All right, welcome everybody to Maths and Ideas from Antiquity to the Renaissance with Rezanegar Astani and I'll hand it straight off to them. Hello everyone. So today I'm going to, as I promised, I'm going to look at some ancient methods of multiplication. and division, the so-called Egyptian algorithms, where some of the stuff that we were talking about are recursive, you can see how they are working, even though they are far, far what you might call rudimentary in comparison to new methods and what is meant by recursive
functions. But nevertheless, what we want to look at is just simply at its most basic level what recursion is and how recursive methods are applied to mathematical operations. So any question before I start from last session? I kind of have a question from last session. Sure. sort of more of a topic you mentioned Hegel someone asked you will I think and
I think it was Maria yeah and I was just kind of curious about what you think about the relationship between Hegelian negativity and I guess philosophy of mathematics generally or how it could apply I just yeah in my notes I was like oh we talked about Hegel a little bit and I don't remember sure sure I mean the thing is that's probably not mathematics but logics I think so you know in basically classical logic negation is quite what you might call to be one
one-dimensional even though it plays a fundamental rule but its rule is usually defined in terms of material implication which means that material implication comes first and the negations comes after it as a kind of what you might call resultant function but the thing is that you see for example in science of logic that Hegel gives a fundamental role to negation you see that as soon as so basically you get you know the bifurication of logic to intuitionistic school and what you might call to be the formal which you can see it in terms of formal mathematical
extension of classical logic. But then something happens at the beginning of the first century through the controversy between probably three people called Mugrov, Hayting and Hilbert, where the followers of Brower, which Hayting and Mugrov belong to, they develop something new, something called intuitionistic logic, which is different from really intuitionism. It has different principles, but nevertheless it is still, you can say that the same, it It holds still the main principles of the intuitionistic school. The thing is that Brouwer and Hayting start to realize that basically all logical connectives
and all logical behaviors can in fact be defined in terms of negation. And that leads them to understand a different, what you might call a different picture of classical logic. And that's what they call the calculus of the problem. So what is the calculus of problem? The calculus of problem is what you might call every formula or every formula needs to be accounted for. You cannot simply duplicate or you cannot add new premises to your lines of proof.
For example, in classical logic you have a turnstile b, then you add a new premise, it would be a comma a small a turnstile b. You still get the same consequence, namely after adding the new premise, your consequence will be still the same. But the thing is that this really only holds in classical logic where negation is defined very rigidly. So they started to think about what is really the function of negation. They came up with this idea that negation fundamentally is what you might call to be the fundamental
architectonic of logic and the ultimate function of negation is two things involution and interaction which is quite similar to idea of Hegelian negation that is being laid out in terms of dialectical interaction so a good example of this what is involution involution you can say that's for example minus parenthesis minus one equals to one. That is like a form of involution. That's basically what you might say. This double negative does something.
You can trace it as some sort of a spatial configuration. Going basically I posits, like you posits, I posits minus one. you posit a negation of that but it always comes from outside. A good example of this is like the game of chess. The game of chess or any form of dialogue in fact, you see that the whole way that a topic of conversation emerges is by way of this going back and forth, switching the roles between the two players. And this switching of the role that allow you to construct a topic, or for example in proof theoretical terms, between falsifier and verifier, the proof of A and the proof
of not A, this interaction between these two processes can be understood as the ultimate function of negation. That allows for the emergence of, as I said, logical behaviors and logical connectives. And then you see that for example in late 1970s, early 1980s, Jean-Yves Girard, the French logician, completely builds on this. It's basically the whole premise of classical linear logic where you have two things. One the so-called resource sensitivities of formulas. For example, in classical logic you might say that I have one dollar bill and I want
to buy a camel and a Marlboro and the camel costs one dollar, the Marlboro also costs one dollar. In classical logic precisely because the resources, namely your formulas, you can simply copy or add them without any restriction. is the you know because of its strong structural laws you can buy both a Marlboro and a camel with your one dollar bill which of course in real life is preposterous because it doesn't work like that. You can only buy a camel or a Marlboro but also imagine that if how you
were supposed to spend your dollar bill and the choice of the cigarette that you want to buy wasn't really yours. It was for example the store that gives you the choice of whether you spend this one dollar bill that you have toward the Marlboro or Camel. So it really depends on this interaction between the two players in an interaction who has the choice of formula and who by virtue of that who posits a counter action so I you know you can think of my spending of a dollar as an action and this store that gives me the choice of what pack of cigarettes I can buy with my one dollar as a counter action and depending on these choice who has a free choice over what
to spend and what to give, this is how negation is being laid out in today's logic, which is the so-called, what they call it, logical or mathematical dualities. And dualities is simply an intuitive way of thinking about it, it's simply an interaction. An interaction means switching the rules between the system and environment, algorithm, algorithm one algorithm to play of one player to falsify or verify proof of A and its counter proof you see in classical logic when you are trying to prove A the negation becomes part of your
proof of A but what if the negation of A or your counter proof could also be understood as actually a logical process independent of your proof of A. And the whole idea is that the proof of A understood without this implementation of a structural rules of classical logic can be said to be a dialectical negation between proof of A and counter proof of A. So this is really, you can see that this idea of negation, once being de-rigidified from its classical conception, and becomes, it is outlined in terms of dialectical interaction,
with this interaction being the switching of the roles between a proof and counter-proof. You can see that it's Hegelian dimensions. And then Girard, for example, builds on this and actually talks about Hegel in Locus Solum, Place Only Matters, when he develops the ideas of ludics. And then you see that the interaction between, for example, proof of A and proof of not A can lead to different scenarios, what in ludics are called designs. And then you get exactly the scenarios that Hegel had in mind. You get like bad infinity, you get absurdity, you get, for example, convergence, what you might call to be agreement between parties, so on and so forth.
I have two questions. Is it on the same topic? Can I follow up real quick? Yeah, I was just asking, you said Brewer and somebody else? Brower, so you know that Brower is the founder of intuitionistic school, and he gives a fundamentally Kantian account of logics and mathematics in terms of his construction out of what you might call to be manifolds of intuition. and so the followers of Brower in 20th century the most famous ones are
hating aren't hating like the hating algebra and called mogul they both contributed massively to information science and computer science So it's kind of like involution and interaction correspond to like self-relating negativity and determinant negation or something like that? Yes, you can say that for example, determinant negation is resolving the incompatibility, semantic incompatibility between two interacting parties. this is the interacting part is you can again you can understand them in really different
terms from like two computers on a network several protocol you can several clients you can think about them in terms of two competing proofs for for example a and that would be the proof of a and the proof of not a you can think about in terms of games you can think about in terms of simply linguistic dialogical interaction so on and so forth this is really fascinating um so i guess what last question on the topic is so hegel you know famously proposes that logic is already metaphysics and and even sort of access to logic is already philosophy in a way
and so the church intuitive logic is kind of realizing logic as well this sounds so so like So what is the relationship between intuitionist logic and metaphysics, I guess? Or what about that aspect of Hegel? That's his ambition to kind of cover everything with logic. Well, you see, I think if you take the Kantian side of Hegel, then you see that if we say that the roots of you know logics or mathematics
is intuitionistic and precisely because really by intuitionism they mean it in the sense that they are derived from the local or the organization of the manifold of intuition and precisely because the manifold of intuition you You can think about it in terms of a continuity with some outside reality that you simply have to take as given. You see that it has metaphysical dimensions, whereas for Hilbert, he doesn't believe in this. I mean, his position is actually very much more subtle. He still believes in some that mathematical construction or logical constructions have
some roots in intuition but he believes in some sort of what you might call a transcendental imposition that's the only way that these manifold of intuition which you might say you might lay it out in terms of local invariances that are derived from sensory experience or interaction with the world there is not such an organization in advance that's that they are derived from you you can have this organization only have you only if you have a priori formal concepts and precisely if you have a priori formal concepts it means that
these a prior formal concepts can be detached from this intuitionistic dimension of experience, de-semantified or de-contextualized from their experiential context and turned into formal elements the formal elements that you can simply manipulate them without you having any connection with you know again with the intuitionistic axioms of math or logics. So it really comes back to this whole idea of we talked about it I think the first session
two approaches to mathematical construction that is mathematical construction is coming from the experiential dimension namely the manifold of intuition which can be sought in terms of some sort of continuity with the objective realm or is it the idea that if you have the manifold of intuition that by itself does not have any what you might call to be axiomatic import for mathematics and logics. The axiomatic import of it is derived from the imposition of pure concepts of understanding or in Hilbertian terms formal concepts which
are purely linguistic in a formal sense on the manifold of intuition. So whether it is the construction of mathematics is coming from below the experiential dimension or is coming from above the formal aspects which are a priori in a full sense and analytic. Cool, thank you. I mean another good way of thinking about this is again comes back to the to the conflict between in chance between analytic and synthetic judgments and a great essay on this has been written by I mentioned I think this guy
per Martin law it's called analytic and synthetic judgments which is the development meant of constructive type theory which is part of you know again revolution 20th century in logics and computer science his name is her and the analytic would be conceptual and the synthetic would be yes analytic would much more in tune with the Hilbertian a school and the synthetic with the Browery in the school and also the the negativity as being laid out in
terms of evolution and interaction you can read Alan Lecomte I typed it ludic meaning and dialogue it's a book not the essays it's really it starts from you know kind of very introductory account of what were the problems of classical logic and how they were redefined in 20th century to the point that negation now understood in terms of dualities can you type out the name of the author oh yes I have I have typed it in the the chat box is Alan Lecombe ludic meaning and dialogue oh yeah there it is all right thanks thank you so any more questions
Oh, Locus Solum is Girard's famous unpublished manuscript where he develops ludics. title of it is a kind of like a bastardization of Raymond Russell I think locus solus the locus solo means place or location only matters it's the idea
that's all you need in order to you shouldn't so basically is the idea that you know when you work with classical logic your logical behaviors are always defined in terms of some pre-existing procedure or rules for example think of classical logic as a game in which some rules have already defined how you you should play this game and then all of your behaviors are constrained by your pre-existing rules, the so-called procedural rules which are quite as strong. Now Giraud in Ludix which is this dialectical interactive account of logic which tries to
re-found logic in some proteological computational fundamental processes is that all of these procedural rules are being suspended all you have is interaction on the location of your syntactic sign that's and the location of the syntactic sign you can think of it as simply a memory as a memory cell that's all of your what you might call to be inferential relation between your syntax, your semantics, your procedural rules are in fact being derived from how two players, the falsifier and the verifier, interact over the same location,
the same in Kersian terms you might call empty sign. Awesome, unless anybody else has burning other questions, do you want to go ahead and get started? I wrote a question in the text bar. Explicit reference to Hegel. Oh, that's Jean-Yves Girard. Oh, sorry, it's Steve.
Okay, so let's start. So I talked about, kind of like a little bit about the definition of recursion versus iteration and what recursion enables us to do in terms of creating these hierarchical structures, self-contained hierarchical structures, where you can single out some long distance rule and reapply to the hierarchical structure and generate more structure.
And then I also talked about the deficiencies of the recursive methods or sequence matchings. Now let's get back to some really, really basic examples. If you remember, let me turn on the iPad so I can start drawing stuff. If you remember we talked about the origin of numbers and you remember I mentioned that the number sense and the origin of numbers has something to do with groupings.
You remember I said that the first forms of numbers you had, for example, different markings without any forms of grouping. For example, these marks correspond to the sheep in the cattle that you have. Now as a number of these markings grow, it's really hard to keep track of them. by virtue of being hard to keep track of them, it's also difficult to impose basic mathematical operations on them, like addition and subtraction. I mentioned that the solution for this tracking,
which you might call to be making computational costs and memory parsing, are these markings friendly, is by dividing these markings into chunks, into bundles. So for example, you have, instead of having like, so say you have 10 marks, you know, in a row, you can turn them into groups of, let's say for example, groups of five or groups of two and then cross them and that's how you show the group. So this idea of grouping is ultimately what addition and subtractions signify, so as multiplication
and division. So Egyptians for the first time are the ones who came up with this idea that multiplication can be laid out in terms of these groupings that reflect additions. So as divisions. Divisions also reflect groupings that subtractions signify. Now let's show how this works. you you
What does it really represent, this operation, to you? 5 plus 5 plus 5? Yes, but can you elaborate in terms of the groups? What it tries to do in terms of groups? Because numbers, as we talked about, ultimately at their foundations are groups, they represent groups. So if you have, so it's basically what it tries to say is that what does it mean to
have three groups of five? You remember that we have these markings. We said that the numbers started to evolve as groups. Now the problem of multiplication, exactly like addition, is about grouping. grouping so you have three times five tries to tell us three what would be three groups of five
now Egyptians had a devised an algorithm that precisely elaborates this what you might call to be fundamental relation between multiplication and addition only in terms of groupings. It's called the method of doubling where you can see what it means for example for higher numbers what it really multiplication expresses. For example let's say you
let's say 35 times 15 so Egyptians are start to think about if this is really a problem about grouping giving 35 groups of 15 so you have imagine that you have some sets within which you have 15 circles. I can't draw 15 circles in this. So you essentially want 3, 5 groups of this set. The best way of how...
So the way that you solve this problem is that to... to see how many groups of 15 you can in the most minimal amounts of computational time and the most minimal amount of effort you can produce. The method is called doubling and it goes like this. So we know that one group of 15 is just 15, right? two groups of it double it and hence the name doubling. Two groups of 15 is 30, right? You doubled. So you have one, two, the next doubling of would be four.
So you see, you do not, no longer, you don't do any multiplication, you are simply adding here. So the doubling of next line would be what? 30 plus 30. You again double it. So 4, the double of 4 would be 8, right? So 60 times, 60 plus 60 would be 120. You again double it. So 8 would be 16. 120 plus 120, 240.
Double 16, 32. 240 plus 240, 480. Now, how can you, you see, you have created enough groups groups to correspond with 35 groups of 15. So you have 30 groups here that equals to 48. have created four groups here that equals to 60 and you need one more group here you have one
1, 2, plus 32. These are your groups. The corresponding number of these is 15 plus 30 plus 480. You can now simply express your multiplication in terms of the addition of the elements that
are inside your groups. So as I mentioned, 35 times 15 is really the problem of what would be 35 groups of 15. So you want essentially to find or construct recursively and you see that you have a doubling procedure that recursively adds to your number of groups until the, basically the solution halts when you have found enough groups of 15. And what is the number of groups of 15 that you want to find? 35.
So you started doubling first one group of 15, then two groups, then four groups, then eight, then 16, then 32. So we have already created enough groups of 15, 35. It's 32, 2 and 1. That corresponds to 35. The number of elements that each of these groups have correspond to 15 for 1, 30 for 2 and 480 for 32. Now you add them and you get your answer.
Was this clear, the relation between multiplication and addition? And then you see you simply started to recursively double the number of groups that you should find to correspond to 35. And the number of elements that these groups contain corresponds to your solution. Any questions? Was this clear? So is doubling, multiplication or addition? Or does it matter?
You see the only thing that you are, they didn't want, they didn't in fact knew that what really multiplication is. So basically they developed a doubling method. And the doubling method, all they needed for them and they didn't really, so basically how Egyptians approached this problem is that they started to devise a table. So if you double, for example, any group, so they started, for example, they have some rudimentary multiplications for the stuff that they wanted, for example, measuring the number of cattle, the farm, the distances and stuff. So instead of, for example, doing the actual doubling, they started to create a table of
doubles. For example, for this amount of numbers, the amount of numbers that routinely you're using, let's say for example 15 is the amount of taxes that you have to pay. Now any number times 15, you can easily already divide a table of the number of doubles. So one group of 15 is this, two groups is that. So even in terms of we, for us, it seems to be a multiplication. But ultimately how they approached it is that they simply started to create tables, like algorithmic tables, tables of instructions for doubles, for the numbers they were routinely
using. But the only thing that even in this method, in like the reconstruction of the Egyptian methods we are encountering, that is really multiplication is the idea of doubling. But the doubling for them is simply a recursive procedure that allows them to find sufficient amount of groups that they want. Ultimately the problem is reversed back and being basically solved in terms of addition. You see you have as we mentioned the problem of multiplication is really the problem of what you might call finding the groups of numbers that correspond
to your multiplication 35 times 15 corresponds to the problem of finding 35 groups of 15 and once you find enough groups this problem can be decomposed and elaborated in terms of addition 15 plus 30 plus 4 180 corresponding to the one group two groups and 32 groups as here we have on both columns you have simply additions
And so we have artifacts that are these doubling tables that they… Yes, yes, they have actually, they have both for multiplication and division, but division is a different procedure nevertheless it is still recursive. It's a method of halving, which basically you cut them to half instead of doubling. So with every multiplication it can be elaborated in terms of doubling. And once you have two columns, then you add your groups and you add the number of elements that correspond to these groups on the first column.
Another example. and you might do this yourself I mean it would be great if you can get a piece of paper and start doing these oops 5 times 4. So we start doing this and see if you can find without using actual multiplication, using
the Egyptian method of doubling you can find the number 125 times 4 you want to start with one group of four you want to find 125 groups of four Thank you.
And yes, Theodore is right, doubling is an additive that is being recursively expressed. So… Is subtraction allowed yet? Sorry? Is subtraction allowed yet or just addition? just addition. Subtraction is only expressed in terms of divisions. So 1, 4, 2, 8. You just be careful because if you do not double on the first column, you definitely get a
wrong answer. Just be very careful with your doublings and the number of adding the second groups elements of the second groups the second column sorry Thank you.
So, you see, because multiplication is commutative, you can do it the other way. other way too. But nevertheless, right now we want to really see the relation between addition and multiplication being expressed recursively. And the more number of steps
that we have, we can see this recursion, how it functions. And what happens if by accident we make a mistake in either the first column where we are doing the doubling or in the second columns where we are adding the elements. anyone got this method can we go to the division now you
you you okay so the division is a bit more trick is trickier and i will give you two homeworks just basic and then you will notice I won't tell you what happens for some of the numbers that simply the Egyptian algorithms of method of so-called hat dob doubling and having doesn't work for them and of course they had you know a way around these problems by adding new algorithms
or table of instructions so the method of division is that the first thing that we need to know about the method of division in Egyptians the numbers for example two divided by I don't know 180 the quotients or the answers to your divisions are not expressed in terms of numbers like a you know one thing or something like that but everything is expressed by something called a unit fraction a unit fraction is as
as its name says it, is our fractions whose numerator is 1 and denominator is any number, any integer. Now let's get back to the idea of, let's say 15 divided by 10. What does this express? that the relation between multiplication and addition could be expressed and be made explicit
in terms of how you recursively add groups of something like 35 times 10, how many groups of 35 you need to find in order to add them so you can get, for example, 35 groups of 10. Now if we are going to go into divisions, then what would be the relation between your division and we know that division can be also understood in terms of subtraction. What does this really express in terms of groups of numbers? What are you exactly trying to do in a division like 15 divided by 10 if you were going to talk about in terms
of finding groups what kind of groups do you and basically are you trying to find in in your division in 15 divided by 10 Theodore, can you elaborate on that? How many groups fit in? Which groups fit in what?
Yes, this is groups as parts of. Yes. What is part of what? How many groups of 10 does 15 fit into? Yes. Imagine a good, back to our pizza example, the recursive, because cutting pizza is essentially recursive method because you have you you need to have the memory of your previous action you know to make for example did to make equal slices so imagine this that you have 15 pizza and you wanted to feed 10 people so everyone
gets equal slices Now, before I start to talk about how Egyptians solved this fraction or division, I want to introduce a method that has been devised by Sylvester, the mathematician of the 19th century, That allows you to express these fractions very easily through unit fractions. And then I will get back to the Egyptian method and then you see that it would be much easier
to use the Sylvester method. And the Egyptian method doesn't work in some cases. So Sylvester method has four steps. For example, the fraction that we have. The first thing that you want to do, and you can write this down, is finding the largest unit fraction. And how do you find the largest unit fraction?
It's the one with the smallest denominator that is less than the given fraction. To find this, namely finding the largest unit fraction less than the given fraction, you can do this by dividing the denominator by the numerator, and take the next integer, which is greater than the quotient for the new denominator. For example, you have four, three. Finding the largest unit fraction that is less than your given fraction, namely four over three,
over three, what you want to do is to divide three by four. So what is three divided by four? Okay. or let's go with this one you essentially you want to round whatever quotient that you get you want to want
to round it to the next integer so another one let's say 10. 10 divided by 7 is 1 something, right? So if you round it to the next integer, it would be 2.
So your unit fraction is 1 over 2. What is 1 over 2? 1 over 2 represents the largest unit fraction that is smaller than your given fraction, namely 7 over 10. example let's say 2 divided by I know 15 now give me the unit fraction for this a unit fraction the largest unit fraction that is a
smaller than your given fraction using this method remember what you want to do is dividing in order to find this largest unit fraction that is smaller than your given fraction you want to divide your denominator by your numerator and then take whatever number that you have to the next integer greater than the quotient. You don't need to calculate it. I mean the best thing is just 15 divided by 2
is seven something right obviously it can't be eight so it's seven something the next positive integer would be eight so your largest unit fraction that is a smaller than your given fraction is one eight So, this was the first step in Silvester method, finding the largest unit fraction that is less than the given fraction. Now, the second step is subtract the unit fraction from the given fraction to see what
is left over. So to subtract this unit fraction from the given fraction to see what is left over. The third step, if what is left over is a unit fraction, you are done because ultimately we want to answer the problem of 2 divided by 15 only in terms of unit fractions like Egyptians. So if you are subtracting these two and your resulting result of the subtraction is just a unit fraction you are done. However if it is not then
you have to repeat the process until you have fully decomposed 2 divided by 15 only in terms of units for actions. Let's start with a simple example. Let's say 3 divided by 5.
We want to express this in terms of unit fraction. So, the first thing is that we have to find the largest unit fraction that is smaller than your given fraction. To do so, we divide the denominator, namely five, by three, your numerator. So five divided by three, what is that? It would be one something.
You don't need to ever... The great thing about this method is that you never know what is after the point. You can just say the one something. All you want to do is that whatever it is, you round it to the next greater integer, would be 2. So your largest unit fraction, that is smaller than your given fraction, is 1 over 2. to solve this as you know
so let's say 5 times 2 10 And basically we want to find the common denominator for both of these fractions in order to subtract them easily from one another. And finding the common denominator for two fractions whose denominators are different, it's just the best way is to simply multiply their denominator. So 2 times 5, 10, 5 times 2, 10. 2, we have multiplied it by 5, so we have to multiply it by 3, right?
6. We have multiplied 5 by 2, so we have then multiplied by what? By 1. So it would be 5. Check the calculator to see it approximately is equal to 3 divided by 5. 110 is approximately equal to 3 divided by 5.
You have 0.6 and 0.1. Why is that you got so much divergence? Or is it really divergence? Any idea? Jake? I got a little lost along the way. I've got roommates came in and out of the house and I think I missed a crucial step and now I'm just trying to puzzle out looking at it. Oh, okay, okay.
Do you want me to go over it then? Yeah, if you didn't mind just going back over the last, like, two minutes, last few steps or something again. Okay. So, and before I start to re-express this, is that the reason these two are different is not that they are different, but the whole idea of solving divisions in terms of unit fractions always can be thought in terms of some sort of approximation. This approximation becomes particularly, you know, divergent from the actual answer if your fractions are small.
But as your fractions grow, become larger, actually you see that the approximation quite asymptotically convergence on the actual answer. So in terms of the small fractions you see that there is a contrast when you do the calculation. But as the fraction becomes larger, this approximation becomes less and less of an issue. So Silverstone method again, you have numerator, you have denominator, you have denominator, Let's say 5 over 9.
In the Silvestre method, the first thing, so 5 over 9 is that 5 divided by 9. To solve this division using only unit fractions, what you want to do using Sylvester's method is to find the largest unit fraction that is given in your given fraction. To do so, you divide your denominator by your numerator. Then you round your quotient to the next positive integer.
So, nine divided by five is what? One, or round down to one. Because it's 1 in 4 9th. Or it's about 1 in 4 9ths. Yes. But what about this? Let's say... 0. 0. 0 So have you noticed that there is something that is now happening that Silvester method doesn't work anymore?
So it only wants to do small fraction smaller than one which is why it deals with these unit fractions Yes Right And is that also linked to why smaller sub one fractions yes urge Yes, yes absolutely So Thank you. So the first thing is that you divide your denominator by your numerator. Whatever is, for example, if it is one something, then you round it to the next positive integer.
It becomes two. Two is the denominator of your largest unit fraction that is smaller than your given fraction. So your largest unit fraction then would be 1 over 2. The second step, you subtract your given fraction, sorry, you subtract your largest unit fraction from your given fraction. If the result of this subtraction is a unit fraction, you are done because that's essentially what you are trying to do just decompose your fractions in terms of unit fractions if it is not a unit fraction then you will continue the process until
you have finally decomposed your given fraction only and only in terms of unit fractions now what is that units fractions are important and Egyptians wanted to express everything with unit fractions well there are so many explanations for this. One, because unit fractions are extremely easy to measure, precisely because you are dealing with units. But there is also a cosmological basis to this, part of Egyptian mythology. Because unit fractions represents ideas of completeness, oneness. So there is, again, as part a mathematical really construct, but also it has a mythological appeal for Egyptians,
and that's why they were using them. Okay, come up with... I think that the clearest example of this is just breaking 5-6 down into the two fractions of 1 1⁄2 and 1 1⁄3. I mean, it might be clearer that way than in the kind of verbal description. We actually get to the unit at the end, like 3, where there... Does that make sense to try to do 5, 6 equals 1 1⁄2 plus 1 3rd, and just... We can do the recursion then, do the first one and then the second one?
Recursion to the first one and the second one. What do you mean by that? I mean if you do, if basically you're doing 5 sixths, you divide 6 by 5 and you get 1 which you round up to 2. So you get, you make that 1 half and then you subtract that from 5 sixths which is 10 twelfths minus 6 twelfths, which gives you 4 twelves, and when you divide again, you get the unit 3, so you put in one-third and you finish the recursion. That's one-half, one-third. Yes, yes, yes. But I mean, sometimes, I mean, you can easily come up with a fraction,
and sometimes after the subtraction, something is actually left which you cannot simplify in terms of unit fraction. then you have to again find the largest units fraction for your remaining, and do the process of subtraction again until you have finally basically turned everything into units fractions. Like, let's come up with something. Let's say for example, or just let's say
7 divided by let's say 13 so the first thing is finding the largest unit fraction that's smaller than your given fraction. Divide 13 by 7 would be 1 something. So your largest unit fraction is 1 over 2, right? Then find the common denominators. Well, this is still a unit fraction. So let's come up with a better number, a number that doesn't in the first stage leads to a a unit fraction let's say
let's say like eight um 1332 feet Right? So what would be our common denominator here? As I said, finding common denominator is just you multiply them, yes.
You have to restart the process by finding the largest unit fraction that is smaller than your given fraction. So let's continue the process. To do that, so we have, what would be the largest units fraction for this? Yes. What would be the common denominator? Okay.
You see, with these numbers, finding the common denominators becomes, sorry, 82, becomes more and more difficult. Whatever your common denominator gets, at some point, once you do this process, you you should definitely get a unit fraction. But what is really important here is that in order for you to find the unit fraction, I mean you express 8 divided by 17 in terms of unit fractions, whatever unit fractions
that you have got, you need to what? You need to use these. You have already used your unit fractions. one was this, one was this, one was this. So basically you shouldn't forget your unit fractions. I think this would be a fantastic homework to start to finally find the unit fraction that corresponds to 8 divided by 17 using Silvester method. because this is seems that I think that there is at least two more steps before you get your final units fraction. So 8 divided by 17 using Silvester method expressed in terms of units fractions.
And again, Silvester method works where your numerator is less than your denominator. That's important to remember. Now let's, so that was Silvester method. In Sylvester method, as long as your numerator is greater than your denominator, sorry, is
less than your denominator. It does not matter whether your numerator is odd or even or denominator is odd and even. You can still do the unit fraction composition. But in Egyptian method it does matter, as we will see it. That if you get odd denominators versus an event numerator, you run into problems or vice versa. So the Egyptian method of halving.
So if you remember, for multiplication, you were doubling. Whereas in this method, you are trying to find a different kind of groups. So for example, 14 divided by 24. Again, we have two columns, one representing your groups and the other representing the
elements corresponding to that group. So let's start with this. That's 14 divided by 24. start to say what would be one group of 24. One group of 24 is already larger than 14. Remember, you want to find how many groups of 24 basically like it like a pizza thing how many 14 a slide
you have like 14 loaves of bread and how can you the slice them or cut them to equal parts giving to 24 people so one group of 24 we want only total of 14 this is already too many so immediately we turn into unit fractions or what in Egypt Egyptians represent them as this line instead of writing one over for For example, 24, they write as this, just a line over 24. So one group of 24 gives 24.
But we only want 14, total of 14. So 24 is already too large. This is already too large. We don't want to do this. So immediately we turn to fractions. We can either take a half of this group or 1 over 24 of the group. Now it's easier to usually do the latter one. So we say that 1 over 24, which we write it like this, 1 over 24 of the group has 1 since the whole group has 24.
Now we can start doubling until we get back to 14 because remember we want to find this. So 21 over 24 has one element. 12, you see, we are having, has two elements. x has four elements. Three, oops, has eight elements.
So we can start our regular doubling method as if it was just a multiplication. we want to find group of 300 so 1 is 16 2 is 32 4 is 64 8 is 128 16 is 256 32 already exceeds 300. So we don't want this.
We halt the procedure as soon as we get to 256. Now, we still need more groups to add to 256 in order to get 300 groups. Now this is exactly the stage that we shift to halving. We say that 16 group or unit fraction 1 over 16 as one element.
have it instead of doubling it eight would be two elements four would be four elements two would be eight elements so as you see We need 8, we need 4, we need 256, and we need 32 in order to get 300.
the corresponding number of elements for these groups that amount to 300 is 16 plus 2 plus Oops. Remember, this is one over two and this is one over four. Plus one, four. Simplified, eight plus one over two plus one over four.
The problem of 300 over 16 is the problem of constructing using this recursive method. many groups of 16 you can plug into 300 one group 16 two groups 32 for 64 8128 16 256 if we double 16 we see
that the number of elements exceed already 300 so that's exact that's the stage that we need to shift from doubling to halving we start with 16 1 over 16 has one element 1 over 8 has two elements 1 over 4 4 elements 1 over 2 8 elements then all we need to do exactly like our doubling method find the corresponding group that is tantamount to that is equal to 300 would be 256 32 8 and 4 then we add them on one side their corresponding column and the answer to our division would
be the corresponding number the numbers corresponding to these groups for four you get one over four for eight you get one over two and for 256 you get 16 and for 32 you get two so your answer would be 18 plus one over two plus one over four hey did you mean to turn off the iPad oh no might need to be refreshed or something hmm let me disconnect the reflector and I think it's kind of got
I don't know Apple to open, but did you just like safely disconnect or whatever unplug it and then like have it resync Turn it off and on entirely maybe Okay, you you mean turn the turning off the iPad entirely Yes, I mean are you connected through that right now or is that kind of a supplementary device? Yeah, okay, how do you how do you turn on your iPad by the way? This is the question you hold the top button and Are we sorry just the just the top button? Okay
Sorry it seems that after a while it just stops responding So I had a question. I think maybe I'm going to... Sure. So you start a new column after 16 because 32 exceeds 300. Yes. So the new column, how did you generate starting from 16 going to 8, then to 4, then to 2? And then how did you generate that? Oh, that would be, you see, you see the whole thing is that, so we say that, you know, so first, you know, because it was 300 divided by 16.
So we say that we start with one group of 16, double it to get 2, double it 16 to get 32, etc, etc. Then we say that we have stopped doubling because doing that again would yield 512, which is too much. Basically, we only need 300. So we have 256 already corresponding to 16. No units fraction, 16. 16 and we add 32 which corresponds to 2 if you remember 1 was 16 2 was 32 so we get 288 we need 12 more right we need 12 more groups to get our 300 but the smallest we have in
the right column is 16 right so we switch to to taking fractional pieces fraction from the smallest one that we have so you see one group has 16 so 1 over 16 of a group has 1 Remember, one is 16, is the smallest group that we have. So, as soon as we get to that number that exceeds our 300, we are starting to take... Are you there Reza?
Is everybody else still on or did we just lose Reza? 16. So if we say that one group has 16, then what would be 1 over 16? 1 over 16 of a group has one element. now it's starting from this doing the having from 16 so you said we use 16 because it's the smallest but the smallest yes okay I think I should know we do we do some more examples that's
use it for a good part of it you do the doubling like the regular multiplication stuff that you were doing but then you see that the number exceeds if you keep doubling then that's at that point you need to switch to the halving method to do those small chunks of the group that you can add to your number 288 256 plus 32 corresponding to the group of 16 and
the group of two the smallest group that you have is one on right left column and 16 on the right column sorry I think you're you're going in and out of signal can you hear me it has one and then you start the having method because you are trying to find the unit fractions hey Rosa you're going in and out at the moment like intermittently like 10 seconds at a time or so and then picking back up do I need to do I need to refresh and restart that might potentially yeah okay okay let me do that then let me do that quickly okay also everybody else
maybe if they turn their beta width down on the top that would help it's in in between the gear and the video camera at the top, you can turn it down and it might help things. And I'm not sure if it makes a difference once you've done it, but if you're gonna turn your video off anyway, then you can turn it all the way down to audio only and that might create a marginal savings on bandwidth as well.
Thank you. methods somehow related to axioms and theorems that in a way that I'm not connecting them to anybody else just a big obvious yes or a big obvious no might be might lead me in the right direction it'd be anachronistic like I don't
believe either the Egyptians or the Babylonians had like the aromatic math so like various set theory yes that theoretic axioms like could describe what's going on but like these are methods that are generated without recourse i think to like mathematical logic right this is more an extension of uh recursion and iteration yeah i mean i think like thinking of these like you know he says like groups of 15 you're counted by powers of two times 15 these are like you know 15s of slaves hauling a pyramid block 15s of oxen twos of twins you know that's kind of the way to think about is that it starts in these material and bodily kind of group counting that's a method like this
Okay. For some reason I can't see any of the icons. Oh, I think that's just because you're on presentation mode, but I must. You don't see any of us down at the bottom? No. It's like a down. That's odd. Here, Jake, take him off on presentation mode. Okay. Okay, let me, let me, yeah, just let me turn, let me reconnect again. One second.
a starting like as I was saying that's the best way to think about this method of having is like for example let's imagine 19 divided by 8 okay so it's like equivalent to this problem of determining how many times 8 goes into 19 so we can see that 19 is equal to 8 plus 8 plus 3 right how many times 8 goes into 19 so you basically want to time 8 plus 8 which is 16 plus 3 which is equal to 19 that it is 8
divides into 19 two times with a remainder of three so what we are here really trying to do when we are saying that 19 divided by 8 is trying to determine how many groups of 8 we need to get 19 Okay, so this in terms of the grouping relations that we get in divisions. How many groups of eight we need to get to 19? And the thing is that in the one that we just looked at was 300 divided by 16.
So that would be how many groups of 16 should we construct in order to get to 300. That's 16 was our benchmark. So we started to say that one group of 16 is 16, two groups, 32, 4, 64, 4, sorry, 2, 32, 4, 64, 8, 128, 16, 256, and then 32 just exceeds 300. So precisely because our benchmark, our smallest group here is 16, that's when we switch
to the method of having by saying that one group has 16 so 1 over 16 of a group has one element and then have it 16 1 over 16 1 over 8 1 over 4 1 over 2 and then you simply add the corresponding number of the groups and the corresponding number of elements to those groups so let me get the iPad again Questions in the meantime.
I understand it, I think, but could you just describe specifically why it is that the largest unit in the left-hand column is also the smallest possible unit fraction in the remainder. Why those are the same? Jake, you were breaking up. Could you please repeat the question? Sorry, any better now? Can you hear me? No. No, not really. Can someone else say something so it might be just my connection? Can you hear me right now?
Reza? Can you hear me Reza? Yes, but kind of I... Let me see, let me see, something is wrong here because all of your voices are robotic right now. No, I cannot hear you. OK, one second.
Hello. Hi, can you hear me? Yes, yes, yes, finally. Yes, sorry for this. It's some problem at my end, I think. So, Jake, you were saying. Yeah, you might have. It's possible that you had answered it answering Theo before, and I just didn't work it out right. but just why the relationship between the largest group in the power series of two that goes into the divisor, excuse me, goes into the numerator should be the same as the smallest possible unit fraction needed to compose the remainder of the answer.
So like why does the one 16th as the smallest needed unit fraction will always match up with the largest group you reach in the left-hand column? So got your resume take one second I you are breaking up again it seems that my connection is there is getting really screwed up is it iPads being plugged in we can hear you well Yeah, yeah, I think Okay, okay, I think we are already I think 12 18 so
Sorry Here is as the habit of like suddenly dropping off Okay, just if you if you can hear me We will pick up on this method and I will make more examples But for the homework, so you had one homework already, then another homework would be basically just give a unit fraction yourself, I mean sorry, give a fraction and then apply this method of Egyptian having doubling and having to it say like you know any for
example like 8 divided by 16 or 9 divided by 45 and see if it can be applied or not you will notice that sometimes you can't use this particularly when you have even an odd number as your numerator and denominators. Are there any texts we should... Did you manage to hear me? Yes. Are there any texts we should be reading?
Okay. I don't think he heard it. I don't think he heard the question. That's a little deliberate. But all right, no, here he comes. If we can't communicate him. Yes. So yes, other than the text, I mean other than continuing the unit fractions and the Egyptians method if you can read the Euclid elements the the text that I recommended it was one from Daniel Macbeth
so the chapter on on Euclid and Macbeth like the second chapter It's not a chapter, it's called Diagrammatic Reasoning in Euclid's Elements. It's an essay actually. Great. I can send you the link, I can CC all of you and send a link to that essay. It's actually on Academia Edu. Let me actually find it and put it on the sidebar here right now. Is it a lot different from the chapter on Euclid in the book Realizing Reason or is
it another version of the same? It's a different version of it really. Well, it seems that I don't even have a connection to type anything. Anyway, sorry for this. Really really sorry for the bad connection. Nevertheless, we will continue our stuff about Egyptian unit fractions. Particularly, I want to. All right. That's what we call an act break, ladies and gentlemen.
Thanks, guys, so much for attending. I'm going to stop the broadcast now. And I'm sure Rez is going to be in touch with all the information we need. See you guys again next Saturday. Bye-bye. All right. Take care, everybody. See you. Thank you. .