 I'm Zor. Welcome to Unizor Education, Mathematics for Teenagers. Today's lecture will be about numerical systems. We have already talked about some ancient people who invented the concept of a number, purely abstract number, and now they can measure quantities. 15 sheep or 15 bulls or 15 stones or whatever else. They are all represented by number 15. Now they have to be able to write it down. Why? Because ancient lawyers had to write down some contracts about the person X bought from person Y, 15 sheep. So they had to put some number 15 in writing. Well, how did they do it? The first and most primitive system is obviously to have 15 written as this. 15 bars. So all these cavemen were using whatever tools they had to put vertical bars on stones. Well, it's very slow and very inconvenient and kind of lengthy. You need too many stones to write it down. So we need some other system and obviously I will be talking about different numerical systems which people have invented throughout the years. But first I would like to invent my own which is just an example of how different people approach different problems. So I don't like that this is too long, right? So I would like to shorten it down. How can I do it? I came up with a purely arbitrary number four and I will group these bars into groups of four. So I have one, two, three complete groups of four and one incomplete which contains only three vertical bars. Right, so I invent a special symbol to represent four strikes, four bars in a group. And this symbol will be a square. So I have another square for the second full group and for the third one. And for an incomplete group I will have different symbol which represents how many bars are in this incomplete group. So if four represent the complete group and square I will use the triangle let's say to represent three bars here. So my number fifteen is represented in this way in my numerical system. Well, what's the number fourteen? Well it will be obviously three full squares and two in an incomplete group. So I will invent plus to represent number two. What else? Number seven let's say. Number eight is a complete group of four and three which is a triangle. Number five would be one complete square which is four and let's say a vertical bar which is representation of one. So any relatively small number I can represent a certain number of full squares and at the end I will have whatever number of bars I have left in an incomplete group. Obviously I will have to have as many characters, different characters as the size of my group is. So if my size of the group is four I need four characters to represent either a complete group of four or incomplete group of three or two or one. So number of characters all together which I have to remember is exactly equal to the group size. So that's good actually because I have to remember maybe a little bit more instead of just one vertical bar I have to remember four different characters but it shortens my notation by the factor of four obviously. Okay fine. Romans did something just one step further than that. By the way before I go to Romans if instead of group size of four I would choose let's say a hundred in my notation. What would I do? Well I would have to invent a symbol to represent one, two, three, four for all these different incomplete groups which are possible up to 99 and then to a hundred. So I have to have a hundred different symbols to represent different sizes of the group. This is a little bit too much. So it's a little bit more convenient because it will shorten my notation by a factor of a hundred but obviously the negative side is that I have to remember too many symbols to represent different quantities. Well Romans did something smarter than that. First they also have decided that we will have a certain group size which will be represented by a one particular symbol. In my case for a group size of four I used square. Romans did for a group size of one thousand they used the letter M. Well probably related to legend, millennium or something. Alright so now we have groups of one thousand and if I have a big number let's say three thousand something. They can put three letters M which represent three thousand and then something. Well something in my previous notation when I had only one fixed group size I had to invent a special symbol for every quantity from one to a group size. Well in this case if we don't have any other rules and agreements about this system it means that now after putting three thousand I have to have a special symbol which represents one, two, three up to a thousand. Well that's not very convenient. So Romans were much smarter than I am and they have decided alright the remainder which is a number from like zero to nine hundred and ninety nine. We will also group into different groups. We will group it in groups of five hundred and they have invented a different character, a different symbol for five hundred which is D. So if you have a number let's say three thousand and seven hundred and fifty two I will use three thousand seven hundred and fifty two in our decimal system. So I put this ten as an index. So seven hundred if you have a group of five hundred can be represented as one full group which is five hundred, right? And then you still have two hundred and fifty two in the remainder. So you have to have maybe if you're not smart like Romans you have to have additionally characters from one to four ninety nine to be represented in one single character, right? But they have decided no, we will break it down again into groups, into groups of one hundred. So whatever is left after three thousand five hundred and two fifty two right in decimal well they will break into groups of the one hundred right now. We have invented another character C for one hundred. So two fifty two is two letters C which means two hundred. Now we have fifty two in the remainder. Well they have invented another character L for fifty. So fifty two can be represented as fifty. And now I have to have representation of only number two which is a remainder. Well they actually have a little bit more. They have X for ten, V for five, and then I for single digit one. So by using the combination of these symbols they can write any number. So in this case I just have to add two ones to get three thousand seven hundred fifty two. Well let's just as a representation do another number in the Roman system. But I will show you a little trick which they did. So this is smart actually what they did. They have used different group sizes. In my case I used only one group size four. And they have decided that they will use big groups like one thousand to cover big numbers. And whatever remainder is they had a smaller group sizes. So again let's talk about M, D, C, L, X, B, and I which represent one thousand in decimal system. Five hundred in decimal system. One hundred in decimal. Fifty in decimal. Ten in decimal. Five and one in decimal system. Right. Using these we will have three thousand seven hundred and ninety eight let's say. Even better. Ninety nine. Well let's try to construct this number from these components. First we have three D groups of one thousand. Now we have seven hundred which can be constructed from certain number of D's right. So it's D that's five hundred. No more five hundred so we have two ninety nine left. Two ninety nine so we have to use C two times right. Then we have ninety nine which is fifty and four ten that's ninety. L four tens. And now we have nine as a remainder. So nine can be written is, I'll continue this as five and four ones right. So now what exactly a trick I was talking about? Very simple. Instead of representing nine as V and four vertical bars for I's instead of that they have decided we will do a different thing. Instead of adding five and four ones they will use subtraction. A smaller representation of one is preceding a bigger one ten in this case. You have to subtract it instead of adding. So this is a representation of nine. So instead of this we will use this at the end. Similarly with this this is ninety. Instead of using L and three X's we will use X and C. This means subtract ten from one hundred and that would be ninety. And then subtract one from ten would be nine. So this is basically three thousand seven hundred ninety nine. Right? I think I'm right. Well in any case trick or not trick this is a system which is one step further from my primitive system when I just divided by groups. The improvement is they have different group sizes. Bigger groups and then smaller groups to represent smaller numbers. Great. And now we will go to something which we conveniently use every day. And we call it usually Arabic system which actually represents only the involvement. Arabs were involved in basically transporting the system from Indians who really invented it to Europeans. So this is system which is a positioning system. Based ten positioning system instead of inventing a special symbol to represent bigger and bigger numbers. They have a very limited number of symbols. Based ten system has only ten different symbols which we all know from zero to nine. And it's not the symbol by itself which represents certain number. It's a symbol which also is positioned corresponding to the magnitude of the number. Number one means just one and symbol one means just number one. But if you put it in a second place from the right and put zero in front of it that means actually ten as we all know. And this means a hundred etc. So we are using this type of notation to represent our weights. So a digit which is on the first place on the right has certain weight. And a digit which is on the second place on the right has the weight of ten times more. And the third one is a hundred times more, a thousand times more etc. So if you have some number what it actually means algebraically is ten. This is five times one hundred plus six times ten plus seven times one. That's what it means. So this is a notation. This is an algebraic representation of this notation. Now I can actually write it slightly differently. It's five times ten square plus six times ten to the first degree plus seven times ten to the zero's degree. Right? Now why is this better? Well it actually represents that this is a base and this, the power is a position basically. You see? Zero, one and two. Zero, one and two. So in general I can tell you that any number can be represented in this system using powers of ten and certain multipliers. Each one of them being from these numbers from zero to nine. Well obviously algebra is trying to abstract and generalize everything. So in theory what we can do is we can invent a system, a positioning system, generalized positioning system which has any number p as a base. Then numbers from zero to p minus one have to be somehow symbolized. Like if p is equal to ten then we need numbers from zero to nine. If p is two for instance which is a binary system used in computers then you need only from zero to one. Only two characters, zero and one. If p is let's say sixteen then you need certain symbolic representation of numbers from zero to fifteen to represent different remainder whatever you have if you divide by sixteen. So and every number m now can be represented as certain multiplier times p to certain degree, certain power of p. Another multiplier on a smaller number etc. And on the rightmost place you will have this. So if you take only these multipliers and write them together put a bar on the top which means it's not a multiplication it's just you write one after another these digits. So this particular way of representing the number means this. And you have to have obviously say that p is the base of your system. So let's just do a couple of exercises. If you have a number let's say seven and you would like to represent this seven written in decimal system as something in let's say binary system. Well in binary system it will be this. Why? Because the one is a multiplier so this has weight of two to the zero degree to the first degree to the second degree. And if you will do this one times to the second degree plus one times to the first degree plus one times to the zero degree it will be four and two and one it will be exactly seven in decimal system. Now what if I would like to represent it in a base three system? Well in base three system I have one zero one and two as digits so I have to represent it in a way like A. Well I actually don't need K in general. Since it's a base three three square is already nine so it's only to the first degree. So what are my multipliers? Well obviously this is supposed to be two times three so it's two times three to the first degree that's six plus one to the three to zero degree. So the whole thing is becoming two one twenty one in three base system. So seven in decimal is two one in tertiary system. Alright fine. What's the most important part about this which mathematicians really have to take care of? If we don't talk about this it's not mathematics it's just some numerical exercise. The mathematicians do the following. First they have to prove that for any number m and for any base p there is such a representation. So always we can find such multipliers which would combine with the whole expression give our number m. So one is an existence theory that this type of representation does exist. Secondly we have to prove it's unique which means if you have another representation of the same number in the same base system with different multipliers. Let's say you have bj times p to the j degree plus bj minus one times p to the j minus one plus etc. Plus b zero times p zero. If you have a completely different representation of the same number using the same base but different multipliers then it's exactly the same representation. Since the k should be equal to j and all these multipliers should be equal to each other for i equals zero one two etc. up to k. So this is a uniqueness. So not only representation should exist but it also should be unique. Alright these are two theorems which I would like you to prove and maybe I will spend some other lecture to demonstrate myself if I have time. Anyway this is a great exercise and this is the real mess. Everything else is just you know some kind of philosophical explanation numerical systems etc. This is not a mess. Proving uniqueness and existence of these two representations is something for real and I would like you to do it. Thank you.