Hi guys, did you know that with these pens I can write on the board? Ahhh, that's not what I'm going to say guys. Can I make a set of pens with these pens?
That's right, like if I take these rulers here I can form a set of rulers and did you know that in the field of Mathematics we can highlight several numerical sets? And this is the content of our class today : numerical sets, so if you want to learn more about this content, I invite you to watch this class, let's go? Take the opportunity to subscribe to Gis' channel and leave a super special thumbs up.
So, when we talk about numerical sets, let's not first talk about sets, right, include the definition of a set. What is it? So it's a collection or grouping of elements or objects as I mentioned at that time, the set of pens the set of rulers I could take the set of bracelets so whether they are elements or objects, right, they have common characteristics, right, they are classified accordingly.
with its characteristics. So it has to have some common characteristic for me to be able to group them into a set, right? So this is the definition, I'll leave it here for you to mark in your notebook, but then when I'm talking about a set, for example, in addition to the ones I've already mentioned, I could say Oh, I have the set of fruits.
So everything that is considered fruit I put in that set. I have a set of triangles, you know, so I put the isosceles triangle, an equilateral triangle, a scalene triangle and I could also say like this I have the set of letters of the alphabet. I could also say, oh, I have a set of numbers and then, when I talk about sets of numbers, we go into numerical sets, because, right, the numbers will be classified according to the type of each number, let's say So, now I'm going to explain our first set to you.
And do you already know what the first set is? Well done to you who said that it is the set of natural numbers. Let's go then?
So as we know, from the history of humanity, numbers arose due to the need to count, as you have seen, the cavemen, they made the lines, the shepherds placed the pebbles on each sheep because they form a relationship, that's not what they did. They would pass a sheep a pebble, the second sheep would pass another pebble, well then they didn't know how to count but they made the relationships and from there counting came about and with the counting the numbers started to appear, ok? But then the numbers that we know, the digits 1 2 3 4 5 6 7 8 and 9, are those primordial ones, right people, because they are primordial and from these digits I can make the composition and find all the other numbers that we know.
But there was also a digit to determine the non-existence of something, which is the digit zero, right, which represents non-existence, so here with this we form our first numerical set, which is the set of natural numbers. And it is represented by the letter N, okay, folks, the set of natural numbers, so when I'm going to represent a set, what do I do? is a way of doing the representation and separates the numbers between commas and then I put them here.
. . Because we know that the numbers are infinite, right people, that from these digits that I have here, these ten digits I make the composition and do all the other numbers that we know, OK?
And then we have the subsets, in fact the subset of natural numbers which is the representation of the letter N with this* here and you know what this means here this* means the exclusion of zero, it means that it is taking 0 out of the set so if you look at this you know that it is the set of natural numbers without the zero Ok so here we have the subset of natural numbers and now let's see the next set which is the set of integers, shall we go? Now see that with the development of humanity, right the emergence of commerce, debts arose, right, and then the natural numbers were no longer sufficient because see, nowadays I I need to buy something and I don't have money, we go there and buy them the same way, you know, the card makes a loan, then there is debt, I have debt to pay, so for example, if I have R$10, okay, and I owe R$15 for a certain thing what I bought I will owe five. So that means my money wasn't enough to pay and I was left with a debt, so that's when our next set came about because the natural numbers were no longer enough.
And then we have the set of integers represented by the letter Z, Z comes from German because a number in German is zalto. So who are these integers see here the integer set of integers is an extension let's say, complement of the natural numbers because if you look here in the set of integers, see we have here who are these numbers that you You just saw the natural numbers, so that's why we say it's a complement, it's an extension. So from the moment the negatives appeared, they joined with the positives, o positive, null, right people, because zero is null, and the negatives, then the formation of all of this is the set of integers, okay?
And then we also have the subset of integers subsets of Z, so if you notice somewhere there the Z with * means that they are all integers, right, they are negative and positive except zero, okay? That this here excludes zero, now if you look at the integers with this + here, what does it mean? It means that the set of non-negative non-negative integers is because they are the positive ones, right people, so here we use this nomenclature Okay, so here I would only have the positive numbers and zero, here it is included because zero is considered natural, okay?
And then if you look closely at the set of non-negative integers, Z +, which is the set of natural numbers itself, so mark this observation. Last but not least we have the set of non-positive integers which are negative, right people, it's kind of funny to say that they are only negative numbers but there's 0 in the middle, oh, it's kind of funny, that's why I say no Positive and the 0 is there. It's OK?
So mark these observations well and now we go to the set of rational numbers. Let's go? So look here at the set of rational numbers that is represented by the letter Q which is a quotient, okay guys?
It is every number that can be written in the form of a fraction, which is all the definition we have here, but look at every number that can be written in the form of a fraction, since, right, people, A, which here is the numerator, can be any integer, okay, but B has a restriction for B. So you were able to write the number in the form of a fraction, it is considered a rational number, it belongs to the set of rational numbers, so as an example look what I brought here six tenths 0. 6 which is the same thing as the fraction 6/10 I have it here, here it can also be simplified, right, taking advantage of the fact that the number 5 will result in 3/5, how do I write the number 5 as a fraction or can it not be written as a fraction?
Yes, I can write the number 5 here as 5/1 or I could put it as 15/3, for example, 15 thirds will also result in five here, this 0444, as you can see, is a recurring decimal. And while I'm at it, I'll leave you with an indication of the class I explained about periodic decimals, okay? And this periodic decimal came from the fraction 4/9 of the generating fraction, right.
And this other decimal which is the same thing as 12 tenths and I simplify, we will have, right people, 12/10 is the same thing as 6/15. So see here all these numbers that are written in the form of a fraction are considered a rational number but now an irrational one. How irrational will it be?
you already know so let's take advantage and see the examples. This set is represented by the letter I, so I stands for irrational, okay guys and which numbers belong to this set? Oh, all those numbers where the decimal part is not periodic because you saw that if it is infinite and periodic, right?
considered a periodic decimal and then a periodic decimal it came from a fraction and if it came from a fraction it is one of the rational numbers, right now if you look at it Look at our famous Pi, I love giving this example, if you look at the decimal part here the decimal part is not forming a period it does not have a repetition as happens in periodic decimals, so if it is infinite and does not have a repetition it is considered an irrational number other examples that we have are also for example the square root of two which we know that the decimal part here does not form a period and is infinite, square root of five we could mention square root of 3 and so on, so we looked at the number and saw that the decimal part is infinite, right and does not form a period is because it is considered an irrational number. So, when we combine the set of rational numbers with the irrational numbers, we have the set of real numbers. Now I'm going to summarize all these sets so you can visualize them better, shall we go?
And this set is represented by the symbol here, which is the letter R, okay guys, and then making a synthesis, because I want to see if you paid attention in class and know how to answer now, so the first set we talked about was natural numbers which are all those numbers used for counting, hence we have the integers represented by the letter Z, who are the integers? You remember the integers are all the numbers that are natural, right, and then I add to the integers, in addition to the natural ones, the negative ones so then I can put one set inside the other, okay, what happened, we have the numbers rationals, which are all those numbers that can be written as a fraction, okay? And then I can take what I can take from the set of natural numbers and integers and put it here because any number that belongs to natural numbers and integers I can write in the form of a fraction, so it is the right rational number and now I I can take the set of irrational numbers and put them here, right, people, because the irrational numbers are a separate set, but what happens when I relate them, because I actually bring together the set of rational numbers with the irrational ones?
What will form by joining these two sets by bringing these two sets together I have my set of real numbers which is our set there before talking about complex numbers, right people, that would be in another class, okay, so we stopped here at set of real numbers , which is the combination of the set of rational numbers and irrational numbers. So this was our class on numerical sets, I hope you understood, guys, I did it separately for you, and if you understood the class and liked Gis's explanation, leave a thumbs up, take the opportunity to subscribe to the channel and share this class with all my classmates and I'll see you in the next class, bye. .
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