Resumen completo PAES M1 Regular 2025

Welcome to the summary of the

PAES M1 2025. In this video you will

learn all the content that is included

in the test, along with learning

to answer questions like the ones you

They could appear in the PAES step by step. The

The methodology of this video is good.

simple. You're going to have a block for

each piece of test content and

later, once you learn the

Full content, you'll find it here

down below in the description with the

exercises that might come up in that

content and you'll learn, like I told you

Previously, to solve them, I would...

passed. We've added something very special.

which are the M30N cumulative guides.

These cumulative guides fulfill the

function of not forgetting the contents

that you have been learning and that

exercise at all times. The video

It has four cumulative guides,

one at the end of each thematic axis and in

each cumulative guide, as its name suggests

It indicates, you accumulate all the materials that

you have seen up to that point. By

For example, you finish some things and the guide

It includes exercises for the entire subject.

not to that extent so you can practice, no

forget the contents and take your

score to the next level. The video

It also includes a summary book

where we summarize each piece of content that

You'll learn in the video. You contain

all the questions we're going to use

and of course the guides too

cumulative ones that I mentioned to you

previously. All so that they have the

better experience, you can print it.

You can have it digitally and for

To get it, simply go to the

video description and we'll go with

all to raise that score. All the

For years we have had the custom of

publish this summary and until the year

In the past we had always published

math summaries, but this year

We incorporated all the Prebo M30M

subjects, so if in this video

We reached 10,000 likes, let's go

lead a summary like this for

each subject, biology, physics, chemistry,

history, reading comprehension and, therefore

assumed, M2. Before we begin with the

In summary, I want to leave you with the following:

invitation, and it is that in the month of

In November we will carry out the

largest ultra-intensive that has been

Made in Chile to prepare for the test.

We are going to carry out this ultra-intensive course.

from November 17th to 30th and

It is designed so that if you, for

For example, if you want to save the test,

you can do. If you want to maximize your

performance in those two weeks, the

you can achieve. And of course, if

you want to fine-tune the final details for

To get 1000 points, we're going to work for

to reach this national level. And just in case you're wondering

You asked, of course, we're going to have

all subjects in this intensive course.

Registration opens on Monday, the 3rd.

November, so if it's that date or

After that, you can register and

We're going to work hard so that

you can upload your score to the next one

level and stay in that race of your

dreams. The ultra-intensive has a

price of $30,000 and obviously you're going to

You will be able to register and you will be able to access

all the contents of the test and

We're going to take that score to the maximum.

Finally, I want to wish you much success.

in your preparation. I know it's a summary

12 hours and suddenly you can see

It's a huge challenge, but I want to

Be disciplined, be

disciplined, that you don't give up, and that

work towards achieving that goal, that

dream you have. I know you're going to achieve it

That score, I know you're going to get that

career and that you will fulfill each one of

your dreams, so I wish you much

success. And as we always say at M30N,

We're going to go all out, you're going to catch up.

that goal.

And we begin with whole numbers.

First, let's remember what

These are the natural numbers.

Natural numbers are those that

They allow us to count elements of certain

sets. They are numbers like one, the

two, the 3, the cu, the 5, the 6, the 7, the

8, 9, and 10, and so on.

Now, what is the set of the

whole numbers?

The set of integers is

composed of the natural elements,

their additive inverses, which we will now discuss

Let's see in a few minutes, and zero. All

This is the set of numbers

wholes.

Now I mentioned something to you about the

additive inverse, which is also

known as the opposite. Look, in

In very simple words, the reverse

additive of an integer is

simply the number, but with the

opposite sign. I changed the sign no

further. As a general rule, if you have a

integer n,

You can represent its additive inverse

as - n. And something very, very important,

I want you to write this down. When I add one

number with its additive inverse, the

The result of that operation will be

zero. Now I want us to look at this in

Spanish and let's look at some examples. Look,

Let's suppose you have these numbers and

you want to see their additive inverses. As

I told you, it's just a matter of changing.

the sign. So, if I have here the

8, its inverse. Ah, I see. -8, it

We have it ready. 3, its additive inverse,

-3. We're ready. And what do we have here?

The -2. What do we do when this question arises?

Is it negative yet? simple. As we said,

We changed the sign. So, the

The additive inverse of -2 is simply

2. And now we have it ready. So, this

of the additive inverse is only

change the sign. Now, something very

Important, if you add a number to its

additive inverse, as I told you, the

The result will always be zero. If you

You add up each of these numbers, always

The result will be zero.

Remember that whole numbers are

can be represented on the number line,

where in the middle of this will be located the

zero. To the right of zero will be

the numbers that are positive and the

To the left of zero will be the

negative numbers.

Remember that zero is not positive.

neither negative.

A very important factor that you must

Remember, and I ask you to notice, that...

the further to the right it is

a number on the number line,

the greater its value will be. For example, I

I know that 5 is a number that is greater than

4, since the 5th is further to the

right. Similarly, if I, for

For example, I put -3

And I also have -4, I know that -3 is going to

be a larger number, since it is more a

the right.

That rule is very important that the

keep this in mind, since in the test you

They're going to ask to compare whole numbers and

If you keep that in mind, you'll never...

to be wrong.

And with the little material we've seen

So far, I want us to resolve

our first question, PES, of this

summary. Let's begin. How many numbers

positive integers are greater than or equal to

that -4

and younger than 3? Already. To solve this

Question, I will use the number line

that we had here. We're going to copy it,

Let's stick it here and get started.

Information always stands out

important. Look, they have to be numbers.

wholes.

positive.

Already. Which ones are positive?

This, this, this, this and this. And so

successively, right? All the

We ruled out the negative ones, and also the

zero.

Now they must be greater than or equal to

-4. All positive numbers fulfill the condition.

This condition, so we're doing well. But

They must be less than three. Therefore

Therefore, five is not useful to us, four

Neither, and not three either. It must be

less than three. Therefore, they are useful to us

only

these two little numbers we have here.

Therefore, the correct alternative is

The letter A. There are only two numbers

that meet these conditions.

And with this, with the little bit of material that

We have seen so far, we have resolved

our first question. PES.

We're going to work 100% so that you can

answer each question that appears in

take the test and raise your score. That this

Let it be a small test of what you are going to

to achieve and we will continue working to

that score, as I told you, reaches

maximum. Now we're going to remember two

very important concepts in numbers

wholes, which are the successor and the

predecessor.

Let's start with the first one. The successor of

An integer is the largest integer plus

close to this on the number line. Is

the one that is immediately after the

right. Look, for example, at number two. You

You see number two and we're going to see its successor, which

It is the one that comes immediately after

the right of this. In this case, the

The successor of two is simply three.

Now, what would happen, for example, if I

I want to see the successor to -4?

Which one would it be? You ask yourself, what is

the whole number that is to the right of this

on the number line? Ah, I see. Perfect.

It is -3 and this is its successor.

It is very important that you keep in mind

that the successor of an integer

It will always be greater than the number

original. And if you want to represent a

successor of an integer n, say,

I have an integer n, the successor

It will always be given by n + 1. And with

You'll always be able to represent that.

Look, for example, if I have the

Four, the successor to four will be 4

+ 1, which is 5, and there you get it.

Now, what about the predecessor of

a number

whole?

The predecessor of an integer is the

nearest smaller integer to this in the

number line. is the whole that is

immediately before to the left of

this. Look, for example, if I have the

two,

His predecessor will be one, which is this one

that we have here. If I have -2, then...

The previous one will be here on the left,

which is going to be -3.

It is very important that you keep in mind

that the predecessor of an integer

It will always be less than the number

original.

So, if I want to represent now

the predecessor of an integer,

Let's say an integer n, we will always

obtain by taking n and subtracting one. By

For example, if I want to extract the predecessor

of four,

The predecessor of 4 will always be 4 -

1, which is 3, which if you look closely is the one that

It's to the left of this one on the straight

numerical.

Now let's look at what numbers are

consecutive integers. This is good

simple. Whole numbers

consecutive integers are integers that come one after the other.

one after the other on the number line. By

For example, one and two are numbers

consecutive integers, since they come one

one after the other on the straightaway. Now, if I

Here I add the three, it would have three whole numbers

consecutive. If I add the four,

would have four whole numbers

consecutive. And likewise if I add the

five. I have five whole ones here.

consecutive.

I could even add zero here.

Here I would have 1 2 3 4 5 6 integers

consecutive. And that's the characteristic

that whole numbers have

consecutive. They come one after the other.

Now, very importantly, always keep in mind

He says that when I have two numbers

consecutive integers,

These will differ in one

unit. Between 0 and 1 there is one unit

distance. Between one and two there is

a unit of distance. Between two and

There is one unit of distance in three. AND

and so on.

Now let's see how to represent

consecutive whole numbers.

Now, what am I referring to by

represent writing with numbers and

lyrics what we say with

words? In this case, for example,

Suppose a statement tells you,

"Hey, I want you to act out three

consecutive integers."

So, how could we do it? Look, in

First, we are working with

our whole numbers and I'm going to say

as follows. I'm going to have a number soon

whole n that will speak for all the

wholes. This integer n will be the

representative of all numbers

whole, of this, of this, of this, of

any.

So, what's the key here?

Remember the definition. Ah, I see.

Perfect. Look, whole numbers

consecutive ones come one after the other.

Therefore, the one that comes after n

+1, right? We saw that we were going

increasing by one unit. And if I want to

another whole one, to this one I had here

I can add one and I have n + 2. Why?

n + 2? Because you added one unit here.

In the same way you did it here.

So, in this way we have

represented

three consecutive integers and this works

for all cases. Because? Because

You can put any value here

a n. Look, let's suppose you put it on,

I'm going to save here, let's suppose that he

you place

one, to give you an example. Already. So,

Here I would have one, here I would have 1 + 1, and here

I would have 1 + 2. And look, I have 1, I have

two and I have here the three and which are

precisely three whole numbers

consecutive.

And in this way one represents those

consecutive whole numbers. I'll do it to you

I have a question, and I want you to pause here.

video and you say it. Which one would come

being the integer that comes after n

+ 2?

We add 1 and it becomes n + 3. And here,

How many consecutive integers?

Would it have?

You tell me. There are four of them, right? And from that

We are representing the numbers in this way.

consecutive integers.

Many times you will come across

exercises where little things appear

how are you. Let's suppose it appears to you in

the three between two bars, or let's say the

-4

between two bars. Rapier

means? that you're going to have to calculate

its absolute value. But what is it?

absolute value?

The absolute value of a number is the

distance between that number and zero

on the number line.

Since it is a distance, it will always be either

positive or zero, but never negative.

So, let's suppose I want

calculate the absolute value of 3.

I wonder, hey, how far away is it

Find the three from zero on the number line

numerical? I'll count one, two, three

units. Ah, yes, perfect.

Simply three.

Now I have the absolute value of -4.

How do we do this? Again we

we asked. We are already located here in

-4. How far away am I from the

zero on the number line? Come on

counting.

1, 2, 3 and four units. Therefore, his

The absolute value is four. So,

remember that whenever you find yourself

a number between two bars,

Let's assume a number n, you are

calculating its absolute value here. Yeah, one

Last example, I want you to solve it

you. What is the absolute value of -3?

You've got it. I'll do it. Now

I have -3. Let's keep counting. 1 2 3 units

to get to zero and we already have it

list.

It's important that you remember the value

positional in integers, already

which is something that has been asked with

frequency in the test. Remember that

We use the decimal base because

We represent all numbers with 10

different digits, from 0 to 9. In this

base, the value of each digit depends on

assumption. Look, let's see the first one.

number. Here we have the 345.

Let's note that here we find the

Hundred, here the ten, and here the unit.

Therefore, we can rewrite

this number in the following way. As

We have a three in the hundred, this

This is equivalent to having 3 * 100. To this

Let's add, since we have a 4 in the

tens, this is going to be 4 * 10. And here

We have a five in the unit, therefore

We add 5 per unit. 1. Let's look at another one.

example. Here we have number 1234.

Here we have the 1000 unit, the

hundred, ten, and one. Therefore

Therefore, we could rewrite it from the

as follows. I have one in the unit

out of 1000, therefore it is 1 * 1000 more I have

a 2 in the hundreds place, 2 * 100 plus a 3 in

the ten, 3 * 10 plus a 4 in the unit,

It's a 4 * 1. And that's how we have

rewritten this value, which as I told you

It's something that has been asked in the

proof. Now let's move on to the next one,

with 102. Here we have hundreds, tens,

unit. This is the same as 1 per

Hundreds, 100 plus the tens digit 0. This is 0

* 10 plus the unit which is 2 * 1 and of that

We have rewritten this value. Such

Once in the test they could reach

ask

by a negative number. And here

Simply put, the only thing that changes is the

The sign that goes before, indicating that it is

a number that is below zero in the

number line or to the left of zero

on the number line.

So, how do we rewrite

This value? According to what we have seen.

Simple. For example, at -256

You're going to put less and you're going to open one

parenthesis. So how do we leave it? Look,

I have hundreds 2 is 2 * 100 plus tens 5

* 10 plus unit 6, 6 * 1 and we have it

ready. And in this way we have

rewritten this value. Keep in mind that

This is something that has been asked in

the test, although they involve powers.

We'll see this later in the

powers and there we'll see how it

we rewrote using mainly

Powers of 10. But that's material

which comes later. Now we'll see

operations on integers.

We will remember the fundamentals of addition

subtraction, multiplication, and division. AND

Let's start with the addition of integers.

Okay, let's start with case number

one. When we add two numbers

of the same sign, the rule is the

following. The values ​​are added together

absolutes and the common sign is maintained.

For example, here we have the sum of two

Positive numbers, 7 and 5. 7 + 5 is 12. And

We have it ready. But what happens when

Add negatives? Well, seeing that

We are in the case of adding numbers of

same sign, we know that if I add two

negative, the result will be a

negative number. But how much?

Simple. I see the absolute value of -3,

how much is it? It is 3. The absolute value of

-7 is 7. So, I add 3 + 7.

Therefore, I'm going to give it a 10. And

Remember that we left you the least because

we keep the sign that both have

numbers.

Now, what happens when we add numbers?

of different signs? Usually this

We also do the process in a more

intuitive, so to speak. However,

Here I'm going to teach you the logic that

We're still following. Already? Well, firstly

In this place, the absolute values ​​are subtracted.

You subtract the smaller from the larger one and you get

retain the sign of the larger number

absolute value.

So let's suppose that I'm going to

I have -5 and I'm going to add 2. What is

Logic? I'm already wondering, what is it?

What is the absolute value of -5? Perfect, it's a 5.

What is the absolute value of 2? It's two.

So, we have everything ready for

solve the exercise. Well, like

We indicate here, we subtract the values

absolutes. The biggest one is the rest

smaller. 5 - 2 is 3 and the

sign of the number with the highest value

absolute. In this case, the highest value

absolute is 5 and corresponds to the number

-5. Therefore, we keep the sign

From that -5 and we have it ready.

We usually do this process by

in a more mechanical, automatic way, by

So to speak. However, it's never okay.

more to remember. And well, here we have the

sum of 10 and -4. We're still in the same situation.

logic. I already have the 10, its value

The absolute value is 10. I have -4, its value

Absolute is 4. Perfect. Here we subtract

absolute values. 10 - 4 is 6. And

we keep the sign of the number that

has a greater absolute value than in this

In this case, the largest absolute value is 10 and

It corresponds to 10. Therefore, we keep

a positive sign, although it is not

It is necessary to write it. Simply

number six remains.

Now we are going to remember something very

important about subtraction.

Suppose I have the subtraction between

two integers A and B in that order,

where it is remembered that A is the minuend and B

is the subtrahend. I want you to remember

This is fundamental because...

They might suddenly ask you about it in the

proof. Remember that if I have the

subtracting a from b, this is equivalent to

add A to the additive inverse of B.

Look, I know that with letters it can be seen very

complex, therefore, let's look at some here

examples. Now let's look at the examples.

We have 5-10. This is very likely the

can solve it mentally, but if

That's not the case, I'm going to show you properly the

logic of how to do it with the

principles we have learned. Already

We have 5-10. Remember then that the

Subtraction can be rewritten as addition

and we already know the principles of

the sum. So, look, I'm going to

rewrite this as a 5 plus the inverse

additive of 10, which is -10. And look, here

I have a sum of two integers of

different signs. I see its value

absolute. Here it's 5, here it's 10. The rest

absolute values. The greatest one

the smallest rest. 10 - 5 is 5. And

I keep the sign of the number that has

the largest absolute value, in this case the

-10. And we've solved it. Let's go with the

following. Look, here I have a -4 - 3.

I'm going to rewrite it as a sum. This is

-4

further. And here I'm going to take the reverse.

additive of 3, which is -3. Today I have here

a sum of two numbers with the same sign,

two negatives, therefore, I keep the

negative. We extract their absolute values.

This is 4. This is 3. 4 + 3 = 7 and the

We have a list. The result is -7.

Now, what happens next?

example? We have here 2 - -3.

So, this is going to be 2 + the inverse

The additive product of -3 is 3. Therefore, +3 is

5. And now it's ready. Having seen

The subtraction, now I want us to remember

Something very important, which is distance.

on the number line. This is the question and

frequently and I want you to please

Write this down too because it's very important.

You need to keep that in mind.

Often they'll give you two numbers in

the number line. For example,

Let's say they give you the 3 and they give you the

-4. And they ask you, "Hey, what's the

distance between these two numbers and what

What are you going to do to be able to calculate the

distance?" Simple. Always. to the number

elderly,

You're going to subtract the smallest one in that order.

That way you'll always be able to get

the distance.

So here the oldest is the one who is most

to the right. We subtract the three

smaller, which is -4.

This is equivalent to having 3 plus the

additive inverse of -4, which is 4 and 3 + 4

It is 7. Therefore, this is the distance

between those values. Now it too

You can see it on the number line. Look,

Let's keep counting.

Of the three. I need to move. One, two, three,

4, 5, 6, seven units to reach the

-4. But so I don't have to walk

counting, simply the oldest

subtract the smaller one. As a piece of advice,

whenever you calculate the distance,

Be careful with the signs, because it is the

most common mistake that is usually

commit. You have to be organized or

organized so you don't make a mistake. And the

You're going to have a good question. Now

I want us to look at something very important

which is the sum on the number line.

Note that this will also apply

for subtraction, since you can subtraction

receive as a sum so that you have it in

account. Look, let's suppose I have the

Number two and I add three units, I

I add three. You know that the result of

This is five. But I want you to have in

It says the following. If I place myself in

a number, for example, two, and it

I add a positive number, which I will

What to do is move to the right in

as many units as the value

absolute of said value. So, in

In this case, I have to move three

units to the right. Story 1, two and

three and we arrive precisely at five.

Now, what happens when we add

a negative number? For example, at two

We're going to add -4.

What's going on? What's happening here

When we add a negative number, it means we...

Let's move to the left. in

Indicate the value in as many units as indicated.

absolute of said value. For example,

We have here the 2. The absolute value of

-4 is 4. Therefore, we're going to move here.

four units to the left.

We counted 1, 2, 3 and 4 and arrived at -2,

which is the result of this operation.

Why am I telling you this is so, so

important? Because it is asked in the

proof. Look, let's look at this question.

And it says: "Consider the following

number line. Which of the following

procedures represents the operation

-5 + -8 using the number line? Already

Let's read the alternatives. To position oneself in the

-5.

Already. And travel in eight units to the

left. And this is indeed going to be

So, if we place ourselves at -5 and...

In this case, we add the negative -8, we

Let's move to the left, since it's

a negative number, and we're going to move

units, since it is the absolute value of

-8. Therefore, the correct one is letter A.

In fact, you represented everything

You could do this on the number line

do it in the following way. Tea

You were located here at -5. You know that

The result of this operation is -13.

And here to get to this -13 that will

to be further to the left on the straight

Numerically, we had to move eight

units to the left

And that's how we arrived at the result

of the operation. Now let's see how

solve operations like these when

we have a series of pure sums and

subtractions.

Well, there are different ways, but

I want you to consider something

fundamental

And that's because addition and subtraction have the

same priority. And in case you have

a series of pure additions and subtractions, the

The key to solving it is to follow the

following. You're going to solve them in order.

And in what order? In order from left to right

right. It is enough to perform the

operations in sequence and you will be able to

to reach the result. If you have already

trained more, there are ways that are a

A little faster, but if you're

newly inverting, there is no

problem in doing it that way. I only know

very tidy. Already? So,

Look, let's solve the sequence.

1 - 2 is -1.

To this we add 3 - 4 + 5 - 6 + 7.

We continue.

-1

+ 3, what is that? It's 2. -4

+ 5 - 6 + 7. We continue. 2 and -4 is -2

+ 5 - 6 + 7. We continue. -2 + 5

- 6 + 7 and we're almost there.

3 - 6 -3 + 7 and -3 + 7 gives us as

Result 4. And in that way we have

solved this series of additions and subtractions.

Here, if you look closely, it was a process that

It depended a lot on not making a mistake when

continue rewriting. Therefore, it is a

valid form, although possibly you

Perhaps you can solve it in a slightly different way.

different. And now I'm going to show you another one

form and you can also reorder

mentally or in writing the terms

grouping together everything that is adding up and

everything that is subtracting. Let's see here

how to do it. Look, what do we already have?

Adding that it's positive. Have

one, we have three, we have

five and we have seven. So, it

I'm going to write here. Maybe you will

Mentally, I'm going to do it

purely for the purpose of the example. I'm coming

Write here 1 + 3 + 5 + 7. And here

I'm going to subtract this and open

parentheses and I'm going to put it here. Which

I'm subtracting, I'm subtracting two.

I'm subtracting five and I'm subtracting

a six. And I write them just like that, 2, 4

6. So, what does this equal? 1 + 3, 4

+ 5, 9 + 7 results in all this,

16.

And from this I will subtract the sum that

We have it here. 2 and 4 6 + 6 12 16 - 12 is =

4. And in this way we have arrived

exactly the same result. Ideal

It's that I do it the way you want me to.

Choose whatever is most comfortable for you.

Now we're going to look at mathematical language

common. Let's suppose we have

that a number x

is increased

in i. If they tell you that, it means that...

You will add i units to the value of x,

You're going to add i. Nothing more than that. Without

However, what if they tell you that a

number x

Does it increase? And we're going to highlight this here.

with red. Increase to and when you

This appears means that the value

It will be transformed into i. That is to say,

We're going to go from having x to transforming it

in i.

S means that it transforms it and is

fundamental because in the questions of

The test comes out exactly like this.

Now, let's suppose we now go with

a decrease. This remains the same

logic. If I have a number x, it is

decreased by i, is decreased by i,

This means that you will subtract i from x,

nothing more than that. On the other hand, if they tell you

that a number x is decreased to y this

a, as I told you before, means

that is transformed, that is, that

value changes from x to directly

Yo. That's the result. Please,

Keep this little chart.

Now, if we have, for example,

the difference between x and y in that order,

Simply subtract y from x, that's all

That's it. This is going to be x - i. And in the

in case the number x is greater than the

number y and they ask you the excess of x

about and, they are simply telling you

asking by how much x exceeds y this is

calculates only by calculating the

difference of x - y.

Nothing more than that.

In this video we won't see how

multiply or divide. However, it is

essential things to keep in mind for the

multiplication and division the rule of

the signs. What does the following tell me?

If I multiply or divide numbers of

with the same sign, the result will be

positive. However, if I multiply

or I divide numbers of different signs, the

the result will be negative. By

For example, I could have 5 * 10 here, the

The result is 50, a positive number. EITHER

It could have, for example, a -3 * a

-4. Since they have the same sign, the

The result will be positive, and in this

Case number 12. Now, here we have a positive result.

For example, a negative number could be 5

* -4 and the result will be -20. Or by

For example, I could have a -3 * a which is

-6.

So, keep in mind that this is

It works for both multiplication and

for the division.

Now we will look at the property of the

lock. Ownership of the lock

The integers tell me that if I

I add, subtract or multiply two numbers

whole numbers, the result is always different

whole. For example, if I add 2 + 3,

This equals 5, which is a number

whole. If I have 5 - 10, the result

It is -5, which is an integer. And if by

For example, I multiply -3 * 5, the result

It is -15, which is another integer. And you

You realize that it always seems to give us...

result is an integer. Always

when performing addition, subtraction or

multiplication of two integers, I'm going to

obtain an integer as a result. And you

You might ask what happens to the division. AND

In division, it happens that not always the

Dividing two integers will give me

resulting in an integer. Look,

Let's look at an example. Let's suppose that I

I divide 2 into 5.

[Music]

And this isn't a whole number, it's a

rational number that we will study further

forward. Therefore, it holds true for addition,

subtraction and multiplication.

a little common language that can

appear in the exercises. If you

They ask for the double of a number x,

This will always be 2 * x. If you

They're talking about three times a number x, it's 3 *

x. If they tell you about four times x, it's 4

* x. The quintuple,

5 * x. and so on.

They might also ask you about the

half of a number x, which is

simply x divided by 2, or one third

of x, which is oxide in 3, or 1/4 of x, which

is oxide in 4, or even a fifth of x

which is x divided by 5 and so

successively.

Now let's remember the papo mudas. He

papo mudas is an abbreviation that we

remember the correct order in which

We must perform the operations.

First we start with the parentheses,

then with the powers, then with the

multiplications and divisions and

Finally, additions or subtractions

which would be addition and subtraction.

Now let's look at some examples so that this...

Let it be clearer. We have this operation.

To perform it in the correct order,

Let's remember the mute papo. What do I

says? Start with the parentheses.

Perfect. We start with what we have

here. We'll end up with 7 - 25 - 21, which is

4. So here we put 4 * 5 - 3.

Next, let's talk about powers. There is no

powers, so we move on to the

multiplications or divisions. In this

In this case, we only have one

multiplication, which is this 4 * 5.

So

- 4 * 5 will be 4 * 5 20 - 3. And

Finally, let's move on to the additions or

subtractions. In this case, this is what

same as 7 men

20 - 3, which is less here 23 and 7 - 23 us

It will result in -1.

And we have now solved this exercise.

Now let's move on to a second exercise that

It's very interesting. Let's begin

solving the parentheses as usual.

We only have one parenthesis,

then we only place 8 - 4 which

It's 4 divided by 2. Let's continue. Here, if

We noticed that there is no

Therefore, powers must be passed with

multiplications or divisions. And it is

It is important that you consider the following.

If I have a series of pure

multiplications with divisions,

Whenever I have to solve it

do in order from left to right,

since multiplication and

divisions have the same priority. AND

in order to solve the exercise,

The convention is that this is done in

left to right.

So, let's begin. We have here 2 + 9

* 4 which is 36

say

2.

Now, 2 + 36/

2

and 2 + 18.

And this is the result of this

operation.

Now, with what we've learned, let's solve

some questions. What is the value of

This operation? Yes, simple. We use the

We'll talk and resolve what's going on.

within the parentheses. Here we have

to solve something. So, we have 1 -

-3

multiplied by -2 - 6 which gives us as

result -8.

We continue. Here we have one

multiplication between two numbers

negative. -3 * -8. The result of this

It's going to be positive and it's going to be 24.

So, how is this turning out? He's going to

be 1 less

the result of this multiplication

which is, and I'm going to write it here in red,

24.

So

It is - 23. And the correct one is letter B.

This was the first question of a

Try it and we've already solved it. Let's go

another question. If five times -10 is

Subtract three times -1, what numbers

Does it get? Already. What is the quintuplet of

-10? simple. 5

* -10

and 5 * -10 is -50.

Here, it will be subtracted

triple of -1. How much is this? 3 *

-1, which gives you the result

-36.

So what are they telling him? A -50

-50

it is subtracted, we subtract it

-36

And here we have to solve this

operation.

Then we'll be left with -50.

Less is more

36

and 50 - 36 is = -14,

The correct answer is B. Also

You could propose this exercise

I'll leave it as an alternative.

so that you keep in mind that always

There are different ways you can do it

you can suggest. You could directly

arrive and place the operation. Look,

quintuple of -10 5 * -10 is subtracted

subtract three times -1. The triple 3 * -1

and here

5 * -10 is -50

- 3 * -1

3 * -1 is = -36.

And if you look closely, you arrive exactly

to the same operation. Do you realize that

Is it exactly the same?

Therefore, -50 - men is +36

and -50 + 36 is -14 and you arrive exactly

to the same result. Now we're going to

remember what even numbers are and

odd numbers. Even numbers are integers

that can be divided exactly by

two, that is, when performing the

division leaves no remainder or the remainder is

zero. Any whole number ending in 0 2 4 6

u 8 is an even number.

An even number can always be written

such as 2 times an integer n. In fact,

This is how to represent a

even number.

So, let's look at some examples. Look,

I believe that 12 is an even number.

Because? Because it ends in two, which is

one of the conditions we saw.

For example, the 126

It is also an even number, and so is it.

You have to consider something very

important, and that is that whole numbers

Negatives can also be even or

odd numbers. For example, -1

It is an even number because it ends in 2.

-14 too.

Now, what about zero?

Remember, the number 0 is also a number.

even, therefore, zero is even. Now,

Another important thing, all of these being

Even numbers can be rewritten

always as the product between two and one

whole. For example, 12 is 2 * 6,

the 126

It's a 2*

63.

Well, that's it. Now, look,

-12 is the same as an integer 2 * 1,

which is -6 and -14 is 2 * -7. From the

Similarly, 0 is 2 * 0. And you

You notice that you can find each even number

rewrite as the product between two and

an entire. Now let's look at the numbers.

odd numbers. And these are whole numbers that

cannot be divided exactly by

two.

In other words, the rest is different from

zero when performing the division.

Any integer ending in 1, 3, 5,

7 or 9 is odd. An odd number is always

can be written as 2 * n + 1, where

n is an integer. Let's look at some examples.

of odd numbers. For example, the 19th.

Why do they end up nine?

They end on the 27th if the 21st

They can also be negative numbers.

For example, -15 is an odd number.

Now, another important thing, as you

I said, you can always represent or

rewrite it this way as 2 * 1

integer + 1. In this case, 19 is 2 *

9, which is 18 + 1, 19. The 27th

* 13 + 1 2 * 13 26 + 1 27

* 10 + 1 2 * 10 20 + 1 21 And in the case

It's a little more difficult to do from -15

to the eye, but in reality it also

It can because it's an odd number.

in that way. In this case it's 2 * -8

+ 1. Look, 2 * -8 -1 + 1 gives us as

result -15.

Therefore, every odd number is always

can be rewritten or can be represented

like 2 * 1 whole + 1. Now let's

learn what even numbers are and

consecutive odd numbers.

Consecutive even numbers are even numbers

that come one after the other on the straightaway

numerical. For example, the number zero and the number two.

are consecutive even numbers, since

They come one after the other on the straightaway

numerical. Again, they are even numbers.

that come one after the other on the straightaway

numerical. If I, for example, add the

Four, here I would have three pairs

consecutive. If I add -2, I would have

four consecutive pairs and so on

successively.

I want you to realize something, and it is

that they will always be separated by

two units of distance.

There will always be two units of

distance between them.

Now, how do we represent

consecutive even numbers? Simple, no.

It's as complex as it seems. Look,

Remember that any even number you

You can write it as 2 * an integer n,

TRUE? And as you know, it's

They differ by two units, it is enough that

For the next one, you add 2 * n

two units and in this way you have

two consecutive pairs represented.

Let's suppose you now want another one

simple.

To this last one we will add two

units because we know we're going in twos

in two. So, we add 2 to this and

We are left with 2n² + 2 = 4. And so

successively.

In this case we would have 1, 2, 3 pairs

consecutive and this is the way to

represent them.

Now,

with consecutive odd numbers is

exactly the same logic. Look, the

consecutive odd numbers are odd numbers that

They come one after the other on the straightaway

numerical. For example, one and three

They are consecutive odd numbers, since they come

They are odd numbers that come one after the other

the number line. And in the same way,

If I add five, I would have three

consecutive odd numbers. If I add -1,

would have four consecutive odd numbers and

and so on.

Now, how do we represent them?

The same logic. I want you to notice that

Here we go two by two. Do you realize?

Let's go in pairs

units, two at a time, right? By

Therefore, it is enough to simply place your

odd number, which in this case you

You represent it as 2 times an integer n + 1,

which is what we saw earlier.

And depending on how many you want to represent,

You add two units. Let's suppose

that you want to represent three odd numbers

consecutive. So, to this

We add 2 and we get 2 * n + 1 + 2 3.

Here we have

an odd number too. So, the

same logic, as we want to add one

Furthermore, to this we add two units and

It would be 2 * n 3 + 2 is 5. And here I have

an odd number, the one that's coming and the

next, which are three odd ones

consecutive and that is a form of power

represent them. Now I ask you

Next, and I want you to leave it here.

the comments. What would the

next odd?

And how many consecutive odd numbers would it have?

In this case, the next odd number would come

being

2 * n, we add 2 to 5 and it becomes 7. And

Here we would have 1, 2, 3, four odd numbers

consecutive.

To move on to the next content,

I want you to answer this question.

What do these numbers have in common with you?

What do you see here?

What do you notice?

Do you realize that each of these

Are these numbers multiples of three? The 3, 6,

9, 12, 15, 18, 21. They are all multiples.

of three. Now, by definition, what

What's happening here? Any integer that

If it is a multiple of 3, it will always be possible

rewrite as 3 times an integer n. In

In this case, 3 is the result of

write 3 * 1.

6 is the result of multiplying 3 *

2. The 9 3 * 3. The 12 3 * 4. The 15 3 * 5.

18 is 3 * 6 and finally 21 which is 3

* 7. And if you notice, it comes true

that we just said. All the

Multiples of three can be written as

that way, like three times one whole.

Having seen this little one,

So to speak, spoiler alert, let's see what

which is a multiple. We say that a

integer A is a multiple of another

integer B. If A is the result of

multiply bun

whole. For example, in the previous case

We saw that 18 is a multiple of three,

since it could be written as 3 * a

integer n, which in this case is that integer

It used to be 6.

So 18 is a multiple of 3.

Then it leads us to see the following

questions that are fundamental.

Is -21 a multiple of 3? Let's respond

this question. If -21 is a multiple

of 3, it must be fulfilled that it will be equal to

3 * 1 integer n. Is there any whole number that I

Will this condition be met? In this

Yes, that's the case, because -7, which is a

whole number, if I multiply it by

3, will give me a result of -21. Therefore,

-21 is indeed a multiple of 3. Now you

I ask the following. Is he a multiple?

of 3? If 0 is a multiple of 3, then it is possible

write as

3 * 1 integer n. So I ask you

Is there any whole number that would allow me to...

Does it meet this condition? And indeed

Yes, because if I replace this n

by a 0,

3 * 0 is 0. Therefore, it is a multiple.

3. In fact, I'm going to tell you that the

cer is a multiple of all numbers

wholes.

Now, 3 is a multiple of 0. If 3 is

A multiple of 0, 3 can be written

as 0 times an integer n. And I ask you

By what integer should I multiply the

Zero so that I get three? And the answer

There isn't one. There is no whole that

fulfill that condition. Therefore, the 3

is not a multiple of zero.

And this is very important. I told you

that zero is a multiple of all

whole, but—and pay attention here—the only one

A multiple of zero is zero. That is, 0

It is a multiple of zero, no other multiple

of cer and is its only multiple since

0 is the same as 0 times an integer n,

where n is any integer. For example,

You could put n * 5 and 0 * 5 here.

0. Therefore, so that it doesn't sound

tangled, I'll say it again and

write it down. Zero is a multiple of all

whole numbers,

But zero has only one multiple that

It's the same zero. Let's suppose that you

Do you want to check what the multiples are?

of an integer. For that, I'm going to

Leave the following note that may

be quite useful. This will serve to

any nonzero integer P. And it is

Simply put, all its multiples will be

These that are in this note that you

let. Look, to make it clearer,

Let's suppose, going back to case 3,

P = a 3 and I want to see all the

multiples. Obviously there are infinite ones

multiples, but I want to write

multiples of three. So I follow this

Point it obviously as far as I want.

write. First we place the

zero. Because? Because zero is

multiple of all integers.

Then we'll add more - 3. What

Does this plus -3 mean? Here it goes

meaning that both -3 and 3 are

Multiples of 3. Now we continue.

We're also going to place

the most - 2 * p, where p = 3.

It's 6. And again, what does he want?

Say this? that both the positive 6 and

-6 are multiples of 3 and so on

successively. 3 * here I would be doing

3 * p which would give us 9, would be more - 9,

both multiples of 3 and continuing here

It would be more - 12 and so on. AND

I'll even add one more here, it would be

plus -5p, which would be plus -5* 3,

which is plus -15 and so

successively. And here, if you look closely, they go

coming out all multiples of 3.

Now, if for example your number P were

equal to zero, because here we are talking

only of non-zero values, it is so

simple. As I told you before,

the only multiple of cer is itself

zero.

Finally, I want to leave you with a little

This note might be quite useful to you in

some exercises. Look, remember that

whenever you add or subtract two multiples

From the same number, the result will be

be a multiple of that number. By

For example, here I'm going to take some of the

multiples of three. Look, I'm going to take a drink here.

The 6 positive and I'm going to take the

-15. Already?

So, if I add 6 to -15, in

In this case, since both are multiples of

3, I will get as a result a

multiple of 3. Look, notice, the

The result of this is -9, which is a multiple

3. Do you realize? So, write it down.

When I add or subtract two multiples of

the same number, the result will be

multiple of that number. Now, it

The same thing will happen for the

multiplication. For example, and here it is

I'm going to do a little more with numbers

small ones. Let's suppose I'm going to

multiply eh

the three with the 12, for example. It's not

the smallest. I said, already with the six.

Already. 3 * 6 = 18. And you realize that 18

It is also a multiple of 3. Therefore,

This also applies in the

multiplication. Write it down here. When

I multiply two multiples of the same

number, the result will be a multiple

of that number. It's quite a point

useful for certain types of questions and

I want you to keep that in mind. Before

Moving on to the next concept, I want you to

Let's make two divisions. Look, let's suppose

that we are going to divide the number 135

in 5. Now, to carry out the division

We do as usual. How many times

Does 5 fit inside 13? It fits twice. 2 *

5 is 10. We subtract 10 and we are left with 13.

- 10 3. And now we go down 5 and we

We ask, how many times does 5 fit into

35? And it fits seven times. So here

5 * 7 35 - 35 remainder 0.

When the remainder of a division is zero,

We say that this division is

Exactly, right? Now, what would happen?

For example, if I had instead of

a 135,

Let's say a 137

and we divide it into 5, we follow the same

logic. How many times does 5 fit into the

13? It fits twice. 2 * 5 = 10, remainder 3.

And now we go down to 7. And how many times

Does 5 fit into 37? It fits seven times. 7

* 5 is 35.

And here, look, we still have a bit left of

two.

However, when the rest of the

division between two integers is different

From scratch, let's say it's a

division that is not exact.

Because? Now let's remember what

means that a number is divisible

on the other hand. Look, if we have two

integers A and B with b not equal to 0, if

We perform the division between a and b and the

The result of this is a whole number,

then we say that a is divisible by

b. It is very important that you have in

account that when a number is divisible

On the other hand, when performing the division we will

to always have to, the rest will be

zero. The rest is zero. How did it happen?

here? For example, in this case 135

is divisible by 5 because the result

It's a whole and we have here that the rest

is zero. However, 137 is not

divisible by 5, since we had a

remainder other than cer. It is very important

also keep in mind that if a

number A is divisible by another number B,

This means that B is a divisor of A.

In this case that we just saw,

We simply had to for the 135

5 was one of its divisors, but not

So with the 137, since it didn't work here.

an exact division.

Now let's look at some examples. The 27th is

divisible by 3. Let's see. If I perform

the division of 27/

3, this will give us exactly a

the whole number will be 9. Therefore,

It is indeed divisible by 3. This

This implies that 3 is a divisor of 27.

Now, 29 is divisible by 5. Let's see.

If I divide 29 by 5, how many times

Does 5 fit into 29? It fits five times and 5

* 5 is 25. Therefore, here we perform

After the subtraction, we are left with a remainder of four and

as we have a remainder of four which is

if not zero, this division is not

exact, therefore it is not divisible by

5.

Now, 0 is divisible by 2. If you

You divide 0 by 2, the result of this goes

to be zero. In this case too

You could write it like this. 0 divided by 2 is

simply 0. Therefore, since it gives you a

whole as a result,

It will indeed be divisible by 2.

Now, let's do it the other way around. 2 is

divisible by 0. If I divide 2 by 0,

Something very important happens here, and that is that

Remember that division by 0 is not

defined. Therefore, since it is not

defined, this is not going to be an integer and

We ruled it out.

Now we're going to review some of the

divisibility rules that will help you

to determine if a number is

divisible by any of these

quickly.

We've already started with the rule of

divisibility by two. If you have a

even number, this even number always,

It will always be divisible by two. By

For example, numbers like 52,

100, 26, or 28 are all

divisible by 2, since they are numbers

peers.

Now, what is the rule of

divisibility by 3? simple. If I

I have an integer and the sum of

all its digits are a multiple of

three, this number will be divisible by

3. For example, if I have the 729

And I add up its digits, I add this one, this one and

I'm going to write this up here. 7 + 2

+ 9. Notice that the result of this

The operation is 18. And since 18 is a multiple

3 will mean that 729 will be

divisible by 3. Another example, 243.

If I add its digits, 2 + 4 + 3, this

It will give us 9 as a result. And being

9 1 multiple of 3, 243

It will be divisible by 3.

Now let's look at the rule of

divisibility by 4. Look, if I have

whose last two digits form a

multiple of four or both are zeros,

I'm going to have to make that whole number

be divisible by four. For example,

If I had the number 264 here,

I see its last two digits. Ah, I see.

It is 64 and 64 is a multiple of four.

Another example, if I have 216,

I look at the last two digits and say,

"Ah, now its last two digits form

The number 16 and 16 is a multiple of 4.

Therefore, this value is divisible by

4." Now, another example, let's suppose the

100. Look, if its last two digits

They are zero, it will also be divisible by

4. And all these numbers fulfill the

condition for being divisible by 4.

Now let's look at the rule of

divisibility by 5, which is good

It makes it easier. Look, if you have a whole and

This integer ends in zero or cco,

It means that the number will be

divisible by 5.co. For example, on the 25th,

the 35th,

the 55

divisible by 5 and also, for example,

100 or also

1000, which is the score that will be

Saar, yes or yes. So, all these

numbers, as they meet the condition of

that end in 0 or cco, will be

divisible by 5. Now let's look at the rule

divisibility by 6. And this has to

fulfill the following. Our number is

Multiple of 2 and 3. Okay, look here

Let's take an example. The 30th. The 30th is

a number that is even, therefore, being

even multiple of 2 and is also a number

which is a multiple of 3. Notice that 30

It is the result of multiplying 3 * 10.

Therefore, being 30, a number that

is both a multiple of 2 and a multiple of

3 will be divisible by 6.

Now, the divisibility rule for 9.

It's simple here. The sum of its digits

is a multiple of nu. Let's look at a

example. Look, the same 729.

The 729, as we saw, if I add up all its

digits, gives 18 and 18 is a multiple of

wildebeest. Therefore, we know right away that this

value is divisible by 9.

And finally, let's look at the rule of

divisibility by 10, which tells me that if

the whole number ends in zero, the

number is divisible by 10. For example,

the 250,

the 500,

100, 100. All these numbers

numbers ending in zero are divisible by 10.

Before concluding this topic of the

divisibility and divisors, I want

that we fill in this next table

together. I want you to pause the video and

try to place all the dividers

positive aspects of these values. It's very

It's important to keep in mind that for

to find the divisors of a number, this

It requires practice, that is, that

Hopefully you have the session beforehand. Without

However, I'm going to do it here too,

But that requires that you have practiced.

Maybe a little bit so you have that

connection there. If you're not going to start building it

as you practice more. So,

Look, I'll go with number 12.

Already. Divisors of 12, divisors

positive. First, let's

place the one. Remember that every number

an integer is divisible by 1.

Subsequently

Number two works for us too, right?

Number three works too, number one works too.

Four, six also works. and also

The same 12 works. I also remember that

every integer other than zero is

divisible by itself.

Now let's move on to the next number, the

18. We are placing subdividers. He

First, the one. The second, the two, the

three. Then comes the six, then comes

the ninth and finally the 18th.

Now let's go to 30. It would be 1 2 3.

Then we have number five. Then we have the

Six, then we'll have 10. We're going to

We'll have the 15th and we'll have the same one here

30. And here we have the divisors

positive out of 30. And now I want you to

Let's solve a question that came up in a

proof. It says, "With respect to the

positive divisors of nu, is correct

to state that there are already positive divisors of

no, we have the one, we have the three and

we have the same nine. So what

What do they tell us about the alternatives? There are two of them,

No? Neither. There are four of them, too. And they are

three and their sum is 13.

Indeed, 1 + 9 10 + 3 is 13.

The letter D is correct. Now let's go

remember what prime numbers are.

Prime numbers are whole numbers

greater than one who only possess

two distinct positive divisors between

Yes, one and themselves. For example, the

two. The number two has the following divisors:

positive to one and to the same two. The three

to one and to the same three.

Five, one, and the same five. And so

successively.

Remember that by definition the number

One is not a prime number. Many times

that mistake is made. And here I want you to

keep this in mind. One is not a number

cousin.

Now let's talk about composite numbers.

Composite numbers are numbers

integers that have more than two divisors

positive.

In this case, for example, the four. He

Four has one as a divisor, it has

as a divisor of two and also has as

divisor to the same four. The number six has

as a divisor of one, two, three and

to the same six. And the number eight, for example,

has as divisors one, two, four and

to the same 8. And if you notice, each one

of these values ​​they have more than two

divisors.

Here I've left you a small table

indicating the numbers, both prime

as compounds, from 1 to 100, where the

The numbers in red are prime numbers.

and the green ones are the composite numbers.

It's also important that you have in

He says that one is neither a cousin nor

compound. Look, one good thing

important, suddenly in questions of

probability they ask you, "Hey, if you

You have a number, I don't know, from 1 to

20, what is the probability that it

"Choose a prime number?" And I'll tell you

I recommend that you always learn the

prime numbers from one up to

Except for number 37, at least it's there.

because they might suddenly appear

questions from those contexts and there you don't

You're going to make a mistake.

Now let's look at the fundamental theorem

from arithmetic, which tells me that everything

an integer greater than one can

to write in a single way like the

product of prime numbers. Let's see now

how to do this process. I'm going to

show two paths. Use the one that best

I'll make it comfortable for you. Look, let's suppose I have

number 72.

I'm going to write number 72 here.

Let's make a little table.

and I'm going to divide it by numbers

cousins ​​to the extent possible. By

For example, I have two, I have three,

I have five, I have seven, and so on

successively.

So, look, let's see. Let's begin

with the two. Is 72 divisible by 2? Yes.

since it is an even number. Therefore,

We divide by 2 and here we will get

36. And I ask myself again, can I

Should we continue dividing by 2? And the answer

Yes, because 36 is an even number.

36/2 is 18. And I'll do it again.

Same question, can I divide 18

in 2? And the answer is yes. By

Therefore, 18/2 is 9. And I'll do it again.

Same question. Is 9 divisible by 2? AND

The answer here is no. Therefore

Therefore, we move on to the next one, the

three. 9 is divisible by 3 and

The answer is yes. 9 div 3 we have left

3. And then I do the same thing again.

ask. 3 is divisible by 3 and the

The answer is yes. We divide by 3 and

We have one left here and we've reached this point.

Therefore, if I multiply all of these

values, 2 * 2 * 2 * 3 * 3, I'm going to

Write here, 2 * 2 * 2 * 3 * 3, this goes

to be exactly equal to 72. Look, here

Getting ahead of myself with the subject, this 2 * 2

* 2 is the same as 2 raised to the power of 3. This

3 * 3 is 3 raised to the power of 2.

And 2, sorry, and 3 raised to the power of 2 is 9 and 8 * 9

es = 72. And we arrive at exactly that

worth. You could also multiply

One by one, there's no problem. Already

We will look at the powers later.

So, here we're going to see the other one

manner. I told you I was going to share with you

two ways, you can do it one

whatever way suits you best. Obviously it goes

It depends on how much you've practiced.

Look, here's another way we can go about it

This is rewriting this 72 of the

as follows. Look, 72 is the

same as a 2 * 36.

Similarly, 36 is the same as

2 *

18. Similarly, 18 is the

same as a 2 * 9. And in the same way

9 is the same as 3 * 3, right?

And then we rewrote it. Obviously not you

I recommend leaving the

multiplications like this, since it is very

It is possible that at the time of writing it

I can pass you a two and you can

to be wrong. Therefore, I recommend you go

leaving it as powers. In this case,

2 * 2 * 2 is 2 raised to the power of 3 and 3 * 3 is

a 3 raised to the power of 2. Don't worry, this

We'll see about that later, so...

Let me tell you, a small spoiler, but here

You already know two different ways to

do it. And in fact, now I want you to

Pause the video and tell me how

120 would remain.

Pause the video, do it yourself, and now...

I develop myself. I'm going to do it now.

using this form we have here.

If you want to do it the other way

No problem. So, 120,

Let's put it here,

120.

and we're going to divide it up. I'm going to...

delete this part

And we're going to use these prime numbers

that we were using before.

Already. So, I'm wondering, is he...

divisible by 2? Yes, because it's a number.

pair. So, this is 60. Then, the 60

is divisible by 2. Yes, therefore, this

It's 30.

30 is divisible by 2. Yes.

because it's an even number and you're left with 15.

Then I can continue dividing by two,

No? Let's move on to the next one. Already. The 15th

It is divisible by 3. Yes, indeed. So

You're left with 5 here, and then the 5 is divisible

by 3 no. Then we moved on to the

following. 5 divisible by 5 and the

The answer is yes. Therefore, 5 say 5 tell

There's 1 left and here are your numbers ready

cousins. You multiply them together, 2 * 2

* 2, which is the same as 2 raised to the power of 3

* 3 * 5. And there we have it ready. AND

We have reached the last part of the subject

of integers, which is the least common,

multiple and greatest common divisor.

Let's start with the first one. The minimum

common multiple of two or more numbers

integers is the smallest positive integer that

is a multiple of those numbers. It is the

smallest multiple that two or

more numbers. Let's see how

An immediate example. The minimum com a

What is a multiple of 6 and 8? Look,

One way you could do this is

start placing the positive multiples that

We have dates for the sixth and the eighth, and we'll see when

We have one in common. For example, already

multiples of six, positive multiples,

the 6th, the 12th, the 18th, the 24th,

30, 36, we have 42, 48 and so on

successively.

Now, positive multiples of 8, the 8,

the 16th,

24, 32, 40, 48 and so on

successively. And here I want you to give yourself

It tells you something, and that is that there are multiples.

what they have in common. Do you realize? But

We're going to look for the minimum, the

smaller. In this case, the multiple

The smallest thing we have in common is the

24.

Therefore, the least common multiple between

These values ​​are 24.

Now, there will also be infinite

multiples that we have in common. By

For example, here we have 48 and there will be

further. However, at the lowest common denominator

multiple one directly searches for the most

little.

Now,

Is this the only way to do it? No,

Obviously, if they have numbers that are more...

larger, more complicated, it can be very

This process is tedious. For some

This process works perfectly for these exercises.

but not for everyone. I would tell you that

The more complex it becomes, the better

Use this second method that I'm going to show you.

show now. Look, what are we going to do?

We're going to place the number six and the

Number eight. And again we're going to call

our prime numbers, two, the

three, five, seven, and so on

successively.

And here's what we'll do:

Look, this is the method. I

I'll ask, are any of these numbers

divisible by 2? If it's divisible, let's go

to divide it. In this case, both are

divisible by two. Therefore, we are going to

Divide 6 into 2 and 8 into two. Here you go

three and here you have four. And now you

You ask the same question again.

Are any of these divisible by two? AND

In this case, four yes, but three

No. But since some are divisible,

We're going to divide. So, what do we do?

Look, we left number three untouched because

It is not divisible by 2, but four is.

We can divide it by 2 and you get two.

TRUE? We divide again by two and

We arrived at these results.

Now I ask myself again, did any of them

Are these numbers divisible by 2? And the

The answer is yes. Therefore, I can

divide by 2 again.

And we've come this far. And the three

We left it intact because it is not divisible.

by 2. Then now that I know that no

I can continue dividing by two, I move on to

three. I can divide three into three and the

The answer is yes. Well, 3 divided by 3

You have 1 left and we're done here. And if

You multiply all these numbers by

Yes, you reached the least common multiple.

Look, 2 * 2 * 2 * 3. 2 * 2 * 2 is 8 * 3

It's 24, which is exactly the number

that we had arrived. Now let's look at another one

example. Here we have the lowest common denominator

A multiple between 2 and 45. I want you to pause.

Watch the video and calculate it yourself.

Yes, I'm going to start putting the

small board. I'll put it here

the 12th

and I put the 45 here.

Therefore, I place my numbers

cousins ​​who are going to help us 2 3 5 7 and

and so on.

Already? Is any of these divisible by

2? The answer is yes. The 12th

There are 6 left and we'll leave this one exactly.

equal.

Now, are any of these divisible by

2? Yes, 6. We divide 6 into 2.

3 remains and 45 remains intact. Now then

We cannot continue dividing by two.

None of these numbers are divisible

times two. Therefore, for the house. We spent

with the three. Are any of these numbers

divisible by three? Yes, well,

Exactly both. If I divide by

three, here's one left for you. We got as far as

Here and here, if we divide by three, we'll get

15 remains. And we can continue dividing

In this case, by 3. 15 is divisible

times 3. Then say 3 is 5. And

We continue.

Already. Look, is 5 divisible by 3? No,

So we'll move on to the next one. The 5 is

divisible by 5, right? So then 5

Divided by 5 is 1. And we're ready. Yeah

You multiply all these values

Now, in this case 2 * 2 * 3 * 3 * 5, 2

* 2 4 3 * 3 9 * 5 4 * 9

* 5 gives us a result of 180

And there it is, ready. If you realize

We could also use the method

I was writing earlier, but here

The numbers were going to increase over time and

larger and you could put a little bit

more complex. And now suppose that

we want to calculate the least common

multiple of three numbers. That

What do we do in this case? I can't wait for

Pause the video and do it yourself because it's

exactly the same logic.

So let's make the table here.

We placed our numbers 24, 32 and 48 and

We placed our friends the numbers

cousins ​​who are going to help us solve

This will be quick.

Already? So,

Are any of these numbers divisible?

x 2? And the answer is yes.

So, 24/2 is 12. 32/2 is 16. 48/

in 2= 24. And we're doing well. Then

is one of these numbers divisible by

two. All. In fact, we divided by 2,

You have 6 left, you have 8 left, and here you have

12. I can continue dividing by two. Yeah.

So here we divide by two, you get

three, you have four left and you have six left. AND

Here, not everything is divisible by 2.

but there are still divisible numbers

by 2. So we divide by two. This

It remains intact. 4 / 2 is 2. 6 / 2 is 3.

Now, here

Let's ask ourselves, can we continue

Dividing by two? Yes, there's one more.

which can be divided by two. In this

In that case, we're going for two. This remains

intact. This gives you 2 divided by 2 1.

We've come this far. 3 divide is not possible,

Therefore, we left it untouched.

Already. So, now we move on to the

Next, and with this one, it's finished.

Both are divisible by three. So

we divide by three. You have one left,

We're finished. You have one left, we're done.

We're ready. We multiply all of this.

That is, I have 1 2 3 4 5 2 2 * 2 * 2 * 2

* 2 * 3. Okay, I already recommend

Leave it as a powerhouse, but that

We'll see later, don't worry.

The same with 2 raised to the power of 5 * 3.

And we multiply this by 3 and 32 * 3

It's 96. Therefore, this is the minimum here.

common multiple.

Now we're going to apply the least common factor

multiple to a question that could

to go out in the test. Look, how

Identify a least common question

multiple in context? The most typical case

It is the following. They tell you there's a certain

an event that occurs periodically and

They tell you that another event is happening

every so often and they ask you when

They will meet again.

In that case you're going to encounter the

a sign that that question is a

least common multiple question as

the one we have here. Let's read it. The

The World Cup is held every 4 years,

while Bad Bunny releases a new

album every 3 years. If both events

occurred simultaneously

In 2022, in what year will they return to

coincide? Yami, do you agree?

with me that the World Cup

Since it occurs every 4 years, it will happen in 4.

years, in 8 years, in 12 years, in 16 years,

in 20 years and so on.

Similarly, Bad's album

Bunny, you know it happens every so often

time, in this case every 3 years. Is

In other words, there will be an album in 3 more years, in

six also, in nine, in 12 and

Let's set it to 15 and so on.

So, what's important here? yes

You realize, here I have only multiples

of four and here I have only multiples of

three. Therefore, since I want to see in what

They're going to meet again this year, I'm going to

find the smallest multiple they have in

common. In this case

It's the 12th. Therefore, they will return in 12 years.

to coincide. And if the events occurred

simultaneously in 2022, I know that if

I add 12 years, we will be in 2034.

where the events will coincide again.

Please note that this question is

it reduced to finding the lowest common denominator

multiple between 4 and 3. Therefore, it

You could do it this way or also

using the table we learned

previously. Both, both ways you

They led to exactly the same result.

Now let's look at the greatest common factor

divider. The greatest common divisor of two

or more integers is the largest integer

positive that exactly divides those

numbers. In other words, it is the largest number

which is a common divisor of those numbers.

Let's look at some examples. Look, the

Find the greatest common divisor of 24 and 36.

I'm going to show a method of how to calculate

This quickly, which is what

we have learned. Look, I'm going to put one

small table with the numbers 24 and 36

and we're going to divide by numbers

cousins ​​following the following way that

I'm going to show you next.

Okay, look, what's the key? Here you

You're going to ask if the numbers I have

I can divide them here. simultaneously

in one of the prime numbers that

We have it here. If that is not possible,

We'll leave it at that. For example,

Can I divide simultaneously?

24 and 36 * 2? And the answer is that

Yeah. We'll have 12 left here, and here we go

to leave 18. Now I ask you,

Can I split again simultaneously?

times two? And the answer is yes.

we can divide by 2 again

simultaneously.

So, 12/2 is 6 and 18/2 is 9. And

Now I wonder, can I go back to

divide simultaneously by 2? And the

The answer is no. Therefore, the two already

We can't use it. Here's the difference from him

minimum with a multiple when we did

the little board. Here we need to verify

that simultaneously, that is, at the same time,

the values ​​can be divided

We have it there. Next up is number three.

We can divide simultaneously by three

And the answer is yes. We divide by

three, you have two left and you have three left. AND

I want you to pay attention to the following. Here

we can no longer divide simultaneously

by three, nor by any cousin

elderly. Therefore, we'll leave it here.

Nothing more than that. Therefore, that's enough

by multiplying these values ​​together

to find the greatest common divisor. 2 *

2 * 3 2 * 2 4 * 3 is = 12. That is the

greatest common divisor of these numbers.

Now I want you to pause the video and

calculate the greatest common divisor between

36 and 54.

Do you have it?

I'm making the table here.

So let's put ours here

prime numbers. 2 3 5 7 and so on

successively. And we'll see if

we can divide simultaneously.

We can now divide 36 simultaneously

and 54 * 2 and the answer is yes.

Here you'll have 18 and here 27.

Now, we can continue dividing

simultaneously by two, right? Since

27 is an odd number. So,

Let's move on to the next one. Can

divide simultaneously by 3 and the

The answer is yes. 18/3 is 6. 27/3

It's 9. And we continue. We can continue

dividing simultaneously by three. The

The answer is yes. Here it will stay

two and here you'll have three. So,

Since we divided by three, here's what you got

two and here you have three. Therefore,

because we can no longer continue to divide

simultaneously by three, as you know

account, nor by any older cousin. It

We'll stop here and multiply these

values ​​among themselves. 2 * 3 * 3 2 * 3 6

* 3 is 18. And we have it ready. Now

I want us to do one last example and

I want you to pause the video and...

You solve it. What is the greatest common factor?

A divisor between these values? I'm going to

I'll do it here. 30 60 75.

Let's place our prime numbers.

We're going to place the two, the three, the five,

7 and so on. And we're going to go

seeing if we can divide

simultaneously.

Here we can divide simultaneously by

two, and the answer is no, since

Here we have a number that is odd, therefore

Therefore, it is ruled out. Fists divide

simultaneously by three and the answer

Yes, that's right. It's going to be 10 here, it's going to be

It will be 20 and here it will be 25.

We can continue dividing

simultaneously by three and the answer

No, that's not it. Therefore, we're going to go with the

five. You mean all these numbers

are divisible by five, end in 00 and

5. So we're doing well. We divide by 5,

You have 2 left. We divide by 5, you get 4.

We divide by 5, you get 5. And if you

fixed here, we can no longer divide

simultaneously by 5 nor by any

older cousin. Therefore, we'll leave it until

there. We multiply 3 * 5 and 3 * 5 = 15,

this being the greatest common divisor.

Finally, we answer a typical question.

PES of how the maximum might appear

common divisor in context. It says: a

The school has 3/4 averages. Fourth A

It has 36 students, B has 45 and C has 54.

For an event, the following will be grouped together:

students in equal teams

amount. in each course to compete

with the others and thus win a prize.

What is the maximum amount of

students who can have a team?

To answer this question, the key is

finds in the quantity of

students in each team must be a

divisor of these quantities of students

totals we have in each course. Look,

an example. Suppose I say that

I want each team to have four

students.

So, if I divide 36 by 4 here,

we reached the point where I can form nine

teams and no student is left out.

But, for example, if I don't place a

If a divisor of 36 is a factor, it will happen that, because

For example, I mean, I already want them to have

five students per team. And if you

You realize, the 36 di 5 is not a

exact division and in fact a student

will be left out.

So here I need to find a divisor of

36, but also as the amount should

To be the same in each course, I must look

a divisor of 45 and also of 54 and that to

the time is exactly the same divisor

that we're going to have for each of these

values. Therefore, we need to look for a

common divisor so that no one is left out. AND

Similarly, here they ask me to be

the greatest common divisor we have.

Therefore, they are asking me for the maximum.

common divisor. Therefore, here I am going to

I'm going to put the 36, I'm going to put the 45 and I'm going

to place the 54.

We're going to apply

the method we've seen all this

a while. So,

already

These numbers are simultaneously

divisible by two. The answer is that

No. Therefore, we're going to move on to the

following. I can go simultaneously through

three and the answer is yes. In this

In this case, 36/3 leaves us with 12. 45/3

There are 5 and 54/3 left, we have 18 left. I can

How can we continue dividing simultaneously by 3?

The answer is yes. Here we're going to

If there are four left, we'll have five here.

And here we'll have six left. And if you

fixed here, I can't go anymore

simultaneously by three, nor by

no older cousin. Therefore, this comes

up to this point. Therefore, we multiply 3

* 3 and this gives us 9, which is the maximum

common divisor of these values ​​and that goes

to be the maximum possible amount that

I can have students in each

equipment.

Before continuing with the summary,

I want to congratulate you on completing

the subject matter of the first topic in the summary,

which was whole numbers, which was the

longer part of the summary. And from

I'm really happy that there is

reached this part of the video. Speaks

much of the discipline you are

having and of course everything that

you can achieve. Therefore, I want

that you continue down that same path, that you don't

Don't give up, keep showing what you're capable of.

You'll be able to when it's your turn to practice with

the exercises. See step by step how it is

Solve each one and practice with them.

exercises so you can reach the

next level. I am completely

I'm sure that if you continue on this path you'll have

tremendous results, that score is going to

go up and you'll be in that race that

You deserve to be there.

Before we continue, I want to leave you with the

Invitation to join the ultra-intensive

M30M, the largest intensive care unit that has ever been

Made in Chile to prepare for a test.

We're going to take care of transforming your

score in 14 days, from the 17th to the 30th

November. And this is going to be a festival,

the biggest festival ever held

in Chile to prepare for a test that will

to be the PES. And what will the

grill? Look, here's the stage

national maximum. What is a grill?

general? If you want to have one

balanced preparation and attendance

Obviously, to the classes you want.

where you will be able to attend M1, to

reading comprehension, obviously

science, history and of course M2,

Where is the trip going to be? Where are we going?

to be in charge of learning the content and

exercise and above all enjoy it

M30M community to make this trip

as entertaining as possible. This is a

main general grill. Obviously

Then we'll give the details of what's going to happen

to be in each block. However,

Let's suppose that you directly...

you want to focus on science, on

mathematics or in any test of

humanities area. Of course they're going to

There will be stages that will run in parallel,

literally while these are being made

Classes will also be held in parallel to

You are where, if you want, you can

Focus 100% on science during the stage

scientist, studying physics, biology,

chemistry and of course also the stage

mathematical, where we are going to solve almost

all the questions that have come up in the

test or very similar to those that will

to go out in the test. And of course

also the humanist stage where you go

to be able to focus on reading comprehension and

in history. Then you'll have for

to enjoy being on stage

whatever you want, and we'll live one

a unique experience to take that

score to the next level.

The price of the ultra-intensive treatment is

For just 30,000 pesos, you'll have

access to the preparation of all the

tests at this great festival we're going

to do, where we are going to take care of

with a clear routine to be able to carry that out

score to the next level. The

Registration opens on the 3rd

November, so if you're watching

The video before, you have to wait until that

Date to register. or if you are

watching the video after the 3rd

In November, you can now register for

Ultra Intensive 30M and we're off to

to be in charge of bringing that score to

next level. You can join by

Click on this QR code. In fact, it's going to

There will be a waiting list if you

Register before November 3rd to

Secure your spot and we'll be there too.

to maybe have one as a little gift. So

Also, if this doesn't work for you

QR code, you can go here to the

description and that's where it obviously always goes

to have the link. We'll take care of it.

that those two weeks, those two

the last few weeks, which are the most important.

impact they can have on your score

really worth it. Make your score

rise like foam and of course we're going

to work so that you stay in that

career of your dreams.

Now we're going to learn how to solve it step by step.

I'll pass on the questions from this content that

They might come out in peace. To do this,

Go down to the description and you'll find them.

to find all the questions and then

return to this part of the video so that

Let's keep learning.

And now let's talk about rational numbers.

First, let's remember what

is a fraction.

A fraction is the expression of

an amount divided into equal parts.

It is represented by two numbers

integers separated by a slash

fraction.

We call the number above the

numerator

and we call the little number below the

denominator with the latter being different from

since it recalls that division by

0 is undefined in numbers

real.

With this in mind, we will now

see how to represent fractions, which is

something that has been asked in the

proof. To do this, we will begin

remembering proper fractions and how

represent them.

A proper fraction is a fraction

where the absolute value of the numerator

is less than the absolute value of

denominator. In this case we have the

fraction 1/

4. Yes. How do we do the representation?

To do this we will work with

little circles.

So, what will we do? In

First, you're going to take your little circle and

You're going to see the denominator. The denominator

It's four. Therefore, our little circle

We're going to divide it into four parts

equal.

After that, let's look at the numerator,

which in this case is one. And we're going to say

as follows. Ah, I see. Perfect. Of the

four equal parts, let's take

a. Therefore, we're going to...

paint exactly as it appears in this

We left the drawing and the rest untouched. Of

That's how we've represented the fraction

1/4.

Now let's look at another example. Come on

Let's see how to represent the fraction 2/3.

The fraction 2/3 to represent it

We follow exactly the same logic. In

First we look at the denominator.

In this case, our denominator is

three. Therefore, our little circle

We're going to divide it into three equal parts.

Next we look at the numerator.

The numerator is two. Therefore,

From our three equal parts, we will

Take two, and those are the ones we're going to

paint. We left the other one untouched, that's all

which can be seen in the figure here. Of this

We have represented the fraction in this way.

23.

We continue. See what happens with one

fraction like this. In this case 4/4 or

4/4.

Look, as always you're going to start by watching

the denominator.

Denominator four. We're going to take

our little circle and we're going to divide it

into four equal parts. But here

Something interesting happens, and that is that

We're going to take all four parts. By

Therefore, in this case we are going to have to

paint each of our four

parts to represent this fraction.

In this way we have learned how to

represents this type of fraction

when the numerator and denominator are

equal.

Now we're going to see how to represent a

improper fraction. We called a

improper fraction when the value

absolute value of the numerator is greater than the value

absolute of the denominator. How can we

How do you represent these fractions? We continue

the same logic. Look, we see the

denominator. In this case, the

The denominator is four. Therefore, we took

our little circle and we divide it into

four equal parts, right? By

For example, this one we have here is

divided into four equal parts,

TRUE?

But we have a small problem, and it is

that the little circle split into four

equal parts, but let's take five

parts. Therefore, the key here is

find it in drawing another little circle

until we can fulfill the agreements

that are needed. So we draw here

The first one, the first little circle tells me,

"Okay, here's a part of it, here you go

"Two, three, and here there are four." It's not enough.

Another small circle is drawn, and then it's done again.

divide into four equal parts and let's go

to take one of those parts. And from this

In this way we have 1, 2, 3, 4 and the fifth one that

We were missing it. And in that way

We represent this type of fraction.

Let's look at another example. Let's suppose now

we want to represent

this fraction 7/4. Once again we have

to take the little circle. We divided it into

four equal parts, but we want

Take seven. Therefore, we are going to have

than drawing more than one little circle. And let's go

counting. Here I have one, two, three, four

parts, but we are missing three parts to

arrive at seven. Therefore, we draw

another little circle, we divide it into four

equal parts and we paint the ones that

are missing. One, two, three. And in this way

We have represented the fraction 7/4.

Like I said, this is something that's coming.

asking in the test and if you handle it

You're going to have another good question in

the pairs.

Now let's see how to calculate the value

numerical value of a fraction. And here

You simply need to perform the

division. In this summary, I will not see

how exactly it is divided, however,

I'll leave you with the results of these

fractions obtained by performing

the corresponding divisions. And how

Are they carried out? Simple. Divide the

numerator in the denominator. In this

case 3/4. If you perform the division of 3

Dividing by 4 gives you the result

0.75.

Now, if you divide 12 by 5, you get...

result 2.4. If you divide 1 into 5, you get

gives 0.2.

And if you divide 1 by three, the result that

It's going to give you 0.33

3

and so on indefinitely. And when this

This happens, as we will see later,

We will say that this number is a decimal.

newspaper. We'll see this in a couple of

a few more minutes.

We will now study the equivalence

between fractions. We say that two or more

fractions are equivalent if

They represent the same amount, although

let their numerators and denominators be

different. For example, these fractions

that we have here, which are 3/ 6/8 and 75/

100, are all equivalent fractions,

because if you perform each of

These divisions, you'll reach the same

result. In this case, if you perform

These divisions, any of these,

The result will be 0.75.

Now, you'll often find yourself

with negative fractions and often

They can also be expressed in different ways.

ways, but I want you to see that this

that you're going to have next

They represent exactly the same number.

For example, if I have -3 here and

divided by 4, is the same result as

have 3 and divide it by -4. Is

exactly the same. And that's exactly it.

The same goes for having the least, so...

to say it, outside the fraction, to carry out

The division of 3/id into 4 and this will be

I simply keep the minus 3/4.

0.75

and each of these operations leads you

exactly the same result. Because

Is this very important to remember?

Because depending on the developments,

often one writes it as one writes

one way or another. However, all

They represent exactly the same value.

Now we are going to study simplification

of fractions, which consists of finding

an equivalent fraction by dividing

both the numerator and the denominator

by some common divisor.

We can find two types of

fractions. Those that are reducible, are

In other words, we can simplify them, and the

that are irreducible, that is, not the

We can continue simplifying. Let's go

See some examples here. To do the

simplification process, we have to

ask ourselves the following. Let's go with the

first part. Is there a divisor that

have in common both the numerator and

the denominator? find a divisor

different from one, because if it doesn't suit you

to remain exactly the same. So,

Is there a divisor we have in

common? For example, the 16th and the 64th

These are numbers that are divisible by 4.

So, what are we going to do? Come on

to divide both the numerator and the

denominator times four. Always for

To simplify, you divide the top and bottom.

So, 16/4 is 4. 64/4 16. And from

We're doing great! Now I'm going back to

ask, is there any divisor that has

What do they have in common, both the 4?

In this case, I was able to simplify.

dividing by 4 again. Both are

divisible by 4. So 4/4 is 1.

All of this divided into 16/4, which is 4. And

Here we reach 1/4. And I return to

ask the same question. Is there a divisor?

that we have in common? If you look closely, the

The only common divisor will be one. By

Therefore, we can no longer continue here.

simplifying and we have arrived at a

a fraction that is irreducible, cannot be

You can continue simplifying.

Now, in an ideal world,

What can we do? Instead of walking

dividing each time as

We find divisors, ideally,

Obviously this requires practice,

It's about you finding the highest common denominator.

divisor to make that process a single

time. In this case, the greatest common

The divisor between 16 and 64 is 16.

Therefore, I could directly do this here.

divide by the greatest common divisor, which

It's 16 here too. and 16/16 is a is 1

and 64/16 is 4 and we arrive exactly at

same result which is 1/4. Very

It's important to keep in mind that no

It is mandatory that you seek the maximum

common divisor. You can start doing it.

looking for divisors as it goes

simplifying. However, it is the way

faster to reach the result.

Obviously, do what works best for you.

comfortable. Look, let's see another example. In

This case, number 120, was born in 15 days. Yeah, look.

One might look at this and say, "Ah, I see,

Look, you know that both are divisible

by 5, then I divide by 5 both the

numerator as the denominator. So,

120/5, what is it? It's the 24th.

15/5 is 3. Now I could also continue

simplifying here the 24 with the 3

dividing by 3, which is the divisor that

what they have in common, or just focus on

that 24/3 is equal to 8. You arrive exactly

the same result by performing the

division or simplifying.

Now you could do it directly

performing the division here, because if you

You state that 120 is divisible by 15 and if

You performed the division here, and it left you with

eight and you arrived at exactly the same

result. We're trying to see

different types of fractions that you

could appear. Many times

Performing the division you will arrive at a

whole number without any problem. No

always, but it can happen.

Now we have the fraction 20/ in 3. In

In this case, the only positive divisor that

What we have in common is one. Therefore,

Here we already have a fraction which is

irreducible, it cannot be followed

simplifying.

We were studying the

whole numbers and now the moment has arrived

to know rational numbers.

Rational numbers are all

those numbers that we can

represent as a fraction, that is,

a division between two integers with the

denominator other than zero. Now, one

very important thing, previously

We study integers, right? But

You have to take into account that all the

Integers are rational numbers.

Because? Because every whole number is

can be represented as a fraction. Let's see

Here are some examples. The 2 is exactly

The same as 2 born in 1. I have the here

division between integers, therefore, it

can be represented as a fraction.

Another example, the cer. Zero is what

same as 0 divided, for example, by 5.

0 divided by 5 0. Therefore, here we have

representing zero as a fraction.

Negative numbers too, because

For example, -5. The -5 can be

rewrite with a -5 divided by 1. And

Here we have represented it as a

fraction. Therefore, all integers

They are rational.

So, what's happening here? And there are

rational numbers that are not integers.

For example, we saw earlier that the

fraction 3/4, if you perform the division

The corresponding result gives you the

number 0.75.

And 0.75

It is a rational number. Because? because

There is a fraction that represents it and of

In that way, we have found a

rational number that is not an integer. Others

rational numbers, for example, 0.1

which is the same as 1/gone in 10. eh

periodic issues that we will now

study. For example, the 0.3 periodic,

which is the result of

perform the division between 1 and 3 or

semi-periodic numbers, which also

Let's study how to start from 0

1, 2, and with the 2 newspaper. All these

are rational numbers, since as

We'll see more about these later.

we can represent it as a fraction. Don't

Don't worry, we're going to learn to

represent these values ​​as a fraction,

But that's the definition of numbers.

rationals and that is the set that

We're going to start studying from now on.

Before we continue, I want you to

Let's remember the structure and the

fundamental about numbers

decimals.

Remember that decimal numbers

They have a whole part and a part

decimal, which are separated by

a comma. One might encounter

decimals that are finite, decimals

periodic, semi-periodic decimals and

decimals that have a quantity

infinitely many decimal numbers, but that

They do not follow a specific pattern. Let's go

study each one of them. For example,

The first one we see here is a decimal

finite, it has a certain amount here of

decimals. We have 1.25,

but you can also observe here a

decimal repeating decimal. Why is this

newspaper? Because the entire decimal part

It has a bar at the top. And

Does that mean? It means that this 25th will be

It will repeat endlessly, following

that pattern. In this case, this would come

being 1,

25

and so on.

But decimal numbers also exist

semi-periodicals,

That is, decimals where we have this

bar, but it's not found everywhere

decimal part. In this case only

He's with number five. What's this all about?

means? This is going to be number 1, 2

and here put 5 5

and so on indefinitely.

There are also other decimal places that

we will study later, what they are

irrational numbers. But as I say,

That comes later, those are numbers.

decimals that have a quantity

infinite decimal places, but not

They follow a specific pattern. These the

We will study this further later. Now we

We're going to focus only on these three.

first.

Now, remember what it is anyway.

the structure of a decimal number. In

the whole part, as always, we're going to

to have unit, ten, hundred, etc.

And in the decimal part, that is, what

It comes after the comma, we're going to have

a tenth, then comes the hundredth, the

thousandth, the 10th thousandth, and so on

successively.

Now, a very, very important consideration.

important.

Here you will often find

fractions that are 2.5, 2.50,

2,500,

2,500 and so on.

It's important that you keep this in mind

following. One usually writes

down to the last digit, which is different

from scratch. For example, here we write the

2.5. However, this is equivalent,

It's the same value at 2.50, at 2.500,

to 2,500. It could even be the same thing

than 2,500

and so on, and they represent

exactly the same value. But

obviously so I don't have to write all this

one only leaves until the last

a number that is not zero, comes

being the five. Obviously the last one

number from left to right until

you get this far and that way you don't

we are writing all these zeros.

In the PAES we will often have

to perform the comparison between

numbers. This occurs mainly among

rational numbers, but also you

It can come out in whole numbers, in numbers

real, etc.

So, what are we going to do?

We're going to have to remember the symbols.

of inequalities, which is going to

to indicate that a certain number is less or

greater than another, or less than or equal to,

etc. And this can often

to be very confusing. Look, first

I'm going to show you what that means.

each of these symbols here and

Later I'll give you a great trick that

will ensure you never make a mistake in

interpret it. Look, if I have two

numbers, a number A and a number B, if

This appears here, it means that A

It's a number less than B. If it appears

This here means that A is greater

that b. Often it will also

A bar will appear down here. And

Does this imply? Look, if this appears

Here, it means that A is less than or equal to

B. On the other hand, if this appears,

This means that A is greater than or equal to B.

And as I told you, this issue here

It can be quite confusing, but there is a

A pretty practical trick. Look, already because

For example, suppose we have this here

And you're having trouble figuring out which one it is.

which is greater, which is smaller, and I confused you, no

I worried you. What you're going to do here is

draw a Pacman. You can put this on it now.

Just keep an eye on it, it's still there. And Pacman always

He's going to point to the one who is older. In this

The case is pointing to B. Therefore,

We know that A is less than B because the

Pacman is aiming for B, then. And it

Right here, look, if I draw Pacman,

Let's draw Pacman here. Look

Here's where Pacman aims for the biggest one, because

then A is greater than B. And nothing.

more than that. Now, if the

Does the bar down here change anything?

Our method? No, look, the only thing that

What we're going to do is take Pacman,

For example, here, and Pacman is...

pointing to A. And what we're going to say

It means that A is greater than or equal to B. And nothing else.

more than that.

Now we are going to study the comparison of

decimal numbers. Here we're going to learn

to order them from smallest to largest.

We will look to compare decimals that are

of values ​​very close to each other. and

We're going to look at different strategies for

compare them. Look, the first case and more

It's simple: you'll find two.

decimals that are positive and have the

same number of decimal places. In

In this case I have two decimal places and

here too. Now, what will the

clue? Simple. You're going to compare which one

has a greater absolute value than another. A

A good strategy to avoid confusion is

Write the complete numbers. I have the

3.23

3.27

aligning the commas together so that we can

to properly establish the comparison between

figures. So, look, listen, look, the

part of the unit is the same, no

We can make a comparison there. The

part of the tenth is also the same,

so we cannot establish the

comparison there. However, the part

of the hundredth

Here we do have a difference, therefore

We establish the comparison. Here you will

You realize, 3 is less than 7, because

Therefore, 2, sorry, 3.23

is a number that is less than 3.27