Hello everyone, welcome to another video.
In the last video, we did the classification of number
system and number system till real numbers.
Today, we will make a flowchart of it.
We will see what we did yesterday in a summarized way.
And apart from the real numbers, we will also see the other
types of numbers that we have.
So, let's see what we did yesterday.
Yesterday, we did the number system.
In the number system, we read the biggest set, the sub
combination.
What was that? Real numbers.
So, first of all, we read real numbers.
Then, what was the further classification of this real
number?
Real numbers were further classified into two sub parts.
No. 1 was rational and No. 2 was irrational.
So, the sub category of this was Rationals and No. 2 was
Irrational numbers.
What were the examples in Rationals?
What will be the form of Rationals? P by Q form.
So, in Rationals, we will have P by Q form.
Q will always be non-zero.
P and Q will be integers.
Q will be a natural number.
Then, what were the examples of Irrationals? I had told you
the examples of Irrationals.
For example, root 2 and root 3.
All the non-square numbers that come under the root, will
be our Irrationals.
I had told you the example of Pi.
So, these were the examples of Irrationals.
What else was told in Irrationals?
Decimals will come under Irrationals, which are
non-terminating and non-repeating.
So, we will do the decimals sub-cases.
What will come under Irrationals?
In Irrationals, we have further categories.
Decimals.
And what kind of decimals?
Non-terminating and non-repeating.
What were the sub-categories of Irrationals?
What were the sub-categories of Irrationals? We had
integers.
We had fractions.
And what kind of decimals?
Non-terminating and non-repeating.
What were the sub-categories of integers?
We had whole numbers.
Which had 0, 1, 2, 3 and so on.
And we had the second sub-category.
Natural numbers.
Natural numbers which had 1, 2, 3 and so on till infinity.
What were the sub-categories of natural numbers? We had
further sub-categories of natural numbers.
Odd numbers, even numbers, prime and co-prime.
We had odd, even, prime and composite.
What were the odd numbers? 1, 3, 5, 7.
Odd numbers which were not in 2's table.
Even numbers which were in 2's table.
Prime numbers which were in 1's table.
Prime numbers which had 2, 3, 5, 7 and so on.
Composite numbers which were not in 1's table.
Even numbers which were not in 1's table.
For example, if we take 6.
Even numbers which were not in 1's table.
Even numbers which were not in 2's table and 3's table.
We had composite numbers.
I hope you remember the flowchart of yesterday.
I had told you to do 5 examples.
All of you have done that.
So, where do we represent all these numbers? I had told you
this in the last video.
We represent all these numbers as number lines.
Either you call it real line.
Either you call it decimal form.
Terminating, non-terminating, non-terminating,
non-repeating.
Either you call it decimal form.
Either you call it fraction.
Either you call it integer, natural number or whole number.
Where can we represent all these numbers?
We can represent them as number lines.
Can be represented on a real line.
Either you call it real or number line.
Both are same.
We can represent all these numbers on a real line.
We can also represent irrationals and rationals.
If I ask you, what is the biggest set? The biggest set is
real.
Then we have sub-categories.
So, we can say indirectly.
If we say for rationals.
So, for rationals, we can say.
All the rationals will be real numbers.
But all the real numbers cannot be rational.
Because some of them will be irrational.
So, you should have this flowchart in your mind.
So, your real side number system will be clear.
Now, we have this main part in our number system.
Real numbers.
The second category in our number system.
That is opposite of real.
What will be the opposite of real? Complex numbers or
imaginary numbers.
So, write it down.
What is the second category?
Imaginary numbers.
When do we get imaginary numbers in picture?
We get imaginary numbers.
Suppose, what did you do in irrationals? Under root 2 and
under root 3 were irrational.
If we do root 4.
I will explain it to you in the next line.
If we do root 4.
We did root 4.
Under root 4.
So, what will be the answers?
We can write 4 as minus 2 square.
Or we can write 2 square.
2 square is also 4.
So, what will be the answer?
Either it will be minus 2.
Or it will be 2.
This is solvable.
But if we have other than this.
This is positive number.
If we have other than positive number.
If we have negative number in under root.
Then what will happen?
We have not learned that.
What we have learned is that.
Whatever we take in cube root.
We can take only positive number.
But if we have negative number.
What will we do?
Negative can also come.
When we have any type of number.
Positive, negative, fraction.
So, there should be some value in any root.
Similarly, if we have negative 1.
Or negative 2.
Or anything in negative.
So, how and what we will write it?
is the equation of minus 1 is called I that is IOTA.
Root of minus 1 is called I that is IOTA.
We take
IOTA because it is imaginary I and IOTA.
It will be similar to remember.
So IOTA is root of minus 1.
Suppose we have to find under root of minus 2.
So how we will find its value?
So how can we write root of minus 2? Can we write it like
this? Root of minus 1 into 2.
We can simply write minus 2 as minus 1 into 2.
Now we can write it as minus 1 into root 2.
We have just read the new notation imaginary IOTA.
So we will write IOTA as root 2. So how do we get IOTA root
2? It is an imaginary number.
Wherever we get IOTA, we will get imaginary number.
Then after imaginary number we started with complex
numbers. It is not necessary that we have only IOTA root 2.
Suppose we have this type of number
A plus IOTA root 2. We have any real number and IOTA root 2
is imaginary number.
Suppose we have any real number A.
So when we have this type of questions or numbers, we give
them a new name that is complex numbers.
So what will happen to complex numbers?
They will be formed along with imaginary numbers. We have
complex numbers.
Suppose we have imaginary numbers. So you can write an
example of imaginary numbers.
Root 2 I minus of root 1 that is I minus root 5.
What will happen to all these?
We will get the sub classification of imaginary numbers.
We will write the standard form of complex numbers.
Standard form of complex numbers will be X plus IOTA Y.
IOTA is under root of minus 1.
Now we have X and Y. These are variables. Their values will
vary on question by question.
So what will be the possible values of X and Y?
X and Y will be real numbers.
When IOTA will be there, then this type of form will be the
example of complex numbers.
Suppose you write an example of IOTA. We have 5 minus 2
IOTA.
When IOTA is there, then this type of form will be the
example of imaginary numbers.
When IOTA is there, then this type of form will be the
example of complex numbers.
Now we have X and Y. The coefficient of I will be imaginary
part of IOTA.
IOTA is imaginary. So whatever will be there with IOTA will
also be imaginary part.
So the coefficient of IOTA will be imaginary part.
The imaginary part of IOTA will be imaginary part of
complex numbers.
Complex numbers are denoted by Z.
So Z is equal to X plus IOTA Y.
If we have a standard form or a complex number, then Y will
be imaginary part of complex number.
If we have a complex number Z, then Y will be imaginary
part of Z.
If we have a complex number X, then X will be imaginary
part of complex number Z.
So X will be imaginary part of complex number Z.
The coefficient of IOTA will be imaginary part of complex
number Z.
For example, 5 minus 2 IOTA.
Now think what will be the imaginary part and real part of
5 minus 2 IOTA.
So what will be the imaginary part of 5 minus 2 IOTA?
What is the coefficient of IOTA?
First we have to convert it in standard form.
In standard form, there is a plus sign in the middle.
But here there is a minus sign.
So what will happen?
If we write 5 plus, then we can write it in minus 2 bracket.
No problem.
Because plus and minus will be same.
So we can write it in this form.
Now see what will happen in standard form?
It will be converted in standard form.
So equate both of them.
X plus IOTA Y and 5 plus minus 2 IOTA.
So what is the value of Y?
What is the coefficient of IOTA?
Minus 2.
So imaginary part of 5 minus 2 IOTA will be imaginary part
of complex number Z.
Imaginary part will be minus 2 and 5 will be real part
without iota.
Now you have understood everything. Now I will tell you how
to represent complex numbers.
We were representing real numbers on a number line.
But where will we represent complex numbers?
We told them that they cannot represent real numbers. That
is why they are imaginary numbers.
So where will we represent complex numbers?
We will represent complex numbers on an x-y plane.
We will name that plane as argand plane.
Now where will we represent complex numbers? We will
represent them on an x-axis.
This is our x-axis and this is our y-axis.
Horizontal axis will be y and x and y axis are
perpendicular.
That is the angle between them is 90 degree.
And what is the origin in the middle? The origin is
represented by O.
Horizontal axis is x-axis and vertical axis is y-axis.
Now we have learnt about number line.
What was there in number line?
Suppose we have a number line.
What was there in number line?
There was zero in the middle.
Then there were positive numbers on the right hand side.
And on the left hand side there were negative numbers.
Then we took two real lines.
One real line was called x-axis.
And the other real line was called y-axis.
Now if you rotate the same line 90 degree, Then positive
numbers will go up.
If we rotate the real line 90 degree,
Then positive numbers will go up.
Then positive numbers will go up on the y-axis.
Similarly, if we rotate the line 90 degree,
Then negative numbers will go down.
Then negative numbers will go down on the y-axis.
And the x-axis is the same as the real line.
We have kept it horizontally.
So it is zero like the real line.
Then there are positive numbers.
That is 1, 2, 3.
And here are negative numbers.
Minus 1, minus 2, minus 3 and so on.
If we had drawn x, y graph in this plane,
Then we would have put points x, y.
Then it would have been x, y plane.
But now we will represent complex numbers in this plane.
So I have told you the name of this plane.
We have named it as Argand plane.
We just did the last question.
The question was, 5 minus 2 iota.
We have a complex number z.
z is equal to 5 minus 2 iota.
Now we have to represent it on an Argand plane.
Or we have to plot this point and tell us where it will lie.
So first of all,
If we compare it with the standard form,
Then we will get the real part.
What was the real part?
The real part was 5.
And the imaginary part was minus 2.
So the real part is 5.
We had denoted the real part with x in the standard form.
So you can think of it as plotting x in the x axis.
So what is the value of 5? You have to plot the value of 5
on the x axis.
And where do you have to plot the imaginary part?
On the y axis.
Now we will plot it on the x axis.
We have to see the 5 value.
So it is written as 1, 2, 3, 3.
4 and 5.
So we will take 5 steps on the x axis.
Then we have to see minus 2 on the y axis.
So the y axis is minus 2.
I had given the lift example.
If you are on the ground level and you have to go to the
minus 2 level.
So what will you do?
What is the minus denoting?
In the opposite direction, go downwards.
And 2 means 2 steps down.
2 floors down.
So if we reach 5 on the x axis, then we have to go down to
reach negative 2 on the y axis. So what do you do?
Go two steps down.
One step is here.
And the second step is opposite to 2.
So it will be minus 2 steps.
Now we will reach the final point.
So what is this point? 5, minus 2.
If we had seen it from the graph, then we would have
written it as 5, minus 2.
But this is our complex number.
So we will write it as 5, minus 2.
This means that the point 5, minus 2, we do not have that.
This is the complex number 5, minus 2.
This is our complex number 5, minus 2.
So I hope you have understood the concept of complex
numbers graphically.
How to represent them?
What are imaginary numbers?
You must have understood the concept of real numbers.
Now I will ask you a question.
Think about it a little bit.
Don't think too much.
Think a little bit.
Suppose we have a question.
Suppose z is equal to 7.
z is equal to 7.
So we have been asked whether this is a complex number or
not.
Is it a complex number or not? So what will the students
say first? Ma'am, this is not a complex number.
This is a real number.
Because what do we have?
In the category of real numbers, this is a natural number,
this is a whole number, this is rational.
So obviously we have a real number.
So I will agree with them.
This is a real number.
Good.
This is a real number.
But we have been asked a question.
Is it a complex number or not? Is it a complex number or
not? We were asked about the complex number.
Is it a complex number or not?
So what will we say?
Will we say it is a complex number or not?
Think about it.
If we see any complex number as a complex number,
its standard form should have been of this type.
x plus iota y.
It should have come in this form.
But we have this.
This 7.
Can we write it in this type? Look at it.
If we make the value of the imaginary part 0 in this.
We can write 7 like this also.
Z is equal to 7 plus 0 iota.
What will be 0 iota?
So, if you multiply 0 with any number, the answer will be
0, then it will be 7 again.
But, see, it is exactly like this, isn't it?
It looks like a standard form, clearly.
So, will this be a complex number?
Tell me, will it be?
Obviously, when we are able to write it in standard form,
then it becomes a complex number, isn't it?
It is a real number, I agree on that.
But, it can also be a complex number.
So, what should we learn from this?
In which category all these real numbers have come?
They have come in the category of complex numbers.
For example, our rationals.
The set of rationals was going to be in the real category.
Similarly, the set of real numbers.
If we always have this portion of y which is 0, then z is
equal to x, isn't it?
And x was a real number.
So, all the real numbers will go in the category of complex
numbers.
So, now we have the biggest set which we are still studying
in the number system.
What will be the biggest set?
The set of complex numbers.
We have studied the biggest set of complex numbers till now.
So, it will be the biggest.
Suppose, if I tell you according to the Venn diagram.
Suppose, we have this whole surface of complex numbers.
This is our complex numbers.
What do we have in this whole packet?
Complex numbers.
They can't come in any packet.
But, I will explain it to you.
This whole universe is of complex numbers.
The small sub-part which will be formed in this whole
universe.
What will it become?
It will become the set of real numbers.
So, the real numbers are nothing but the sub-parts of
complex numbers.
They are in the sub-category.
So, the biggest complex is the real number.
Then, there are two rational and irrational numbers in the
real category.
Then, there are different types of rational numbers.
Then, there are irrational numbers.
Then, there are root of 2 and root of 3.
So, all these will come in this whole universe.
I hope, you have understood the whole number system.
There are no questions about the number system.
I am just telling you the theory.
Basically, you can generalize or categorize any number.
So, you should know this much.
In the next class, we will see the questions and do the
rest.
Thank you for watching.
That sounds interesting.
Count me in. I'll make sure to spread the word to all my
knowns.