Hello everyone, welcome to the first lecture series of
Aptitude.
Today we will do Quantitative Aptitude.
The first topic of which will be our number system.
Which is considered to be the foremost important and
easiest topic in Aptitude.
If we don't know the numbers, then how will we do
Mathematics? You can understand that numbers are the
foundation of Mathematics.
Or the foundation of Mathematics.
So numbers are the foundation of Mathematics.
Without numbers, we will not be able to start Mathematics.
Similarly, when we are knowing numbers, we should also know
their category.
Which number belongs to which category?
So today we will learn from this lecture, what are the
categories of numbers, what are the categories of decimals,
what are the categories of real numbers, complex numbers.
We will learn all these today in this lecture.
So let's start.
The first numbers that you will think of, the first thing
that will come to any child's mind is, the counting
numbers. 1, 2, 3 and so on.
So the counting numbers from which the child starts
counting, or when we start counting from 1, 2, 3, all the
numbers that come after that, which category will they come
in?
They will be called our natural numbers.
Because we are coming naturally, and counting numbers are
also called natural numbers. Then if we add one more number
to this, which is the day of India, of Aryabhat Saheb Ji,
that is 0. If we add 0 to these natural numbers, then this
whole set will be
called the whole number set.
So let's write.
What did I tell you about natural numbers? 1, 2, 3 and so
on. All the numbers that come after 1 crore, 2 crore, all
the numbers that come after that, what will they become?
Natural numbers.
Look at how they are proceeding.
In 1, if we are going one step ahead, then 2 is coming.
In 2, if we are going one step ahead, then 3 is coming.
So if we proceed one step ahead in numbers, from 1, then
they will be called our natural numbers.
Similarly, what will be the whole number?
Instead of starting from 1, where will we start?
We will start our steps from 0.
From 0.
So we will start from 0.
Then we will go one step ahead.
What will come next? 1.
Then we will go one step ahead.
2.
Then 3.
And so on.
So what is this category?
It is the category of whole numbers.
So natural numbers will go from 1 to 0.
And where will the whole numbers come from?
From the whole numbers, there is one more addition.
That is 0.
So if we add 0, it will become the whole number.
Now we will proceed.
The whole numbers and natural numbers, whatever you call
them, they also have sub-categories.
You must have heard of odd-even rule.
Odd-even number.
Where did it come from?
It is from mathematics.
What is an odd number?
What is an even number?
Then you must have heard of prime number.
So we will study all these in their sub-category.
In the sub-category of natural or whole numbers.
So first of all, you will study odd number, then even
number.
The numbers we started with, where did we start the natural
numbers from?
From 1, 2, 3.
We started with 1.
So if we take 1, then how will it become?
Odd number.
Then after this, instead of taking one step, we have to go
two steps ahead.
If we take one step ahead, we will get 2.
But if we take two steps, we will get 3.
So what will happen to 3?
Odd number.
So after leaving two steps, from 1, after leaving two
steps, whatever numbers we get, how will those numbers
become?
Odd numbers.
So what will happen to odd numbers? 1, 3, 5, 7, and so on.
Odd numbers have come.
Now, even number.
What is even and odd?
Opposite of each other.
From these two, what we get?
A whole set of natural numbers.
So everyone knows the table of 2.
2 ones are 2, 2 twos are 4, 2 threes are 6.
So whatever numbers we get, 2, 4, 6, 8, 10, whatever
numbers we get in the table of 2, in which category will we
apply them? We will call them even numbers.
So what will happen to even numbers indirectly?
All the numbers which are being divided by 2, or we can
say, all the numbers which are coming in the table of 2.
So 2, 4, 6, 8, 10, and so on.
So see,
I told you that if we start with odd numbers from 1, then
if we take steps of 2, 2, then we will get even odd
numbers. Similarly, if we are saying even numbers, that
from where is our even number starting? If our even number
is starting from 2, then we
will take two steps ahead. One step is here, if we take one
more step, then 4 will come here.
Similarly, if we take two steps, if we start from 2 and
take two steps ahead, then what we will get?
We will get even numbers.
So odd number becomes even number.
Third is prime number.
What is prime number?
Prime, what will happen to even numbers indirectly?
All the numbers which are coming in the table of 2, and the
opposite of odd numbers, all the numbers which are not
coming in the table of 2, we will call them odd numbers.
Now what is the scene of prime number?
Whose table is the prime number coming in?
It is not coming.
So prime numbers are prime.
Everyone will take prime, Amazon prime.
Similarly, you can relate prime numbers.
Which is only and only.
There is a special thing in prime.
So the speciality of prime numbers is that it will either
come in its own table, or in the table of 1.
In the table of 1, all will come.
So only the numbers which come in their own table, and not
in anyone's table, that will be called our prime number.
So first of all, what will be the prime number?
It will be 2.
We do not write 2, because we exclude 1 from it.
We start with 2.
So first of all, what comes after 1? 2 comes.
So 2 comes only in its own table, and not in anyone's table
except for itself.
So how does 2 become?
Prime number.
Then comes 3.
Whose table does 3 come in?
Except for itself.
It does not come in anyone's table.
It comes only in its own table, and in the table of 1.
So 3 becomes the second prime number.
Then comes 4.
So 4 comes after 3.
Whose table does 4 come in? 4 comes in the table of 1.
4 comes in its own table.
But 4 also comes in the table of 2.
2, 2s are 4.
So how does 4 become?
Non-prime number.
So 4 will not be our prime number.
Then comes 5.
Whose table does 5 come in?
It comes in its own table, and only in the table of 1.
It does not come in anyone else's table.
So 5 becomes the third prime number.
Similarly, how many numbers are there in 1 to 10? 2, 3, 5,
7.
How do we have 2, 3, 5, 7?
All prime numbers are between 1 to 10.
Then after 10, 11 comes, 13 comes, and 17 comes, and so on.
In this way, our prime numbers also keep increasing.
New scientists keep searching what will be the last prime
number, what will not be the last prime number, how far is
it going.
All this has become our prime number.
After the prime number, we must have heard another number.
Composite number.
What does composite mean?
Composite means, which is made by combining two things, by
combining more than two things.
So what will be the composite number? You will read the
factors later, so I am not using that word. Factors,
divisible, etc.
I am telling you related to what we are reading now.
So in our own table, means in the table of two other
numbers, we will be able to write it. So what will be that
number?
Composite number.
Like I wrote 4 now.
I said 4, 4 is not a prime number.
But if we write 4, then how can we write it? 4 is equal to
2 into 2.
So this is being written in the multiple of another number.
It is being written in the multiple of another number other
than ours.
So what is this?
Composite number.
How did we get 4?
Composite.
Similarly, 5 is a prime number.
We cannot write 5 in any other multiple other than that.
Now let us see 6.
How can we write 6?
2 into 3.
We can write it.
Other than 6, we have got other numbers whose multiples are
also 6.
Composite number is the smallest composite number in a
number.
Similarly, if we see in prime, everything starts from 2.
So, how do we get 2?
We get 2 as the smallest prime.
And this is the smallest prime.
On top of that, see in which category do we get 2?
In odd or even?
We get it in even number.
So, we have only one even number, which is prime.
So, what is the specialty of 2?
There are two specialties.
One is the smallest prime number.
And second, it is the only even prime number.
Whose is this talking about?
This is talking about 2.
2 is the only even prime number and it is the smallest
prime number.
Similarly, what is the next number? 3.
3 is also small but bigger than 2.
So, we will not call it smallest.
So, what is 3? 3 is the first odd prime number.
Similarly, we will do the next category.
Whose is this related to?
This is related to only natural number and whole number.
Now, next we will have natural and whole numbers.
See, we are moving forward one by one.
One step ahead of 0 is 1.
Then one step ahead of 1 is 2.
But we move ahead in life.
But in life, we sometimes go behind.
We also have downfalls.
So, if we have to go behind 0, then what will happen?
So, there also some numbers are there.
In our real life, we are not able to see what will happen.
For example, we can see 1.
One biscuit.
One toffee.
But can we see minus 1 or negative?
Can we see minus 1 toffee or minus 1 biscuit?
No, we can't.
So, if we are going one step behind 0, then what numbers
will come?
Mathematicians have named them as negative numbers.
So, whichever step we go one step behind 0, then negative
numbers will start from there.
So, when they come in negative picture, then we will
automatically call those numbers as positive numbers.
So, from here, we will start talking about negative numbers
and positive numbers.
About good people and bad people.
So, let's move ahead.
What will we have next?
First category was natural numbers.
Then whole numbers.
Then their subcategories were prime, odd, even and
composite.
So, what we have now?
Third category.
After whole numbers, we have third category.
That is integers.
In integers, we had 0.
Now, we have read whole numbers from 0.
0, 1, 2, 3 and so on.
We are moving ahead one step at a time.
Now, if we go one step backward.
We have to go one step behind 0.
So, we are going one step behind 0.
So, we will denote the sign to go back.
We will denote it as negative sign.
And if we are going one step behind 0, then it will become
negative 1.
Did you understand?
We were going one step ahead from 0.
So, it was positive 1.
If we go one step behind 0, then it will become negative 1.
If we were going two steps ahead from 0, then it was
positive 2.
If we have to go two steps behind 0, then it will become
negative sign. And if we are going two steps behind 0, then
it will become negative 2.
That is why this series of negative numbers starts from
opposite direction.
We write 1, 2 like this.
But how will we write this?
Minus 1, minus 2, minus 3.
So, it will go like this, minus 3.
And it will also go one step behind 0.
Did you understand? What are integers? What is the set of
integers?
First, we need neutral 0.
Then we have positive numbers.
Positive 1, 2, 3 and so on.
Then we have negative numbers.
Negative 1, negative 2, negative 3 and so on.
These two numbers are endless.
From both the sides.
These two numbers are going on endlessly.
Because we don't know the initial number of these numbers.
If you say that 1 crore is the biggest number.
It is not bigger than 1 crore.
Then I will say, add 1 rupee to 1 crore.
If you add 1 rupee to 1 crore, then it becomes 1 crore 1.
So, it will keep increasing.
If you give me any number, I will add 1 rupee to it and
give you a bigger number.
Everyone knows addition.
Similarly, if you go to the negative side.
Then I will subtract one more number from it.
Give me any number, smallest.
I will subtract one more number from it.
Then I will go further back.
So, what are these numbers?
What is this system?
They are endless.
They have no end.
That is why we say that they are going on infinitely.
They are going on endlessly.
So, we have integers.
Integers are zero positive numbers.
If I write integers.
What will happen to integers?
There will be three categories in integers.
First of all, zero.
Which is neutral.
It has no sign.
Neither positive nor negative.
It is neutral.
Then we have positive numbers.
Positive numbers like 1, 2 and so on.
Positive numbers.
What else do we have?
Third, negative numbers.
Negative numbers include minus 1, minus 2 and so on.
Let me give you an example.
How can you associate integers in real life?
Look at the lift in front of you.
I hope you have seen the lift.
I hope you have used it.
Nowadays, we use it in all the metro stations.
If we look at the ground level in the lift.
What does the number denotes on the ground level?
Zero is written on the side.
When we are on the ground level, zero is written.
Suppose we have to go to the second floor.
Which button do we press?
For 2, it goes up.
When the lift from zero is going up.
Each floor is being added.
Positive numbers will keep going up.
1, 2, 3.
How many floors will be there?
Now suppose we have to go down.
How do we denote down?
Suppose we were denoting down to go back from zero.
Here, the positive numbers are going up.
Now, if we have to go down one floor.
There is a parking below.
What do we do in that?
What do we press for that?
You must have seen.
It includes minus 1, minus 2.
Such numbers are denoted for the side below.
Minus 1 and minus 2.
I hope you understood the topic of integers.
What are integers?
What do we get after integers?
Till now, we were talking about one quantity.
That we have to go one step ahead.
Or we are giving one toffee.
Or we are giving one chocolate.
But, if your younger brother asks you for something.
You give him the whole thing.
But, if your younger brother asks you for a chocolate.
Do you give him the whole chocolate?
No, right?
You give him half of it.
Or, you give him a smaller one.
Or, you give him only one cube of it.
So, what happens?
That whole thing is not like a whole number.
So, what do we get from that?
The concept of fractions starts from there.
For example, we have a chocolate like this.
And, we have small cubes like this.
Suppose, we have 5 cubes.
If we are giving only this side.
That means, we are giving half of the chocolate.
Suppose, we are not giving even this much.
We are giving only this side.
4 cubes on top.
So, what will be the fraction of this?
We have 10 cubes in total.
Total pieces are 10, but we are giving only 4 pieces to our
younger siblings.
So, how do we do this mathematically?
How many pieces have you given to your sibling?
So, how many pieces have you given to your sibling?
Write it above and how many pieces are there in total?
Write it in the divide below.
So, you have given this much fraction of chocolate to your
sibling.
So, from here, the concept of fractions will come.
So, write the next heading after integers.
Fractions What will be the example of fractions?
In fraction, one will be above and one will be below.
Suppose, we write 1 by 2 which is half.
The number above is called numerator.
And the number below is called denominator.
So, the fraction numbers are of this type.
1 upon 2
Or we can write 2 in fraction.
In fraction type.
Why not?
If we write 1 below 2 in divide.
So, anything divided by 1 will be itself.
So, this is in the form of fraction numerator and
denominator.
So, how did we get 2?
We can say it in fraction number.
Similarly, 3 by 4.
In fraction, it is not necessary that the number above will
always be small.
Or the number below will always be big.
It can be vice versa.
See, this was 2 by 1.
So, what was 2 above?
It is bigger than 1.
And the one below is smaller.
So, the number above can be bigger or smaller.
The same thing is below.
The denominator can be bigger or smaller than the numerator.
So, whatever number we can write in the form of numerator
upon denominator.
That number will be called fractions.
Did everyone understand till fractions?
Now, after fractions, as you have written 1 by 2.
What is 1 by 2? 1 by 2 is the total. What did I say?
The denominator is total.
And the numerator is what we are giving or what we are
using.
So, we have the numerator.
Which we are using.
So, we have 2 in total.
Suppose, we have 2 blocks in total.
From that, how much we have to use? Suppose, we have to use
only 1 block.
So, this portion is called 1 by 2.
That is, we have to use 1 portion.
Out of how many total portions?
Total out of 2 portions.
Now, if we see it like that.
If we divide it.
1 by 2 is a fraction.
I will tell you in the new slide.
It is a fraction.
Suppose, 1 by 2.
What does it mean? 1 by 2 means.
We will write 1 inside.
And 2 will be our divisor.
From which we divide.
It is called divisor.
And we have 1.
Dividend.
Now, divide 2.
1 cannot be divided by 2.
1 does not come in its table.
So, what do we have to do? You must have learned division
in your childhood class.
We will put a 0 in front of it.
We will put up 1, 0 after 1.
If we are putting 0 here.
Then, what do we have to put in the quotient?
We have to put a decimal.
So, we do not leave anything in front of the decimal.
What is a decimal?
It is a point.
So, I will get 0 point something.
Now, see.
10 has become.
10 has become instead of 1.
So, we will divide 10 by 2.
What will come?
What is 2 times 5? 10.
So, we will divide by 5. 2 times 5 is 10.
So, 10 has become.
What is the remainder? 0.
So, if we have to write the fraction in decimal.
If we have to write the fraction in decimal.
Then, what can we write 1 by 2?
We can write 1 by 2 as 0.5.
When we have to use a point like this.
Then, we call this as a decimal form.
Now, what will be your decimal form? 0.5 In this decimal,
see.
There is no number in front of 5.
We can write 0, 0, 0 in front of 5.
It does not matter.
But, there is no number in front of 0.
So, what is happening?
The decimal is getting stopped.
After 5, it is stopped here.
So, what are the decimal forms like this?
What is the name of stop in English?
Terminating.
So, this type of decimal.
After 1 number, or after 2 numbers, or after 10 numbers.
It stops.
So, those decimals are called as terminating decimal.
Terminating means it is terminating.
These days, we say that jobs terminate us.
It is quit.
It is ended.
It is stopped.
It is finished.
Bye.
So, we have terminating decimal.
Which is after 1, 2 or some finite number.
Finite means after some number digits.
It is stopped.
That is the decimal form.
So, we have terminating.
Now, if we see the opposite of this.
Then, non-terminating.
What will be non-terminating?
Which will not stop.
Which never stops.
So, we will have non-terminating.
You understood 2 terms.
Terminating and non-terminating.
So, you wrote the new heading.
After fractions, the next heading will be.
Which number was it?
Fractions is 4th.
So, we will have 5th number.
5th number would be decimals.
Now, all the categories of decimal.
What will be the type of decimal?
Something like 0.123.
Or 4.749.
All these types of numbers will be in decimal form.
We can go from fractions to decimals.
And from decimals to fractions.
You have learnt decimals from fractions.
Now, how will we go from decimals to fractions?
I will tell you that.
How will we go from decimals to fractions?
Suppose, we had a number.
0.123.
What should we do?
Remove the decimal.
So, if we remove the decimal.
When we divided.
When we put the decimal.
So, we did divisor.
See, in the last question.
We put 0 with 1.
When we put the decimal above.
So, we put 0 in front of 1.
Similarly, when we remove the decimal.
So, we have to write 0 in divide.
First, we will write 1 to remove the decimal.
And all the numbers which are in front of the decimal.
All the digits which are in front of the decimal.
Digits means each number.
Which are in front of the decimal.
We will put 0 in front of 1.
So, here is 0.123.
So, what is after the point? 1, 1 number, 2, 3.
So, how many digits are there after the decimal? 3.
So, we can write 3 zeros.
We removed the decimal.
We wrote 1.
All the digits which are in front of the decimal.
We put 0 in front of them.
So, this is the fraction form.
We can write fraction from decimal.
So, this decimal was 0.123.
So, we got the fraction form.
So, we can convert fraction to decimal.
And we can convert decimal to fraction.
This is decimal to fraction.
And fraction to decimal.
Now, do the sub-categories.
Sub-categories of decimal.
Now, I have written the sub-categories of decimal.
Sub-categories of decimal.
Sub-categories of decimal. So, we have got this type of
numbers. Terminating Decimals And what was just opposite of
this?
Suppose, no number is finished.
For example, you must have read a number Pi.
You must have read Pi.
2 Pi r radius 2 Pi r perimeter formula of circle. So, what
is Pi?
When you use its decimal form in questions, you will use 22
by 7. Or you will use 3.14.
You can see 3.14 is finished after 4.
But, we use it to solve questions and numericals.
But, if you divide 22 by 7, as I have taught you to do with
fractions and decimals.
Then, you will get never ending.
It will continue to be 3.147 and so on.
Scientists are finding out the numbers.
So, how do we get Pi? 22 by 7 or 3.147 and so on.
So, what will be the decimals of this type?
How will we get the decimals of this type?
Non-terminating decimals.
What is this?
Non-terminating decimals.
Now, look at an example.
1 by 3 You have a new example.
There is a number 1 by 3.
It is a fraction.
We have to convert it into a decimal.
Do it.
What will be the result?
The number above will go inside.
And, the number 3 will become a divisor.
So, 3 and 1 are not going from the table of 3.
So, we have to put a point and put 0.
You can put as many as you want.
Now, you can put 1 or 2.
It doesn't matter.
Suppose, we put 1.
So, 3.
What is the smaller number than 10 in the table of 3? 9 is
there.
3 x 3 is 9.
So, it becomes 0.3.
What is the remainder? 1.
When we put 1 point in the decimal upper quotient, we get a
break.
We can put as many zeros as we want.
If we put 0 again, it becomes 10 again.
It is repeating the same.
So, it will go from 3 again.
3 x 3 is 9.
Similarly, the remainder will be 10 again.
Similarly, 0.3333 will go on increasing from the table of 3.
We can write 1 x 3 as 0.3333, and it will go on increasing.
The probability of incrementing will be 0.3333.
This is how we can write 1 x 3.
Look at this.
It is not finishing.
3 x 3 is going on increasing.
So, in whose example it came?
Automatically it came in the example of non-terminating.
Because it is not getting terminated.
Similarly, if we see, this is the example.
Non-terminating, non-terminating, but what is this
happening?
It is repeating.
3 is repeating itself again and again.
So, this is the example.
Non-terminating will also have two categories.
That is, non-terminating, non-terminating is the decimal,
but it is repeating.
So, non-terminating, repeating decimal, one category is
this.
Whose example is yours? 1 by 3. Whose example you have is 1
by 3.
That is 0.3333 and so on.
Now, whatever number is repeating after our decimal,
recurring, let's use one more word for repeating.
Recurring means it is repeating.
Its synonym is recurring.
So, instead of writing it so big, what do we put on it?
Whatever is repeating, we put a bar on it.
For example, if 3 is repeating, then how can we put 0.3 on
it?
And whatever number is repeating, we put a bar on it.
That means, what will the bar denote after the decimal?
That this number is repeating.
Suppose, we have a number 1.272727 and so on. It is just
repeating.
Now, you will say,
Ma'am, one number is not repeating.
222 is not repeating.
777 is also not repeating.
But, you see this pattern.
Make a packet of 27.
So, keep looking at the packet.
272727 These two digits are repeating together in the same
sequence.
272727 So, we have two numbers which are repeating. So, how
will we write it in the form of a bar? 1.27 is repeating.
And what will we put on it? A bar.
This is our example of non-terminating repeating decimal.
Non-terminating is repeating.
When we took the opposite of terminating and terminating,
we got the opposite of repeating.
We can also get the case of non-repeating.
So, what will happen?
Another case of non-terminating will be non-terminating but
it is non-repeating also.
First of all, it is not even terminating.
And on top of that, it is not even repeating itself.
So, this type of example is done.
You can make any example.
For example, 1.47 321 437 205 6 There is no fixed pattern.
There is no packet.
It is repeating itself.
So, this is the example of non-terminating non-repeating
decimal form.
So, we have done fractions, decimals, categorically,
terminating, non-terminating, sub-categorically,
non-terminating, repeating and non-terminating
non-repeating decimals.
Now, we have the collection of all these.
In which collection will all these go? Let's see.
Next, write
Rationals.
6th Rational Rational numbers
Rational numbers are those numbers which we can write in
the form of P by Q. For example, we read in fraction that
you can write in fraction in the form of P by numerator and
denominator.
We read in fraction.
But, in the category of fraction, all those examples will
not come.
For example, we have some numbers which we cannot write in
fraction form. But, they will come in the category of
rational.
So, what are the rational numbers?
Write.
Rational numbers are those numbers which we have in the
form of P by Q.
P by Q form.
Now, it is written in divide.
So, we can never divide with zero.
So, the numerator below should always be non-zero.
So, Q should never be zero.
And, P and Q should be integers.
Q can be integer or P can be integer.
The definition is that we take P as integer.
Both positive and negative.
If there is a negative sign, we prefer to take it in
numerator.
And, Q is a natural number.
We will not say whole number.
If it is zero, then Q should be a natural number.
And, P should be integer.
So, the numbers which come in this form are called rational
numbers.
For example, 7 by 5 7 is non-zero below 7 is above P is
integer 5 below 5 is a national number, so 7 by 5 is a
rational number.
If we write 1 in the divide of 0, there should not be 0 at
the top and bottom, but there can be 0 at the top towards P.
Suppose we have P as 0.
If we write 1 in the divide of 0 in any number, then we
will get that number only.
What can we write 0 as? 0 by 1.
So 0 by 1 is also in the form of P by Q.
So we have got the rational number.
If we want to write it in negative, then we can write it as
minus 2 by 7.
So this is also in the form of rational number.
So all these fractions, natural number, whole number,
integers, all these categories will also be in the form of
rational numbers.
Now the main topic is about decimal.
Which type of decimal will be in the form of rational
number and which type of decimal will not be in the form of
rational number?
So this was the category of decimals.
So only this category and this category, non-terminating
repeating category.
Only these two categories will be in the form of rationals.
The remaining category is non-terminating non-repeating.
This will not be in the form of rational numbers.
This will be in the opposite of rational numbers.
What will be the opposite of rational numbers? Irrational
numbers.
So the third category of decimals will be in the form of
irrationals.
I hope you can see it clearly.
I have written it on this side as irrationals.
So non-terminating non-repeating will be in the form of
irrational numbers.
And all the numbers that we have read from natural numbers
to non-terminating repeating numbers, all the numbers that
come up to this point will be in the form of rational
numbers.
Only this category will be in the form of irrational
numbers.
So after rational numbers, you will get 7.
After rational numbers, you will get 7.
Irrational numbers.
So all the numbers that are not in the category of
rationals, means real numbers, all the numbers that we have
read so far, if they are not in the category of rationals,
then they will automatically lie in the category of
irrationals.
For example, we have squares.
If we have a square root, like 4, we already have a square
root of 2, but 3 is not a square root of 3, then a number
like root 3, a number like root 5, means if we have
non-square numbers in the root, then they will lie in the
category of irrationals.
What are the other numbers?
What was the other category of irrational numbers? If we
have non-terminating decimals, and which category of
non-terminating decimals?
Non-repeating category.
Then those numbers will also lie in the category of
irrationals.
For example, we had Pi.
What was Pi? 22 by 7? 3.147 and so on.
So, it is non-terminating.
It is not getting terminated.
And after its decimal, the digits are not getting repeated.
So, Pi also lies in the category of irrationals.
I hope you have understood everything about rationals and
irrationals.
We had a combination of natural numbers, whole numbers,
prime numbers, odd numbers, integers, fractions, decimal
forms, rational and irrational numbers.
Now, if we combine all these, we have natural numbers,
whole numbers, integers, fractions, terminating decimals,
non-terminating decimals, repeating decimals, and the rest
of the numbers are irrationals.
If we make a collection of all these, what will we get?
When you used to make a small real line in the form of
copies,
you used to make a real line like this.
What did you write in the middle? You used to write 0 in
the middle.
Then 1, 2, 3 and so on.
And here, minus 1, minus 2, minus 3.
So, the collection of all these, the collection of
rationals, irrationals, etc.
is this.
It is nothing else.
It is the small real line that you used to make in the form
of copies.
It is this.
Real line.
So, the collection of all these is real numbers.
All these are your real numbers.
Revise till here.
After this, I will tell you in the next class.
The concept of complex numbers or imaginary numbers will
start.
As we move forward in the classes,
I will tell you about it.
Till now, what should be your concern?
What will come in the categories of real numbers?
I hope you have understood all the terms that I have used
in today's number system.
You must have understood the examples.
Now, I would like to give you a homework from my side.
Try to make 5-5 examples of 1-2 subcategories.
Then, you will find out which category this number will lie
in.
Thank you.
See you in the next lecture.
For every friend you refer, you earn points that you can
redeem for discounts or even cash rewards.
And also you'll get the job guarantee after completing this
course.
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