 My name is Zor. Welcome to Unizor Education. This lecture will be about harmony. Harmony in numbers. Well, since we start Algebra, we start with numbers. Well, let's think about how history was developed. As far as numbers are concerned, the first numbers which people knew were natural numbers. 1, 2, 3, etc. Why are they called natural? Obviously because you can find them in nature. You can count sheep, one sheep, two sheep, three sheep, etc. Or you can count bulls. One bull, two bull, three bulls. Or you can count stones or any sort of cows. So gradually people develop this concept of number. Number is an abstract thing. Natural number, obviously. Well, today, as far as numbers are concerned, we know lots of other numbers. We know zero. We know negative numbers. We know rational, irrational numbers, complex numbers, etc., etc. But let's start one by one, step by step. Natural numbers and people's life were related, obviously. But then people were just thinking, well, if you can add 5 to 3 and get 8, can you get it backwards? Can you get it back from 8 to 3? Let's say if you bought 5 sheep and you had the material ready, you got 8. What if you sell 5 sheep? Basically, you have to subtract, right? So the idea of subtraction was absolutely natural. But what's interesting, you can subtract 5 from 8, but you cannot subtract, let's say, 11 from 5 or 8 or any other number which is smaller than this. You cannot subtract a bigger number from a smaller one. So subtraction has its limitations. Now people don't like to have limitations. It's like if I'm telling you, yes, from number 3 you can move to number 8 by adding 5. And basically you can move from 8 back to 3 by subtracting 5. So sometimes you can move back and forth. Sometimes you cannot. Well, how can we explain it and what exactly is the solution to this problem? People don't like when something is not possible. If I'm telling you you can travel from A to B, but you cannot travel back from B to A. It doesn't sound very good. By the way, my family left the Soviet Union in 1979 for the United States. And at the time we were told we cannot move back, we cannot go back to Russia. And believe me, it was not very harmonious. We said goodbye forever to all our friends, etc. So harmony is something which has its inner beauty, inner symmetry, etc. And operation of addition among the natural numbers is not that harmonious because you cannot really reverse it always. So people came up with other numbers. And that's exactly the point which I would like to make. Theory is developed not only based on certain practicalities but also based on the necessity of its inner beauty. So how the theory actually was developed in this case? First, number zero was invented. Because obviously if you can move three steps or eight steps you should also be able to move no steps at all. So number zero was a natural addition to natural numbers to signify basically Well, I went to market and I didn't find anything and I bought nothing. Basically you have to express it somehow in your numbers. So that's what zero is all about. And then people came up with negative numbers. Negative numbers is something which you really need to always be able to inverse your operation because now with zero and with negative numbers the whole set becomes complete as far as operation of addition is concerned. Because now you can add two numbers and you will get exactly the number from the same set. And you can always reverse your operation. For instance you can add negative eight to move backwards. So not only you can move forward from three to let's say eight but you can always move backwards by the same number of steps without leaving the set of numbers which you have. So you have certain set of elements and you can have an operation and what's important is you can always have an element which being applied to any other element leaves it basically unchanged like three plus zero will be still three. And you can always apply any element to any other element and you can still be within this group. So reverse operation is always very, very important and unit operation which doesn't change is also very important. Now the theory becomes symmetrical and in a way more beautiful. At least for some people. Well if you remember there was a lecture about abstraction and in this lecture I was actually explaining about something about group concept. So my point is that now all integer numbers positive, negative and zero they become a group. Group is a beautiful mathematical object because it has the symmetry because it has its harmony etc. So the point of this lecture was basically that the progress, mathematical progress sometimes is moved by practical reasons but sometimes it's moved by purely necessity of the theory itself. So theory becomes more harmonious, more symmetrical, more beautiful at least in some designs. Thanks for your attention and next time we will introduce rational numbers.