let's see. Put in a parenthesis there. Put in a parenthesis there. Okay, that's now grouped to the right. Okay, we get a completely different answer. So again, it's a convention to see how we group these operations. And actually, people hadn't codified this very well. I mean, before I was building Mathematica and things like this, the question of what does this really mean? Was something where you ask people, reasonable people could disagree about what that meant. I'll give you another one that's same type of thing. 2 minus 3, minus 4, minus 5. Which way does that group kind of exercise here? Well, if we were to say this groups. It groups to the left. It's assumed to group to the left. But if we had it grouped to the right instead, it would give a different answer. So again, it's sort of a convention about how things work. So another big convention, people, the people, you know, learn in school, is if you type something like two plus three, let's type a nice times sign times five. What does that mean? Well, you know, it could mean, let's see how we would group that. It could mean, well, let's just group it to the left, two plus three times five. Or it could mean two plus three times five. And it is an utterly arbitrary convention that got made in the, I guess in the 1600s, that if I type two plus three times five, I mean, do the times operation first and then do the plus operation, even though I wrote it out in this order where the plus seems to come first. So when people learn all these mnemonics, which I don't now remember, I'm sorry to say, for what operation do you do first? One has to remember one is basically learning a piece of frozen history. There's nothing fundamental about the fact that you do the operations in a particular order given this notation. It's just a kind of convention about what that notation means and how to interpret it. And so, for example, if we did this and we put in a wedge, the convention is, let's put in a wedge there. The convention is that that means that the wedge binds tighter. It has higher precedence in the language of computer languages and formal grammars. It has higher precedence than times, which in Turn has higher precedence than plus. So this expression groups like this, and that's a convention that it does that. And by the way, when we put in parentheses, we're kind of forcing a particular grouping. We're saying, okay, like in this case here, for example, we're saying, okay, the default is to say three times five is computed first and then we add the two. But actually we want to do it the other way around. We want to do the 23 first and then we want to multiply by five. Now, in fact, if you go sort of computeristic on this whole thing, we have to stop it actually evaluating these things. But then we can say something like this. We can say, show us the expression tree of this. And what we'll see is that the thing is, it's made this tree where plus is at the top, says do the plus. We do the power at the bottom and it's the times of the power, the plus of the times of the power. We can also see that if we just look at the full form of this expression, we'll see that it's written out as power inside. If we were to do this without parentheses. Let's see what. Well, this particular one, I got it. I put in the parentheses in the right way. But if we were to do something like this, we would see in the full form of that we have to hold it, because otherwise if I don't use hold, it'll just evaluate it. This is two, the plus of the two and then the times of the three and five. And if we show, you know, the tree version of that, we can see it is a tree where the plus is on the outside and then there's times on the inside. If we did this the other way around, we'd see the thing work differently. So let's say it would be something like 3 times 5 plus 2. Again, we'd see that would be a plus on the outside of a times and so on. And the same thing with the tree. Okay, so that's kind of how the. Okay, somebody is pointing out, okay, I don't think in British English math education that it was the same mnemonic, but apparently it's PEMDAs. What does that stand for? Parenthesis, multiplication, division, addition, subtraction. Really, I wonder if it's that way around. In any case, the thing I'm sort of trying to explain is it's a convention that it works this way and some extensions of that convention, like what you do with iterated division or what you do with powers, those conventions had to be made very, very recently. And these other conventions, like times comes before gets done before. Plus those come from the 1600s. But I guess I got to make the convention for wedges and things in much more recent times, so. Okay, well, let's talk about. We talked a little bit about things like, we've now got notation for addition, et cetera, et cetera, et cetera. Let's talk about algebra for a second. So one of the things we might say is, we could say something like, okay, we could test, is five equal to four plus one? And it says, yeah, five is equal to four plus one. We could work out what's four plus one? And it's five. And so we could say, let's test, is four plus one equal to five? Great, it's equal to five. Okay, next question. If we know we've got four plus something and that's equal to five, how do we represent that? How do we represent asking the question, what plus four? Four plus what is equal to five? And so people had this good idea in the 1400s that. One could start using letters to represent these kind of unknowns in an expression like this. I have to say, I think one of the reasons it took a while for people to have this idea is that until the 1200s, people have been using Roman numerals. So in Roman numerals, this would be, you know, if I say Roman numeral, I wonder what it's going to do if I say Roman numeral 4 plus Roman numeral. Well, plus Roman numeral 1 or something. I'll see something like this. And then if I have plus Roman numeral 10. Oops, I've got an X right there in my number system. And so the notion of using a letter to stand for a number would be not an obvious thing to do if you were using Roman numerals to represent your numbers. But as soon as Hindu Arabic numerals came in, it started to make more sense to say, let's distinguish the thing that we don't know, the unknown, from the things that are just numbers. So then we could start imagining, well, we have something like this. This is a, you know, four plus X is equal to five. Okay, what's X? The point is that now we can start talking about things where we don't have to say. We could start just having things like four plus X. We don't know what X is. Okay, we say four plus X is equal to five. Let's say we want to solve that equation. Four plus X equals five. Solve that for X. Okay, now we Discover that, oh, X can be equal to one. But the important thing is that we've got this kind of slot that we can put in here that represents an arbitrary thing and represents an unknown. And that was kind of a. A big idea of the 1400s and the kinds of equations, the kinds of things where you could say, this is equal to that. Well, you know, we could start saying 5 plus 3x here. What value of x gives that. Oh, interesting. Well, there's a minus sign there. So that's going to be important in what we're about to talk about. I should have perhaps mentioned that the other type of numbers that comes from a long time ago, comes from antiquity. People like Pythagoras were very interested in them were rational numbers, numbers that are the ratio of two other numbers. And so, for example, we can say, well, what do we get if we say. What do we get if we say six divided by two? Great, we get three. What do we get if we say six divided by seven? Well, that's not a. That's just a six divided by seven. We don't know what that number is. Now, much later in the 1600s, people had the idea of decimal numbers. The idea of extending this notion of positional notation to fractional numbers. So that's, you know, the 1/10, 100th, et cetera. That was a very late idea. That idea emerged only after the invention of things like logarithms and so on. Before that time, people were representing numbers just by giving them in the form of fractions like this. And so, for example, the Egyptians, famously, they didn't know about numbers like 2/3. That wasn't a thing for them. The only kinds of numbers they talked about were numbers that were one over something. So I think. Let's see if we've got one of these. I think we probably have a thing. Let's see. Here we go. So if we take 2/3 and we write that as an Egyptian fraction, the Egyptians always were writing numbers as one over something plus one over something. So two thirds better be equal to one half plus one sixth. And it is. But to the Egyptians, our current notation of 2 divided by 3 represents 2/3. They didn't have that. They were just saying, let's write out everything as one over something plus one over something. So let's try a different fraction here. Let's try something like, I don't know, seven eighths, let's say, okay, seven eighths is apparently one half plus one third plus one 24th. Let's work that out, see if that's right. Yes, it is. And let's maybe try a bigger. Let's try something else. Let's try a number like. Let's try something really kind of huge like this. Okay. In this Egyptian way of. Of doing things, you get a pretty big number there to represent this. But this is clearly a way to represent rational numbers. It isn't the way we use, but it's a way that you could use. The Egyptians used it, the Babylonians had a different way of doing this. I don't know whether we can do that here. With more like our notation with powers of 60 and inverse powers of 60. All right, well, let's see. We're going on talking about different kinds of numbers. The next kind of thing that happens is we got these equations and we've got equations, and they're things like 4 plus x equals 5, solve for x. Okay, given that we're doing that, what about other operations? What about 2 times X plus 6 equals 7? Okay, we can say, solve that for X, and okay, the answer is X equals a half. One feature of equations like this, equations where we're just multiplying and adding and so on, we could write out all kinds of fancy equations where we're multiplying and adding things and so on. And one feature of those equations is anytime we do that, and we put X's in there, anytime we do that, the value of X we get out is always a rational number. It's always a ratio of two whole numbers. And so that's a thing. But now let's start multiplying more X's in here. So let's say we say X times x equals 4. Let's say, okay, so if x times x equals 4, what's x equal to? And then, well, the answer that Pythagoras would have known is x equals 2. The idea of x equals minus 2, as I said, that comes from 1500. That's a much later idea. But so for right now, let's just concentrate on x equals 2 here. So we're now saying we can solve this equation X times X or x squared equals 4, and the solution for that is x equals 2. Okay, so then. But Pythagoras thought that every equation you could write down you could solve using rational numbers. And so then he wrote down the equation X times x equals 2, and he was very confused because he couldn't find a rational number. You know, there's no, forget the Egyptians with the big numbers in the denominator. There isn't. He couldn't find a rational number that could be multiplied by itself to give two. Even if he had something, you know, let's say we've got. Let's try and approximate it. Let's say I'm going to use a tricky fancy thing because I know how to do it. Let's try and do this. Let me see, let me try this. I'm just going to do something fancy just to show this. Okay, let me do this. Oh, no, actually, there's another way to do this. Ooh, there's a better way to do it. No, actually, I'm going to do the thing I was just about to do. All right, let's do this. And actually, I might talk about continued fractions, which were another way of representing numbers that became popular later on. Okay, so here we've got a sequence of rational numbers and let's compare those to the square root of 2. So let's take those things. Let's take the numerical value of those things and let's look at the numerical value of the square root of two. Okay, so what we see is these rational numbers are getting really close to the square root of two. And Pythagoras, who knows whether he worked any of these out, Maybe he got as far as 41 over 29, I don't know. And he was like, well, that might be almost square root of two, but wait a minute, it's not exactly the square root of 2. And in fact, what he eventually found was a proof that the square root of two can't be a rational number. And that was a big and confusing thing at the time. So what is the square if the square root of 2 can't be represented as a ratio of two numbers, what is it? And that was a thing that was kind of left hanging for a very long time. It's a what we would these days call an algebraic number, but it's a number that is, in a sense, you can just say, well, the square root of 2 is the solution for X in the equation. X squared equals 2. That's the definition of the square root of 2. And maybe there's nothing more that we could say about that. Well, okay, so let's ask the question. Let's take equations of the form. Let's say x squared plus 5, x plus 1 equals 0, for example, or solve that for X. Okay, that's the famous quadratic equation. That's the equation where everybody learns the. The formula for. If we put in A, B, C here, everybody learns this formula that the solution to this equation is Given by this formula here. This was discovered by the Egyptians, for sure, and was something where. Now notice, with this formula, we can see something which is every quadratic equation. Every equation of the form something times x squared plus something times x plus a number equals zero. Every quadratic equation can be solved in terms of the square root of a number. So if these numbers here, a, B and C are integers, we are guaranteeing that every quadratic equation, whatever integers we feed in here, the solution to that quadratic equation will always be a. A formula that just involves square roots. So that's kind of an interesting result that you can solve any equation. You know, we know that x squared if we just have x squared equals 2, then that the solution for that is x is square root of 2. But this is telling us that the bigger equations, the equations that involve other linear terms and so on, they can all be solved in terms of square roots. Okay, so then we get to the 1500 roughly, and a chap called Cardano, another person called Tartaglia Ferrari, a whole cast of Italian characters, and they got interested in solving cubic equations. So that's equations of the form x cubed. So X times x times x, it's called x cubed because the reason it's called x squared is that if you have a square and its sides are both of length two, then the area of the square is two times two. So that's why one talks about two squared being two times two and so on. Two cubed would be. You've got a cube whose sides are all of length 2. So the volume of the cube is 2 times 2 times 2 or 2 cubed. Okay, so then Cardano. So Cardano got interested in solving cubic equations. Well, cubic equations are a giant mess. Let's see. I think we might be able to say this. There we go. Okay. This is the solution to that cubic equation. So it has all kinds of messy things in. Has a variety of roots. Let's take a look. One of these will probably be a. Oh, boy. Okay. We're trying to solve this cubic equation. It's a total mess. You might have thought, oh, it's just going to involve cube roots. Well, it does just have cube roots in here. But the thing that Cardano noticed is that essentially as soon as you start talking about solving cubic equations, you can't get away from having negative numbers. And when we talk about, oh, we've got five takeaway six. Okay. People had invented, for example, when if you're doing accounting and you say, I've got $5 gilders whatever. I've got five, I've got to pay you six. How many do I have left when I've done that? Well, accounting. If you're doing accounting and you're trying to keep records of what you're spending and so on, you might end up wanting to say, well, I'm making my account book and I want to represent the fact that I owe you $1 or something at the end. I don't have at the beginning. It's just like, well, I start off with my $5, I pay somebody $1, I get left with $4, et cetera, et cetera, et cetera. I've just got some number of dollars. What does it mean if I've spent more dollars than I have? How do we represent that? Well, people like Luca Pacioli, who was a friend of Leonardo da Vinci's and was sort of the originator of things like double entry bookkeeping and the kind of notion of making sort of general ledgers for accounting, this idea of what do you put on your accounting ledger if you actually owe a dollar and it's not like you have a dollar? And even in modern accounting, actually, I think we have. And gosh, I haven't used this in millions of years, but in Wolfram language, we have an accounting form which shows the representation that's usually used in accounting data for a negative number. So if we were to make a table of numbers from N from minus, well, let's just say the accounting form of that n from minus n from minus 5 to 5. Okay? That's the way one would typically write out a negative number. Perhaps not so much very recently, but that's the typical convention in accounting. I don't know whether Luca Pacioli used that convention. Wouldn't be surprised if he did. I mean, the notion. Maybe he used red ink, hence the term in English, about that in accounting. But anyway, this is a possible notation for a negative number. The subsequent notation that I think Cardano already used was to have a minus, a negative sign. Okay, so big achievement of Cardano, big argument over who invented this. He figured out how to solve cubic equations. In fact, he figured out a general formula that lets you. For any cubic, I could just fill in ax cubed plus BX squared plus cx plus D equals zero, and the result is a total mess. But this is what Cardano figured out, and big achievement. But one of the things that happened here was he realized you really have to. As soon as you're sort of enumerating roots of a cubic equation, you really have to start talking about negative numbers. But actually, he ran into another thing as well. If you look carefully at this formula, you will see a little I here, okay? As many of you will know, that's the square root of minus one. Okay? So here's what Cardano ran into. He said X times X equals, okay, X times x equals 2. Great. You know, we can solve that and we can say X. Actually, let's do X times x equals 4. We can solve that. That's all good. The Results are either 2, 2 times 2 is 4. Great. Minus 2 times minus 2 is also equal to 4. But now the question is, what happens if x times x is equal to minus 4? What value of x? It's kind of the same thing that Pythagoras ran into. What rational number is it that gives me square root of two, that gives me a two when I multiply it by itself? What number is it that gives me -4 when I multiply it by itself? Okay. And so then what happened is that the idea got invented of imaginary numbers, numbers which just have this thing I. And the idea is that two times I I is just this thing that we invent which has this property that I times I is equal to minus one. And so we can start saying, you know, five plus seven I. And we can look at that number and we can say, what's that number squared? And we can do all sorts of arithmetic, all sorts of standard arithmetic that we can do on ordinary numbers. We can also do on complex numbers or imaginary numbers. Imaginary numbers are just the things like 7i, for example. Complex numbers means the real part plus the imaginary part. Okay? So if we look back at our cubic equation here, and we look at this cubic equation here, well, we got this big messy solution here. If we look at what that actually is, what we'll discover is that big messy solution. If we say, what's the numerical value of that? It'll be minus that number, that real number. And then here it's got two complex numbers involved. But there was this formula, and one of the things that happens is that even when. So look back at the big formula here. The big formula here involves I. And even when the actual answers are all real numbers, it's still the case that the formula, in the intermediate stage of getting those numbers, you end up dealing with complex numbers. That's something that's not still not terribly well understood. You know, in the end, you've got a cubic equation, it's got real numbers in it, it's got integers in it. Then you've got this formula for working out the solution. And in the formula, you're doing things with complex numbers. And then when you work the formula out at the end, you get ordinary real numbers without any imaginary parts. Sometimes, often, most cubic equations don't have real roots. Most cubic equations have complex number roots. They usually have a real root, two complex roots, and so on. So, okay, so just for fun, we can look at what happens with equations of the fourth degree, which were actually solved around the same time as equations of the third degree. That is, well, we have. Let's see, let's fill this in. C squared plus D times X plus E equals zero. And let's take a look at that equation, what the solution to that equation is. Okay, it's a monumental mess. And let's make my screen bigger. Just to enclose it. These are all. These are the different solutions. So remember, you know, we went from the quadratic formula, which we can sort of more or less remember. I don't think anybody was, except for maybe the guys who were originally arguing about who invented it, memorized the formula for the solution to the quartic equation to the equation of the fourth degree. But notice if we go and dig in here, if we look in great detail, we will see somewhere Here we'll see 2 to the power 1/3. That means the cube root of 2. That means the solution of 2 times 2 times 2 of x times x times x equals 2. I think if we dig carefully here, we will see. Do we see fourth roots here? I don't think we do. In this representation, there are other representations where you do see fourth roots. But the thing that's worth realizing is if we were to ask kind of, how can we solve this equation? In principle, we've got this formula here that involves square roots and cube roots and fourth roots. It's just taking roots of numbers. And that eventually allows us to solve the equation. Okay, so that was 1500. Then a question that exercised people for 300 years was, well, what about equations of the fifth degree? What about quintic equations where we have an X to the 5 in here? Can we go and do the same thing again? Is there a solution to the quintic that involves just square roots and cube roots and fourth roots and fifth roots? That was a big question for a long time. So in the 1830s, Galois proved that it wasn't possible, proved that there is no formula for the solution to the quintic equation in terms of just roots, that there's no way of writing A pure formula like that. It was actually an important result because it was kind of an impossibility theorem. A little bit like the Pythagorean result that there's no way to find a square root of two represented as a rational number. The Cardano result, there's no way to solve the cubicle without involving complex numbers. And then the Galois result, there's no way to represent the solution of an equation of the fifth degree without going beyond square roots and cube roots and so on. And normally people just say, you can't solve the quintic. That's not really right. I mean, if I type in your average, I'll type in some random quintic and, well, some quintics. If I just type a quintic at random, my chance of hitting a quintic that easy to solve is rather low. But, okay, what we do in Wolfram language here is we represent the solution to the quintic as these approximate numbers. But if I were to say, for example, let's say I just say, give me the possible values here. And I were to say, let's just add all those up. Ah, I want to do this. I want to say, what is all of that? Okay, it's minus six. Well, that's interesting. Notice that six is the coefficient of. This turns out to be a general fact that the sum of the roots is. That depends on that coefficient there. So there are things we can do with these numbers, even though these numbers we could represent approximately, but we can't, like, pick it up and say, oh, it's the fourth root of whatever, Even though we can't pick it up and say there's a formula for it. We can say, I've got this number. Here it is. I've got an implicit representation of it. That number is a root of this equation. It is the third root of this equation. For example, I can pick it up and do things with it. Like, I could take that number. If I take that number, I can perfectly well say, what's that number? To the fourth power plus five. Okay? And I can treat that number just like I could say there's an X there. I. I could treat that number like this. I can then say, let me work out what the numerical value of that is. But I'm just treating this as an arbitrary number that's just sort of hanging out there. So in any case, one of the things people say is you can't solve quantic equations. You can solve them implicitly where you have these kind of algebraic numbers. This is Something we figured out for mathematical and modern language many years ago is kind of the implicit representation of root objects, as we call them. Actually, there's also another solution you can give in terms of elliptic functions, which are a kind of a fancy kind of generalization of sines and cosines. Maybe we'll talk about those maybe a bit, in a bit. But the idea is there is a formula for the solution to the quintic, but it doesn't just involve square roots and cube roots and so on. It involves exotic other functions. By the way, I should say there's a formula for the solution to the cubic that involves sines and cosines, trigonometric functions. You can, instead of using square roots and cube roots and things, you can write it out in terms of trigonometric functions. And those sort of unravel, just like the complex numbers eventually unravel to give you other things. So the trigonometric functions unravel to give you other things. The analog of that for the quintic is the elliptic functions, which don't necessarily unravel. For a quintic equation or actually an equation of any degree, there are representations of the solution, either implicitly or in terms of these higher order elliptic functions, hyper elliptic functions, and so on. So in any case, but let's come back to what we're talking about, complex numbers, the thing of the 1500s, that was. And we can do all sorts of things with complex numbers. I mean, we can. Complex numbers have a lot of the same properties as ordinary numbers. So, for example, what are some properties? Well, actually, now that we're talking about algebra, we can talk about some other things. So, for example, if I say Y plus X, that we don't know what X and Y are, but whatever they are, we know that Y +X is equal to X +Y. And that's a general property of + that we can sort of algebraically describe independent of what these values actually are. We can fill any values in here, but just as a matter of algebra, these two things are equal. So we can do all kinds of stuff with algebra now. So we can say X squared minus one. Okay, great. That's equal to minus one plus X squared. But now it's also the case that this is algebraically equal to X plus one times X minus one. So if we say something like, is X squared minus one equal to one plus X minus one plus X? Actually, Wolfram language will at first just say, I don't know, if we substitute a particular value for X, like we say x equals 6, it'll definitely be true because there'll just be two numbers and they're equal. But we can also take this and we can just say, simplify that. Tell us a simpler form for this. Okay, it's true. Great. And in general, once we have algebra, okay, when we have a number like 56, there's just one way to write out that number. Even writing out a fraction like 2/3. We already saw that the Egyptians had a different way of writing out two thirds. We could write out two thirds as. Or, you know, we could write out seven eighths as the sum of these Egyptian fractions. Or we could write out seven eight as our seven divided by eight. We could write it as instead here as just this plus, you know, this plus this plus this. So again, it's not self evident. What. The fact that we choose to write numbers as. We choose to write 7, 8 as 78 rather than 1/2 plus 1/3 plus 1/24 is again one of these sort of conventions about what we do. By the way, I might mention that one of the things that's always interesting with fractions is, is if we say something like 4 over 6, we could write 4 over 6 as 4 over 6, but we choose to reduce that fraction to lowest terms by dividing through by two. And in fact it becomes a question which fractions can we do that with? Like if we say table of I over J. Well, let's say I over seven, I goes from one to seven. Okay, so some of those, like one. I'm sorry, I shouldn't have done seven, silly me. Ten. Some of those, the only way we can Write them is 3 over 10, but this is 2 over 10. And we can alternatively write it 1 over 5. And so this is, so there's, you know, you could ask a question like for how many numbers, how many of the, of the x of the I over 10 I, whatever we call this N over 10. For how many of the N 10 fractions can we reduce them and how many of them get stuck as N10? So we can go ahead, let's do this for 100, for example, let's see, we can just count, okay, we've got more of these numbers here. And for example, let's say, I'll tell you what, let's just do this. Let's say we want the numerator of that. So that will have reduced it to lowest terms. We get the numerator. And then we ask how many times do we get a numerator equal to one here? Okay, so let's Go ahead and do this for different numbers N. And let's make a table for every number N of how many fractions of its fractions? How many of them? Okay, so this is the result. So for which of its fractions get left as one over something? Okay, so here we're going. Oh, whoops. You know what I should have done? I should have said n minus one here because we don't really want to divide a thing by itself. So this is saying one over two. Okay, this is one of those. In the case of. What am I doing here? This one here is. Which one am I getting? What am I getting here? That should be. Oh, what am I getting? Table of the numerators. Yeah, I think I did this right. Oh, I see. I see. That's one over two. I'm confused here because I should have one over three. Oh, no, No, I'm sorry. I did this wrong. I did this wrong. Very silly. What I should have done. I got myself totally confused here. What I should have done was, let's say I make the table of M over 10. M goes up to 10 minus 1 9. Okay? So the real question is, which of these numbers still has a numerator of 10? I was going Egyptian on us and looking at which numbers have a one in the numerator. That's not what I wanted to do at all. What I want to do instead was I ask which numbers don't reduce which numbers just stay with a 10. Which numbers are not reducible to lower terms? So that would be the denominator here of that equals N. Okay, so this is. Am I going crazy here? Let's just look. Denominator of that line. 143. Okay, great. Denominator of I over N. Okay, what am I doing wrong here? Let's see. So let's say this is for 10. I over 10 equals 10. And let's go. Oh, let's just do these. Denominator of I over 10I goes up to nine. Oh, oh, oh, oh, oh, oh. Silly me. Silly me. I want to count. Yeah, yeah. What I'm doing here wrong is I'm counting the number of ones. I need to count the number of times that it's true that that's equal to one. Okay, so there we go. So this is now the number of fractions that don't reduce to lowest terms. So there's one half, there's 1/3 and 2/3. There's 1/4 and 3/4 there. We could just give if we wanted to instead of just counting the trues we could just select, we could just say for which of these is it true? So we could select in. Well, it doesn't really matter. We can give the list of what numerators. But anyway, this is a number of fractions with denominator N that just stay with denominator N. They're not reducible. Okay, and so if we plot this, let's see. Well, you can just say list, line, plot. It's a pretty wild creature. Looks like that. And so this is a complicated thing. It actually has a name. It's called the Euler phi function, sometimes called the tottient function. And if we work out the Euler phi function of N or N goes up to 50 or whatever, we should see the same numbers there. It has a different convention for one here, but we see the same numbers showing up here. And one thing, if you really start looking carefully, you'll see that for prime numbers this thing is equal to just the number minus one. And those are the maxima here are the prime numbers. So that's 43, there's another maximum at 47, another maximum at 41, and so on. We can talk about. Well, okay, let's talk about prime numbers for a second. So prime numbers again, known in antiquity, known to Pythagoras. What is a prime number? Well, if we say, what can we, if we have the number 20, for example, we can write that number as 2 times 2 times 5. I hope I know my arithmetic. And yes, 20 is 2 times 2 times 5. And many numbers have the property that you can factor them, you can write them out as factors like this. And we can even say, if we say factor integer, we could even make a table here of a bunch of numbers and their factors. And we could say N goes up to 20 and we could make a nice grid here that. Let's just make a nice looking grid. And this will now be on the right. We're showing kind of representation of the factors of the number. So this is showing that 6, this is 2 to the power 1 times 3 to the power 1 or 13 is just 13 to the power 1. 14 is 2 to the power 1 times 7 to the power 1. And the way we can think about this is which numbers don't have this property that we can write them out as a multiplication of other numbers. So you know, we can do it in a kind of a cheating way. 13 is 13 times 1. But let's say that doesn't count. And let's imagine that we are trying to make, let's say we're trying to make, we're trying to take, let's say 14 squares and we're trying to make a rectangle out of 14 squares. Well, we can do that. We can say it's a 2 by 7 rectangle because 2 times 7 is 14. And so we can arrange squares in a rectangle to get. Or we can arrange dots in a rectangle and we can have 14 of those in that rectangle. But there are some numbers where we can't arrange them. They're numbers. You know, we often talk about a number like 16 is a square number because it can be, you know, if we make, I don't know, let's just make a table of. Let's do this. We can make a. We really do it pedantically here. We can make a table of 16 little dots and we can partition them and then we can make a grid of those. And we can plainly see that our 16 can be arranged in a square. If we say, well, how about 14, can we arrange that in a square? And the answer is, well, yes, if we have 14, then we can arrange it in a square which is 2 times 7. And if we have 20, we can arrange it in a square. Actually, sorry, in a rectangle. If we have 20, well, we can arrange it in a rectangle in different ways. We can arrange it in a rectangle like that. We could arrange it in a rectangle like, like this and so on. But the question is, can we arrange it in a non trivial rectangle? And. The question here is, and that's the numbers that we can't arrange in rectangles. Those are the prime numbers, the numbers that we can make up as units, that we can use as units to build other numbers out of. And it's a fundamental theorem, often called the fundamental theorem of arithmetic, that every number, every ordinary number has a unique factorization. There is if we try and represent that number in terms of primes, there's a unique way that, that that number can be represented in primes. It's 2 to the power 3 times 7 to the power 1, where 2 is a prime and 7 is a prime. There's always a unique way to do that. And that's a fundamental fact about integers. It doesn't happen to be true for other kinds of numbers, but it is true for integers that you can find these discrete set of primes that, for which the. Where every integer can be written as a product of those primes. Now, to tell whether a number is prime, one very simple way to do that is to Just say, well, can you divide it by anything smaller than it? If you can't, if you have the number 56, you might say, well, can it be divided by two? Yep, it can be divided by two. So it isn't prime. And so then we're. And one way we could test is something prime. We could just say, try dividing it by a whole series of numbers. Let's try just dividing it by first 10 numbers. Okay. There are cases where it divides exactly where it doesn't get left as a fraction. So this number isn't a prime. Let's try 57. Okay, 57. I didn't actually know if 57 was a prime. Immediately 57 isn't a prime because, look, it's divisible by three there. And let's try 59. Okay, looks like 59 is a prime. How far do we have to go in doing these divisions? Well, whatever. We definitely don't have to go further than eight, because by the time we've got eight times eight, we've got 64. So if 59 isn't divisible by eight, there's nothing. If it was going to be divisible by anything, it's got to be divisible by something less than 8, because by the time we've got to 8 times 8, we've got a number bigger than 59. So this is kind of the. So that's how you work out whether a number is prime. Actually, there are vastly more efficient ways to work out whether a number is prime. There are all sorts of cool tricks. There's this thing called Fermat's little theorem, which is not his last theorem, but his little theorem, which actually relates to this Euler Phi function that I was talking about before. And there's this property. You fill a number into a particular formula, and if it's prime, that formula has to work out in a certain way. And so that's a sort of a cheap way to test to get a good estimate of whether a number is prime. If you try that test a bunch of different times, you can get a higher and higher probability the number is prime. And then people figured out how to get a deterministic version of things like that. But anyway, you can work out by something much more efficient than just doing successive divisions whether a number is prime. And there are all sorts of things to say about the distribution of prime numbers. And it's been a big driver for mathematics to work out what the distribution of prime numbers is when we make a big table of prime numbers, you know, we can make a table of the first. You know, let's say we make a table of the first thousand primes and then we. Let's say we plot those first thousand primes. Endless work has been done on. Oh, that looks very regular, doesn't it? Let's go down to 100 primes, it'll look a bit less regular. Endless work has been done on trying to figure out what you can actually say about that curve. And for example, roughly how fast does the curve go up and all these kinds of things. So this is. So that's the story of prime numbers. Actually, in the time of Pythagoras, prime numbers and perfect numbers were really neck and neck as being equally important perfect numbers. So if we look at the. Divisors of a number, so divisors are all the numbers less than that number that can divide that number. So trivially, 1 and 10, divide 10, but also 2 and 5. So if we say divisors of 100, we'll get a whole bunch of numbers. And one of the things that Pythagoras really was interested in was numbers where if you add up all the divisors, so you take the total of the divisors, how does that compare to the number itself? And a perfect number, I think we actually have. Yeah, we have a perfect number is a number which has the property that the sum of its divisors, except for itself, is equal to the number itself. So if we go here and we say, let's work out the divisors of 6, the divisors of 6 are 1, 2 and 3, and the trivially 6 and the sum. So if we say, let's take the total of most of that list and we get six. Okay, so for what numbers, is the sum of their divisors equal to the number itself? And so we could just go through and we could find the total of most of the divisors of M of N. And let's work that out for a table of that for N goes up to, let's say 100. And these are. So these. This is. Yeah. Okay. Actually what I should do is subtract the number itself here. Actually, let's take the number itself. No, we can subtract the number itself. Okay. So here, most of the time we don't have. Actually, this is a funky curve. I wonder what this even looks like. I don't even know what this looks like. I'm not sure. Okay, so this is a curve that says, what's the difference between the number and the sum of its divisors. And occasionally it's zero. That is what Pythagoras called a perfect number. A number that's equal to the sum of its divisors. And if we look here, let's find which of these. Let's find the first few perfect numbers. Let's say, what's the position in that list of the perfect numbers? Of the zeros. And we'll find 6 and 28 are the first two perfect numbers. What's the next perfect number? Well, the next perfect number is. Oh, why is that? Oh, I was typing the wrong place. That's why I was not seeing it. Okay, the next perfect number is. So that was. The third Perfect number is 496. Okay. The fourth perfect number is probably a lot, quite a lot bigger. Let's make a table of perfect numbers. And N. Let's go N goes up to, let's say, 10. So those are the first 10 perfect numbers. They get pretty big pretty quickly. And these perfect numbers are related to primes of the form 2 to the n minus 1. There's a whole story about that, that Fermat was. Pierre de Fermat was much involved in. That's a different story. But what we see is there are perfect numbers. So this number here has the property that the sum of its divisors. Let's work it out. Let's prove that that's a perfect number. Let's work out its divisors. If I could spell, and there are its divisors, and let's take the total of those and, okay, we have to subtract the number itself and then we'll find out it's equal to the sum of its divisors. Okay, so one thing Pythagoras wondered is, look at these perfect numbers. He only knew, I think, the first three. Maybe these perfect numbers are all even numbers. And it's probably the oldest unsolved problem in mathematics. Are there any odd perfect numbers? Are all the perfect numbers even? Clearly, the perfect numbers are few and far between. But the question is, could there ever be an odd perfect number? Nobody knows the answer to that. It's been an unsolved problem for 2000 years. So it's kind of interesting to see how quickly one gets to that. But perfect numbers are about sort of making numbers up from their parts from their additional. But parts under addition, usually called aliquot parts. And back in antiquity, people were no more interested in primes than they were in perfect numbers. Primes turned out to be much more fertile from the point of view of thinking about mathematics. Well, I should wrap up in a moment here, but I think what we were talking about here was the representation of numbers in terms of primes. And one question you can ask is. If we ask to factor. Let's say. I was talking about. We're talking about things like complex numbers. Let's see, how do I get a good one that isn't. Let's see. Come on. Oh, that's what I need to do. Factor integer. What was I doing? I was doing the wrong thing. Okay. Funky fact about complex numbers. They also turn out to have the property that. Is that right? Yes, I think they're a unique factorization domain. Somebody. I think that's right. Yeah. Anyway, this is a way to factor 8 plus 2 I as minus I. These are essentially the. The Gauss, the primes. So we can ask something like, is 7,5i? Is that a prime with respect to. In terms of complex numbers? So we could say. Here we can say factor integer. That thing. I think we default to. Yeah, we do. So that shows that this is actually seven, five I is actually I plus one times I six I. So if we ask about one plus six I and we say, is that a prime? I think it will default to. It will say, yes, that's a prime. If we say 1 +7 I, for example, chances are that's not a prime. And again, we try and factor that, we'll see that that can be factored into complex numbers. So that's a feature of complex numbers. It's not a feature of other kinds of ways to construct numbers that they have this way of. Of having the notion of primes. But primes are kind of. The big result is that every whole number, every integer, can be uniquely factored into primes. It's not like a rational number where you can write it out as a sum of other rational numbers like the Egyptians did, or another way. There's a unique way of representing any integer as a product of primes. That's often called the fundamental theorem of arithmetic. So while we were talking a little bit about algebra, I thought I was wondering whether I would make it as far as calculus. If you guys have thought this was fun, I thought this was fun. Maybe I should go on another time and see if I can make the bridge from algebra to calculus and beyond. I think I'll be able to do that. I would Predict it's another 45 minutes to get seriously into calculus. But I think what I was showing you before was I was trying to show you that when we have Different. When we have an expression like x to the 4th minus 1 or something, there are different ways to write X to the fourth minus one. And that's a feature of. We can, we can represent the same algebraic expression in different ways with integers. There's one obvious way to do it with algebraic expressions. There are different ways to do it. A very common way to think about it is in terms of so called polynomials, which are sums of the form x to the 1 plus x to the number plus x to the 1 plus x to the 2 plus x to the 3 and so on. Kind of the analog of writing out a number as 1 plus something times 10 to the 1 plus something times 10 to the 2 and so on. You write out a polynomial in which you have. We could actually do this in a funky way here. We could say, we got a list of digits here and we could say, put an X there. Oh, come on. We could, we could write it out like this. This is kind of, Instead of a 10 here, we're just putting an X there. This is kind of a. This is sort of the algebraic equivalent of a number written in positional notation. It's called a polynomial. And we get to do all kinds of things with polynomials. And there isn't just one way to write out this polynomial. We could write it out in different forms for any value of x. These forms will always be equal, but they have different structures. And that's kind of a big thing. When you do algebra, you're typically transforming from one of these forms to another of these forms and there's no necessary. There's not really a best form. You might say this form as an expanded polynomial is the best form. But for other purposes, you might say this factored form is the best form. There are all sorts of best forms. All right, well, okay. Sounds like people were having some fun here. So. All right, I'll go on and I'll do this another time and we'll make it to calculus and beyond and see how far we can get with math. See how many. See how many weeks, months, years of math classes we can compress into a short amount of time. I would say that the key thing that's letting me explain all this stuff is that I'm just typing stuff into orphan language and we're seeing what happens. That's kind of why we're able to, I think, as easily as this, sort of explain what's going on. Anyway. Well, that was fun for me. Thanks for joining me. And I'll. We'll find the time to Schedule another of another of these live streams and maybe I should upload this. Let me go ahead and do that. Actually, I am going to. Let me share again for one second and let me go ahead and upload this notebook that I just created to the cloud so you all can can play with it. So let's go ahead here. And it says upload to the cloud. I'll just publish it. I'll publish it on my account here and I'll just say publish. And you should get that. And I can put this in. Well, here we can do very, very weird tech. If I was doing this in person, I might, might just show you that barcode. Oops, there we go. If you go to that on a phone, you should be able to open this notebook. And I will also put the. Maybe somebody can add this to the chat session and I can just to show you what this does. If I go there, I can go, oops, here we go. This is just a web browser. I opened up our math story in a web browser here. If you go there and you can, if you have a Wolfram ID and an account, you can go and start just. Actually this is a public copy. So you can say make your own copy. You make your own copy and you can start editing this and I could change that. Oops, did I make my own copy? See the original, you are viewing a copy of the original file. Okay, so here I can just edit that to a 5 and recompute it and once it's loaded in. And so you can do that too, you can create a cloud basic account if you don't already have one and play around with it and. Okay, so next time we try to get to calculus. So see you another time. Bye for now.