 I'm Zor. Welcome to Unisor Education. Today we'll talk about real numbers. We call it real. Basically, there are numbers like any other numbers. Let me just remind a little bit about history of numbers. I did talk about this before, but very, very briefly. Remember, we started with natural numbers. Somebody decided to count sheep or whatever, one, two, three, etc. And then they come up with an operation in addition. By introducing this operation, they actually made a very simple thing. If this is a set of all natural numbers, and operation of addition just take two numbers together, and they come up with a third number. Well, then obviously the reverse operation of subtraction was invented, and the problem was that the natural numbers were not sufficient to satisfy always the operation of subtraction. You can subtract 3 from 5, but you cannot subtract 5 from 3. It will be negative number. So that's how negative numbers came. Then operation of multiplication was invented, like 2 times 2 will be 4. Then obviously the reverse operation, division, 4 by 2. And then they realized that negative or positive numbers are not sufficient to satisfy division. We cannot divide 3 by 5. So they invented rational numbers. So my point is that every time people invented certain numbers and certain operations on these numbers, if these operations eventually led them to basically inability to perform these operations within the framework of existing numbers, they invented new numbers. They couldn't do division among integer numbers. Well, they invented rational. So what's the operation we know right now? Among numbers we know addition, we know subtraction, we know multiplication, we know division. And there is another operation which is a power. Like for instance 2 to the power of 3 is 2 times 2 times 2. It's just a shorter way if you wish, but still 3 times. You can consider this as an operation between number 2 and number 3. Like same way as with multiplication. If you want to multiply 2 by 3, it's actually 3 times addition. So here's an interesting observation. 2 times 3 and 3 times 2 is exactly the same thing. These are equal. It will be 6. In this case, which means operation is commutative, as we say. In this case, operation of power is not commutative because this is not equal to 3 to the second. Because this is 8 and this is 9. So the operation of power is not commutative. But this is just a side note. What's interesting is that we do have a new operation. An operation of power and we have rational numbers. Rational numbers we always denote as p over q where p and q are integer numbers. But operation of power means that any number can be brought into any power, right? Well, with 2 and 3, as we say, it's simple. Great. Let's derive certain properties of this. If 2 to the third degree is 3 times multiplied by itself, 2 times 3 if I will multiply 2 times, let's say, 4. What that would be? Well, this is 3 times 2 multiplied by each other. Then I multiply again by 4 times. All together it's 7. Which means this is 2 to the seventh degree. And if you know it is 3 plus 4 is 7. So the general property of the power is, if I will multiply it by another a to another degree, then degrees, these powers can be added together. Well, this is the property of power of the number. Okay, great. Fine. We found this from basically talking about integer numbers in integer powers. But we would like this property to be always true. Well, let me just give an example. What if I will use instead of b, instead of natural number, I will use 1 half. I would like the property like this to be always true. But according to this rule, it's a to the first degree, which means a multiplied by itself once, which is a. Right? Fine. So what is a to the power of 1 half? Well, this is the number. If we multiply it by itself, we will get a. Well, we know this operation in algebra is a square root. So basically a to the power of 1 over 2, 1 half is a square root of a. Okay, fine. But now let's talk about when this operation can be defined. We know our rational numbers. That's where we are right now. And what I would like to say is that introducing of this new operation of power actually brings us outside of the domain of the rational numbers. A to the 1 half is not necessarily a rational number. Well, if you have 4, for instance, then you know the square root of 4 is 2 because 2 times 2 is 4. So you can say that 2 is 4 to the power of 1 half. Fine, great. But what if a is equal to, let's say 2? Is there a rational number which is a square root of 2? Well, the answer is no. And just as a small exercise, let me prove. So I'm going to prove that A to the power of 1 half, where A is 2, is not a rational number. Okay, here in the proof. Very easy. Let's assume that square root of 2 is a rational number which means it can be represented as a two integer numbers p and q with a division in between. And let's assume that this is not a reducible fraction. If you remember reducible means that if there is a common denominator, I can always reduce by this. Let's say 4 by 8 is equal to 1 over 2. 4 is a common denominator. So I reduce it. And this is obviously true. We can check the use of multiplication. Okay, so it's irreducible. There are no common denominators between p and q. Fine, great. From this, with very easy calculations, we come up with a second equation. q times square root of 2 is equal to p. Right, because that's what actually the division is. It's a reverse of multiplication. So I multiply this by square root of 2. So I should get p. Great. Now let's square this whole thing. q square times 2 is equal to p. What does it mean? It means that p is even number. It's divisible by 2, right? These are all integer numbers now. p and q are integer. This is irreducible fraction, correct? But they're all integers. Okay, p is even. If p is p square, sorry, p square is even. If p square is even, then p is even. Because if p is odd, odd times odd will be odd number. So we cannot get even p square if p is not even. So this is obvious. But if p is even, it means that it can be represented as 2 times m. Where m is some other number, right? All even numbers are divisible by 2. So the result of this division will be m. But if p is represented in this way, then p square is equal to 4 m square. And going back to this, if I will substitute p here, I will get q square times 2 is equal to p square, which is 4 m square. Reduced by 2, both sides, it will be q square as equal to 2 m square. Again, all are integer numbers. m is integer, p is integer, q is integer. Everything is integer. Which means that q square is even and therefore q is even. To compare this, p is even and q is even, what does it mean? It means that the 2 is common denominator. Both are even, which means we can reduce this fraction. And we have assumed that we cannot from the very beginning. So that's the contradiction. So we have proved that this cannot be a rational number. Very easy proof. Same thing probably can be done in many other cases, especially with square roots or some other roots, etc. Alright, so as usually, we have an operation, in this case it's operation of power. And we wanted this operation to be performed on any number which we know at the time. And right now we know the rational number. And as we saw, 2 to the 1 half, power of 1 half, which is actually a square root of 2, is not a rational number anymore. As usually, mathematicians are saying, okay, our rational number is not sufficient for operation of power to any degree. And so what do we do? We invent the new numbers, as usually. So the square root of 2 will be a new number, not rational. So all these numbers which can be obtained using operations of power and many other operations are called irrational. So for each one of them, we can prove that they are not rational. Similarly, for instance, with a variety of just now. But in any case, the point is that the set of rational numbers is not sufficient. We have certain operations, excuse me, which bring us outside of this set of rational numbers. And all these outside numbers which we come up with, like square root of 2 for instance, we call irrational. And real numbers are rational plus all these irrational numbers which we can come up with using called the different manipulations. By the way, square root of 2, or just any square root of any number, is not the only way to get irrational number. There are many others, and we will probably talk about this, but what's important is that the number of irrational numbers is significantly greater than the number of rational numbers. And that's what I would like to do right now. I would like to count numbers which we have. What doesn't mean counting? Well, can I count, let's say, all all numbers? Count means I have to put a number on every element of the set which I'm trying to count. So this is a set, any set. And these are elements of these sets, all these elements. And I can say that this is element number 1. This is element number 2. This is element number 3, etc. Which means I'm putting my elements of this set into correspondence with natural numbers, 1, 2, 3, etc. So if I can count all these elements of my set, it means that my set is countable. Well, okay, let's understand. Let's think about the following. What sets are we talking about? Well, first of all, if you remember, we started with a set of natural numbers. And they are countable because they are really the representation of the counting. So the set of natural numbers, 1, 2, 3, 4, 5, is countable by definition. All right. Then if you remember, we added the number 0. So these are my natural numbers and this is element 0. So this is the next set which we are talking about. Is this set countable? Well, obviously yes, because we can put into correspondence to the 0. We can put number 1 to the 1. We can put number 2 to 2, number 3, etc. And that's how we will count. 1, 2, 3, 4, 5, 6, etc. Okay, great. It's easier with this guy. How about all integer numbers? Positive and negative. So we have 0, 1, 2, 3, minus 1, minus 2, minus 3, minus 4, etc. How can they count these? Actually quite easily. This is number 1. And then we go to the right. This is number 2. Then we go to the left. Number 3. To the right. Number 4. 5, 6, 7, 8. And as you understand, eventually any number, positive or negative, will get counted. So eventually, all the negative and positive integer numbers will be counted and will find their correspondence to the natural number. So I can say that the number of integer numbers, positive and negative and 0, has the same power, the same cardinality as the set of natural numbers. They are infinite, but they are infinite of the same type, so to speak. They're all countable. Well, it's not really obvious in the beginning because, you see, if you look at, for instance, all integer numbers, obviously all natural numbers is a subset. So it looks like the whole set has, I would say, the same number of elements, although that's not really precise, as its subset. It doesn't sound maybe nice, but probably it's because we really shouldn't really say it's the same number of elements. We should say that these are infinities of the same type. This is infinite and this is infinite. But since we can put into correspondence one to another, these are the same types of infinity. Well, obviously we can put some more examples, but probably an interesting one is rational numbers. Rational numbers is basically pairs of integers. One is in the numerator and the other is in the denominator. So it looks like, again, quite double, basically, all the numbers. But again, it's still countable. How can it be counted? Very easily. We know that all integer numbers, positive, negative, zero, etc., are countable and all integer numbers in the denominator are also countable. Well, so how can we count all of them? Well, number one from here, number one from here, number two from here, and number two from here. Again. And that's how we will count. Since we can count all integers, then we can count pairs of integers using this system. So it looks like the rational numbers, which seems to be a much more powerful set, much more humorous set than all the natural numbers, is still countable. So it's still of the same type of infinity. And here is what's an interesting point. If we will add irrational numbers, numbers which we can add by doing many different things, like, for instance, square root of 2 and many others, this is not of the same type. This is a much more powerful set. So if this is a countable set, real number is used continuum. This is the word which we are using to represent the power of this set. We cannot count it. It's much more humorous than the number of natural numbers. And interesting representation of irrational numbers, rational and irrational numbers using decimals. Now, you know that you can use decimal numbers to represent, for instance, this as 1 over 1,000. Right? So, I was talking in one of the previous lectures that any rational number can be represented in a decimal system like this using a periodic fraction. Periodic means that it will have a certain number of digits in front and then a certain other number will be periodically repeated, which means that the whole number will be 0.12345674567 4567, etc. up to infinity. So all rational numbers can be represented in this format. It's a periodic decimal fraction. What's interesting about irrational numbers, irrational numbers not only infinite, like a periodical number, for instance, they're also a periodical. There is no such thing as a period in representation of the real numbers in decimal system. So it's an internet number of digits and there is no sequence which we can say, okay, this is a repeatable pattern, so to speak. With rational numbers, always there is a repeatable pattern after a certain number of digits in the beginning. There is always a certain number of digits which is a period. It's repeatable one after another up to infinity. With irrational numbers, there is no such period. Okay, and geometrical representation of numbers. So, geometrically you can have a line you can put a point, call it 0, you can have a certain unit of measurement and then you can use this unit to measure from 0 to the right and call these points positive integer numbers then measure to the left would be negative numbers and that's how we put all integer numbers into correspondence with certain points on this line. Now, obviously, number 2 and 1 half can be represented by middle between 2 and 3. And any rational number let's say p over q but if p is smaller than q then it's somewhere between 0 and 1 we divide this into q parts and take only p of them. That will be p over q. So, this is how irrational numbers are represented and obviously it can be anywhere on the line. What's interesting is that if we take any point on the line, the result is some rational number which is very close to this point. Well, just for example, let me just take this segment from 0 to 1 and blow it up. If this is 1 and this is 0 and I want to represent this particular point with certain rational number. Well, I not necessarily can do it, but I can do it as close as I want. For instance, I can divide it into 10 different pieces. Whatever 10 pieces are, I'm not sure there are 10 of them. So, this is 1 tenth and this is 9 tenths. Now, this is let's say 7 tenths. All right, so my point is approximately represented by 7 rational number 7 tenths because 8 tenths is already greater, is already to the right of my point. So, approximately it's 7 tenths. Can I do it more precisely? Yes, very easily. Let's divide this piece between 0.7 and 0.8 into 10 parts and again it will be, let's say, between 5 and 6, so it will be approximately 5 but not exactly 6. But it's much closer to this point. Now, divided this segment into 10 more pieces we can get the next decimal digit and that's how we will gradually represent our point closer and closer. So, we find a rational number which can be as close to this point as possible. Now, will we hit the point? Well, sometimes yes. If this point is already a rational number then eventually we will get it. But what if this point is irrational number? As we know, it's an infinite sequence of digits which we will not be able to approximate to express exactly we can only approximate it. All right, it means that there are certain points on the line which we cannot represent as a rational number. These are points which are irrational numbers. So, rational and irrational numbers together they represent all points on the line. So, the power of this particular set, the power of the set of points on the line is the same as the power of all the real numbers which is called continuum. All right, so that's very interesting and just as an exercise about continuum let me ask you this question. If I will take this segment on the line and let's say this segment on the line where can we find more points here or there? Well, obviously you think that there are more points here than here. It's nature of thinking. However, I can always put these points on these two segments into one-to-one correspondence and if I'm putting two sets into one-to-one correspondence they are infinite so I can say that they are having the same type of infinity the same power as each other. It's very easy to prove. Let's connect these two and these two. Now, this is my very important focal point. Now, let's take any point on this line, on this segment. If I will connect this, that and my focal point I will have this one crossing, right? So I can say that this point on the top segment is corresponding to this point on my focal segment and for every point I can find the corresponding and if these are different these are different. So it looks like for every point from here I can find the corresponding point from here. So this is a one-to-one correspondence which means figuratively speaking philosophically speaking that there are as many points here as in this a smaller segment. Well, obviously the next step would be the following. If I have a semicircle, the number of points in this semicircle is again bigger, smaller than let's say the segment because just proved that every segment has exactly the same number of points. Well, it's exactly the same type of infinity number of points, let's say. Well, obviously this is also the same very easily if I rotate for instance a segment from here to here and this is my correspondence. Obviously every line represents the one-to-one correspondence between the point on the segment and the point on the curve on the semicircle. So again, the number of points in the semicircle is exactly the same as the number of points in this segment. And by the way, the number of points in this segment that we have just proved and the last exercise in this infinity thing is what if I would like to compare the number of points in a segment with number of points in an infinite line, infinite in both directions. Well, this is obviously seems to be like more points like, right? Now it's still exactly the same continuum and here's how we can prove it. If I have a line I can say that the number of points in this line is exactly the same as the number of points in this semicircle. How? Well, this is the center of a circle then every point on the line connected to this center will cross the semicircle at a certain point, which is the corresponding. So this is one-to-one correspondence. Every point on an infinite line by connecting it to the center of the semicircle will give you the corresponding point in the semicircle. However remote this point is it's always crossing somewhere because this is a complete semicircle which means only these lines will not cross my my infinite line. All other lines will cross it somewhere. So again, the number of points on the line on an infinite line in both directions is the same as the number of points in a semicircle which isn't turned the same as any segment, etc. So it's all the same type of infinity. It's all continuum. So the number of elements in all these sets set of real numbers set of points in a segment or a semicircle or a circle or whatever or an entire infinite line is exactly the same type of infinity. It's called continuum. And by the way out of curiosity, if you would like to think about how many points are on the plane it's still the same continuum. All right, now have we found every number which we really want to do every operation which we want? The answer is no. We have expanded from natural to integer to rational to real numbers but there are still certain operations which we cannot perform. For instance this operation, square root of minus 1 there is no real number which is being squared give us the minus 1 because all squared numbers are positive. So this is a subject to the next topic and with the introduction of these numbers then we can get something which is complete harmonious, if you wish, symmetrical, etc. Every operation which we know and we know operation of addition, suppression multiplication, division power only then these operations will be complete when we will introduce something like this then any power can actually be applied to any numbers. But that will be a topic of the next lecture. Thank you very much. That's it for now.