[Music] In everyday life, you may need to count objects, animals, sheep. 1 2 3 4 5. This is actually the most natural thing. We call all these numbers natural integers. But sometimes there is nothing to count. Nothing. Well, nothing is zero. 0 which is also a natural whole number . It's even the very first. The set of natural numbers is naturally denoted by n. If 0 is prime, then what happens when we have less than nothing? What are we going to talk about? You're welcome. Because nothing is not nothing. The proof is that we can subtract it. Nothing less nothing equals less than nothing. So if we can find less than nothing, then nothing is already worth something. Well, for all this, we have to introduce negative numbers such as -5, -200, -14. We find a small minus symbol in front of them . Negative integer and positive integer form the set of integers of relative integers and is denoted by z. It comes from the German sal which means number. So, if the integer is integer, the sequence will tell us that it is not destined to remain so. Let's take an integer, we 're going to share it, cut it, fraction it, decimate it. After this cruel torture, we obtain numbers from a new family, decimal numbers. 1.5, for example, is a decimal number. It is composed of an integer and half of an integer. And the set of decimal numbers is noted d and we understand why. But sometimes the sharing goes wrong and the result becomes more difficult to represent. For example, let's divide an integer into 3. So let's divide 1 by 3. What happens? We find 0.33 3 et cetera with an infinite sequence of 3. There is, however, something rational in this division because we understand this number. We know all these decimals which are three, but we cannot write them in decimal form precisely because for decimal numbers, writing must stop. Well, the number 1/3, the result of the quotient 1/3, is part of the set of rational numbers and is noted q. It comes precisely from the Italian t. We have used the term split above. Indeed, all numbers in the set of rationals can be written in the form of a fraction with the numerator over the denominator. The result of 1/3, for example, is written 1/3. And so, in general, any rational number can be written in the form of the quotient of A/B. A and B, each being a relative number. Let's take the number set delirium even further and solve, for example, the equation x car = 2 or what number must be multiplied by itself to find 2? The solution is √ because 2 which is therefore written with a kind of extended r above the 2 is approximately equivalent to 1.414 2 1 3 5 et cetera. But this sequence never ends. This sequence of numbers never stops and we do n't understand it. in the sense that the decimals of this number follow one another without logical sequence. Unlike earlier with rational numbers where we had an infinity of 3, I understand well, there are other rational numbers where when we do the calculation, we end up with decimals which repeat themselves, like 2 4 8 2 4 8 2 4 8 and it repeats like that to infinity. But not here, there is nothing rational anymore. Well √ because 2 is precisely an irrational number that the Pythagoreans in the 6th century BC long denied and even tried to hide. So this number is part of the set of real numbers and is noted r as its name indicates. Note that among the real numbers, there are what are called algebraic numbers of which we have just given an example. √ squared 2, it is an algebraic number. These numbers are solutions to polynomial equations with integer coefficients. X squared = 2 and is precisely a polynomial equation. But there is worse. There are what are called transcendental numbers. These are all the numbers that are not algebraic and the writing is as mysterious as the previous ones. But what's more, it cannot be tamed by any equation with an integer coefficient. no polynomial equation with integer coefficient. The number pi are transcendental numbers. There is no equation of the type a x^ n + b x^ n - 1 et cetera whose solution is pi. For example, these equations are polynomial equations. You can see, I put all kinds of them in each time with x to the power of something. Well, there is no such equation to which pi is a solution. This is why we say that pi is a transcendental number. And one might think that there is nothing more complex than transcendental numbers, and yet it is the complex numbers that will take us even further towards mathematical abstraction. For example, let us cite the most famous of complex numbers, the solution to the equation x² = -1 or what number must be multiplied by itself to find -1? a very strange equation which for middle school students has no solution because it does not verify the rule of signs. Less by less makes more, more by more makes more. How can you get -1 by multiplying a number by itself? Well this is true, it does not have a real solution, but on the other hand it does have a so-called complex solution and this solution is the number i. So, it's a very strange number, it's true, but by this number, we can generate the whole family of complex numbers which is noted C and all of whose elements are of the form A + iB, our I which I have just spoken about. A and B on the other hand are real numbers. So let us note finally that whole numbers are part of decimal numbers which are themselves part of rational numbers which are part of real numbers, which finally themselves are part of complex numbers. If I take for example the whole number 1, well it is an integer. OK? But it's a decimal. 1.0 proves that it is a decimal number. He is also rational. I can write it as 1/ it is also a real number and it is also a complex number. Well, if you're only in middle school, you can tell your math teacher that you know a complex number. This is a We will not go further into the families of numbers, but you should know that the story is far from over and that it is still getting more complicated. Then come the hypercomplexes or quaternions noted H, the octaviones noted O, the pedadics noted QP. [Music]