 I am Zor. Welcome to Unizor Education. I would like to put a little bit more solid foundation under the concept of integer numbers. We talked about the fact that natural numbers do exist in the real life. And then we have introduced the concept of zero and negative numbers primarily based on their properties. For instance, zero is a number which if added to any natural number will retain. The result will be the same as the original natural number. Well, this fact of defining an object based on its properties is not really mathematically strict or correct in this case, I would say. And here is why. For instance, I am saying, okay, there are people who have two eyes and they do exist in real life. Now, let me define a new concept, a person with one eye in the middle of his forehead. Well, I can define whatever I want, but we understand that these people do not exist. And in the same way, how do I know that zero is something real which I can deal with? Any principle of defining an object based on its properties requires, well, two major things if we really want to deal with these new objects. Number one, we have to prove that they exist. And number two, in many cases, we would like to have them in certain unique incarnations. So there is only one zero, for instance, as an element of some other set. So this type of definition of zero as something which added to five results and five is not mathematically rigorous. I would like to put a little bit more solid foundation and approach differently a concept of integer numbers more constructively. Well, basically I will construct these numbers from existing elements whatever we have right now. I assumed we have right now a concept of natural numbers quite well defined, which means we know that they exist. We know the order that one precedes two, two precedes three, etc. We know how to add numbers, natural numbers. And we know that any two numbers, any two natural numbers can be added together and you will get another and we know which one, natural number. Operation of subtraction is not always defined. We can subtract five from eight, but we cannot subtract eight from five. So it's partially defined operation of subtraction, only in case the number from which we are subtracting is bigger than the number we are subtracting from. No, it's twice worse. The number from which we are subtracting should be bigger than the number we are subtracting. Okay, so anyway, I presume that these properties of natural numbers we do know. And based on these properties, I would like to construct relatively rigorously integer numbers, all integer numbers. So here's how we can do it. And by the way, this is not a unique approach and there are probably some other mathematically solid approaches on how to construct integer numbers. This is something which I just came up with as a good way of explaining mathematically better in some way than before. What's the concept of integer number actually is? And what are these numbers? Okay, here it is. We know about natural numbers, so I'll just put them natural. One, two, three, etc. So we know these numbers and we know their properties. Integer numbers, as I'm defining them right now, by definition, integers are strings of the form open curly bracket, a sign, plus or minus, and some natural number and closing curly bracket. And also another string will be an element of this set of integer numbers. So all elements, all strings of this format where instead of one, two, three, or three, seven, we can actually substitute any natural number. All these strings which look like this, I call integer numbers. Well, I can again call whatever I want, but first question, do they exist? Of course they exist because I constructed them. This is a constructive approach to creation of integer numbers. So we have a set of strings, and this set of strings I call integer numbers. Well, that's only the beginning. Now I have to define operations. And again, my goal is to define operations in such a way that any two integer new objects, any two integer numbers can be subtracted, added, subtracted to each other without any restrictions. And I have a well-defined operations on all these elements of this new set of integer numbers. All right, fine. So let's define these operations. The rules. Basically, I'm defining the rules. Rule number one, if I add element zero to any other element, by definition, the result will be the same as any of these elements, which means zero plus, now this plus is addition. This is something which I define as an operation. This plus is just a symbol. It's just a sign. It's a character basically as part of the string. So if I define operation of addition on any of my new set of integer numbers with element I call zero, I will have the result, my original number. In particular, actually, it means that zero added to zero results in zero. And minus 37 added to zero results in minus 37. That's what my first rule actually states. Zero added on any side to any other element including itself results in that element which we are adding it with. Okay, that's my first rule. Great. Let's continue. The second rule. If I'm adding two elements of my new set with opposite signs, so one is plus and another is minus, by definition, I'm getting this element zero, which I have started with. This is a rule. This is the definition. This is an axiom, if you wish, of how I define operation of addition between any two numbers with different signs and the same natural number inside. Well, as you understand, this will be a prototype of my reverse operation in the future. Okay, that's rule number two. I finished with zeros. Rule number three. If I'm adding two elements with pluses inside, rule is I have to get the inner natural numbers which are inside, add them according to operation of the natural numbers as we know it, and that will be what? One sixth, right? And that would be the result with the same operator, with the same sign inside plus. Plus, plus, and plus. And similarly, by definition, again, if it's two minuses, I will exactly rule according to the same principle. I retain the sign, inner sign, minus, minus, minus. And the natural numbers inside should be added together and that will be the result. So, which rules I still have to define? Obviously, with opposite signs. One is plus and another is negative. Okay, so what's the definition of this? Let me just wipe out this one. So, my rule number five is if I have two numbers with opposite inner signs, what I have to do is the following. I have to compare these natural numbers. Well, which one is greater? In this case, it's 123. By the way, if they are equal, you remember there was a rule number two that the result will be zero. So, they are not equal and one is therefore greater than another. I perform appropriation of subtraction on these two natural numbers. From the greater one, I subtract the smaller one. I always can do this. So, in 123, minus 37 is what? 86. 86, right. And then, I retain the sign of the bigger element. That's my rule. Since the bigger element is 123, which has an inner plus, I retain the inner plus here. Similarly, if I have minus 123 and plus 37, I retain the sign of the bigger one. So, operation of subtraction results in the same 86, but the sign will be minus in this case. That's it. Basically, I have defined all the operations on elements of my new set, which I call integer numbers. So, integer numbers are these. Everything, whatever I have inside the curly brackets, with the brackets themselves. The strings, basically. Strings are my new numbers, my new elements of a new set. And they are operated upon using this type of operations. Okay, let's continue. I have not defined an operation of subtraction yet. This is easy. If from any number, that's rule number six. If from any number, let's say, minus 123 doesn't matter. I defined operation of subtraction. Again, this is the operation. This is just an inner character, which looks the same, but doesn't really matter. And if I want to define operation of subtraction, by definition, this is an operation of addition of the opposite element. And we know what opposite element is, because we have defined it before. Opposite element is the same element, is the same natural number inside. Inner number is the same, but inner sign is opposite. So that's my definition. So there is nothing, basically, to prove. And we know how to do this, because this is actually... We know that we have two different inner signs. So it's difference between bigger and smaller, which is age of six, and the sign is retained from the smaller one. So it's minus age of six. So we have defined operation of subtraction. We can subtract any number. What else remains to be actually noted? Obviously, laws of commutation and association are very easily provable, if I define everything, because basically I have defined based on the properties of natural numbers and operations of addition between natural numbers. And so obviously, all these rules are transformed and they fall from the corresponding rules for natural numbers. Commutative means you can A plus B is equal to B plus A. And association is when we have three numbers, adding together the order doesn't really make sense. And again, everything is very easily provable. So these operators are commutative, associative. So basically my set of integer numbers is very well defined right now. I have constructed, so it's a constructive approach, I have constructed these new elements of my new set of integer numbers using strings of characters. And I have defined operators of addition and subtraction for all of them. Now I can subtract any element from any element because I have just redefined it as an operator of addition of the opposite element. The opposite always exists. So any two numbers can be and can be subtracted one from another. Okay, now here's a very important issue. Usually people who teach integer numbers, they're trying to say, okay, we have natural numbers, one, two, three, four, etc. Let's add to this set new numbers which is zero and minus one, minus two, etc., with these properties. As I was saying, this is not mathematically rigorous type of definition. This is one of the examples of a better or more solid definition. So what's important now is to understand that in this particular approach, it's not like integer is a big set and natural is a subset. That's not the case in this particular case because integers, as I defined them, are strings with curly brackets and then plus or minus sign inside and some natural number. And natural number is just a number by itself. In this particular approach, this is not true. This is something like this. It's two completely independent sets. So what's the relationship between them? I mean, we understand that there should be a relationship and natural numbers are like positive integers, like. So how can we do this type of thing? Okay, here's what I'm saying. Within the set and integer numbers, there is a subset which I can call pseudo-integers, pseudo-natural. I mean, I just came up with this, pseudo-natural. And I'm saying that there is a one-to-one correspondence between these two sets, a natural number and a subset of the integers which is called pseudo-natural numbers. So one-to-one correspondence. But that's not it. One-to-one correspondence is not sufficient. Here is much more important toology. If you take this element and it's corresponding this, and you take this element and this is the corresponding element, what if you will add these two elements together? And for instance, the result will be this element which has the corresponding element, pseudo-natural element here. What's important is to show that the sum of these is also this. So the sum of two images is image of the sum. That's basically the short version. Let me just repeat it again. The sum of two images of two original natural numbers is equal to the image of their sum, of their original sum. So this is something which we really can prove. And here is how. But before I prove that, let me just say that it is extremely important to prove this property because then we can deal with pseudo-natural numbers which are elements of integer set in exactly the same fashion as we were dealing with natural numbers. So instead of doing arithmetic with natural numbers, we can do arithmetic with pseudo-natural numbers. But what's important is that if you cannot subtract 5 from 8 here, you can subtract, but let me just repeat this picture again. This is integer numbers and these are pseudo-natural numbers. You have two elements here and you have among natural numbers two elements here. One is image maps here and another is maps here. If you cannot subtract, let's say this from this, because it results in non-existing natural number, you can subtract the images of these two guys and you will get just another integer number which obviously is not part of the pseudo-natural because it's outside of this. For instance, if this is 8 and this is 12, this is an image of the 8, this is the image of the 12, you cannot do that in the natural numbers. But you can do this and that would result in some number outside of the pseudo-natural subset. So that's the advantage. You can do everything you can do with these numbers, you can do everything you can do with this plus something which you cannot do there. All right, so let's just talk about this correspondence, what corresponds to what and how natural numbers are mapped into pseudo-natural. Well, this is elementary actually. If this is the natural number, let's call it m. These are integer numbers, but I'm basically saying this is the correspondence. So any natural number, 1, 2, 3, 123, corresponds to an integer number with a plus inner sign and the natural part of this exactly the same 1, 2, 3. Now, how can we say that sum of two images is image of sum? Well, let's just illustrate it using some example first. So let's say you have two integer numbers. 1 corresponds to plus 5 in curly brackets and plus 8 in curly brackets. These are natural. These are integer. My pseudo-natural. Well, let's summarize. According to the rules of natural numbers, a arithmetic, this is the team. Now, how did I define an operation of addition between two strings? I said if they have the same sign plus the sign will be retained, then I have to take two natural numbers inside and add them according to the rules of natural numbers, which is the team. But this is exactly what I would do if I mapped the team directly. I would get curly bracket plus the team curly bracket, right? So if I do summation first and then I do the image mapping from a natural number to a pseudo-natural integer, or if I do mapping first and then summation within the set of integer numbers, I will get exactly the same thing. In some way, it resembles accounting. If you remember, you have to have some kind of balancing. If you have some kind of a matrix or a spreadsheet using computer terminology, you have numbers here which are totaled into this one. Then you have numbers here which are totaled into this one. Then you can summarize vertically. This will be there, total, this will be there, total, this will be there, total. And you can summarize vertically the column of the totals. So the total of all the local totals should be equal to totaling these totals. These are row totals, right? So this is the row total and this is the row total. So the sum of the row totals should be equal to sum of the column totals. And this is your balancing. So it's supposed to be exactly the same number, whether you summarize first this way and then this way, or if you summarize this way first and then this way. This is your basic balancing, basic accounting, if you wish, etc. This is very similar because that actually makes the whole picture harmonious in some way because we can deal now with these numbers in exactly the same fashion as we deal with these guys. Now let's talk about subtraction. That's easy. If I want to subtract from 5, I subtract 8. Well, this does not exist, obviously. We can't really put anything into a correspondence, but here we can. If you remember, what I was saying was subtraction is actually addition of the opposite element, right? So if I subtract this, it's exactly the same as I take this and then I add opposite element to 8. Now, how to add these two elements with different inner signs? Well, I know that I have to take from a bigger, the smaller, and that would be the inner number, and the sign should be of the bigger one. So these guys exist in integer world and there is nothing in the natural world which corresponds to them. So these are negative numbers. And I call these guys with plus positive and I call these with minus negative. That's basically the definition of what is a positive and what is a negative integer number. And zero, you know, it has already been defined. Well, basically that concludes my presentation of a little bit more solidly defined concept of integer numbers. Defined as strings with certain properties. Jumping a little bit forward, I'll probably do exactly the same as an extra material for rational numbers because we have exactly the same situation. Like here we have with subtraction, we have the same situation among integer numbers now with multiplication and its reverse division. You can multiply five by eight, but you cannot divide five by eight. So we need new numbers, rational numbers. So rational numbers can be introduced in exactly the same way using certain concept of strings which I will do in that next lecture. And then jumping even further, when I will introduce complex numbers, I'll just use this, this particular type of constructive introduction of the concept of complex numbers. And well, I think it's a little fun to go to something.