Hey guys
Hello everyone my name is Hina Kauser, welcome back to my channel
In this video
I'm going to discuss three main point i.e Rational number, Irrational number
and Real number , how they are related to each other and also
how we are going to represent these three numbers
over the number line. Hope you have
already watched my previous video,
if not, please go through the description. So let's go ahead
In this slide,
you can see that three main point
I have written over here
Rational number, Irrational number and Real number
Rational number: any number that can be
represent in the form of p/q where
p and q both are integers
and q does not equal to 0,  are known as Rational number
Second point is - Irrational number
Any number that can not be written in
the form of a ratio of integers i.e p/q
Or you can say that apart from Rational number,
numbers are known
as irrational number
now one more statement you can add
as a characteristic of Irrational number-
these are non terminating and non-recurring value
eg. few example I have written over here
like root 2, root 5
pie and root 13 etc
now next one is Real number
Real number is defined as
the collection of Rational number and
Irrational number are known as Real number
in the terms of symbol representation
like you can write as Q U P = R
Q is a set of rational number and
P is a set of irrational number
and "U" is union operation used in mathematics term
i.e Q U P =R
So let's move to the next slide
this is graph representation of real number
This is real number line
you can see that there are few number
over here, few are negative
few are positive, zero and
few number are represented as rational term
as p/q , form
if you hide this portion then
you're left with Natural number, and if you include zero along with these three number
then you left with the whole number
and if you hide these rational value
like p/q form number then you're left with
the integers numbers and if you add these
three p/q form number along all these number
then you're left with Rational number
now the question is what? where is the Irrational number?
So I will tell you how
to locate Irrational number over the
number line along with the solution of the question
let's go back to exercise, the first questions of the exercise
is what? state whether the following statement are true
true or false? justify your answer
so the first part of this question is what?
Every Irrational number is a Real number
Every Irrational number, they are
talking about the set of all irrational number
so let's go to the solution of this questions
here is the solution of first part
that every Irrational number is a set
of real number, is it true or not?
so you can see this is graph
representation definition of real number is
what?
Rational value and Irrational value collectively known
as a Real number, you can see that
Irrational number is a part of real number
i.e Q U P = R
where P is subset of R
then you can see that
every Irrational number are Real number
So the answer is correct
because it satisfies the condition, as per the definition
so this one is correct or true
let's move to next option i.e every point on the
number line is the form of root m
where m is the natural number
Is it true or not? every point on the number the line
we talking about the number line
and every point is in the form of root m
and we need to check that this value
is true for which,  for m is natural number
so let's go to the solution
I have draw a number line
to the number like -3
-2, -1 , 0 , 1, 2, 3,.....and so on
we need to proof
or you can say, we need to check
whether the questions condition is right or wrong
about condition
let a value m=4
we know the root 4 is equal to plus, minus 2,
not a single value
so 2 and -2, two possible value
and we can say that -2 does not belongs to natural number
Natural number is what the
number you used to count in your real
life, is known as natural number and -2
is not a natural number,  it is integer value
and second value is 2 belongs to N
here we come to the contradiction
as per the statement like every value can be
shown as root m where m is
natural number but solution is the
saying that there
is two possible value
if the value is terminated value then you got
that there is minus value
i.e -2 does not belongs to N,  this is not a natural number
so that's why, due to this contradiction
of this definition we can say that
this is incorrect because root m
can be shown on the number line
but it's not true that
m is always a natural number, it can be a negative value
or you can say that answer could be integer
not a natural number, so this is incorrect
now the third part of this questions is what?
every Real number is an Irrational number
again they are talking about full or complete set of
that number, every Real number is an Irrational number
so let's move to the solution of this question
as I have discussed that
symbol representation of these
three number type
i.e Q U P= R
so we can say that P is subset of R
P is a part of R and
R does not part of P, R(real number) does not
belongs to P, P belongs to R
so due to these two statement
what we can say that every Real number is not an Irrational number
so that's why, we can say ... this option is also incorrect
let's move to the next question
so now the next questions is what?
Are square root of all positive
integers Irrational? Square root of all
positive integers are irrational?
If not, give an example
so first we need to analyze
that the statement is true or not, if not
then we need to explain with example
so let's go to the solution
this is second question solution like
square root of all positive integers are
Irrational or not?
so what we can say that suppose m
is a positive integer or you write as m
belongs to Z plus, Z plus is set of
positive integers
as Z={1,2,3......}
so let smaller value like
m=2 so root m
you can write as root 2
and root 2 is an irrational number, because the value
of this one is non termination and non recurring value
so what we can write that the value which is
non-terminating and non-recurring value
now we are taking next value of m
i.e m=4, so what we can write here
root m= root 4, as we know that
root 4 would be equal to plus, minus 2
so what we get or resulted value or came here
like plus, minus 2 which is terminating value
or you can say that plus, minus value is
a Rational value, or Rational number
so you can write, why this is Rational number?
because plus, minus 2 you can write as
divide by 1 and plus, minus 2, 1 belongs to Z i.e Integers set
and 1 does not equal to "0"
this three condition is satisfied by
the solution of root m is equal to root 4
due to these three condition satisfied by the
solution of root m= root 4
so that's why, here we comes to the contradictions
this one solution is Irrational number and
this one is Rational number
and the statement is what? Square root of all positive
integers are Irrational number
but we get.. both, Rational and Irrational number
so that's why, this answer is not and here are the example
you can write to the
solution of this questions, I hope you understand
and let's move to the next one-
now next one is what?
we need to show root 5 on the number line
so I'm going to discuss how to
locate Irrational number over the number line
so comes to the solution of the third questions
I used over here some colorful pen to differentiate
the line first and second one and also
there are few value given over here
it will help you to differentiate, so the
first thing what I can do that we need to draw a number line
simple number line i.e 0, 1, 2, 3...
-1,-2,-3....
as per sheet size ( your sheet size you can draw)
first we need to understand,
pythagoras theorem
before this we just need to understand this triangle
so I just taken over here 1 unit (side size) triangle
like we need to draw like this one (as shown on screen)
0 to 1 i.e 1 unit
and just mentions this point as O and this one as A and this one as B
we just draw a perpendicular line over the
O to A line, i.e AB and this
one is also is 1 unit
value so we are comes here like
right triangle (angle A=90 degree), value i.e triangle OAB
the triangle is right triangle and name is Triangle OAB
so as we know this in pythagoras
theorem is what?
base square plus height square is equal to hypotenuse square
in this triangle, what we have
OA square + AB square= OB square
OB is what? this
is hypotenuse value, OB you can write as
in root form is equal to
root of (OA square + AB square)
and now as we have a value
of OA=1 unit,
and AB is also 1 unit
OB , this is questions mark(?), as we don't know
the exact value of this so
OB is equal to =root of (1 square + 1 square)
and here the value is (1+1) i.e  OB = root 2
so what we can do that once we get the
value of OB, i.e root 2
so OB we can consider
as a radius of circle and we
can draw an arc
by considering OB as radius of the circle
wherever this arc touch over the
number line, that point will show the root 2 value
I hope you understand this solution, how to
we can locate the Irrational number over the number line
let's move to the next value, how to locate over number line
so what we can do that on the same graph
or you can say that on same number line
we just draw a perpendicular of 1 unit
i.e BC on OB
OB value is root 2, we have
calculated and one more we just draw a perpendicular on OB,
BC value is 1 unit
so now the right triangle is what?
Triangle OBC, you can write over here
so in this, what you can
write OC as Pythagoras theorem
is OC = root of  (OB square + BC square)
since OB= root 2,
and BC= 1, and
OC= root of (root 2 square + 1 square)
and OC= root of (root 2 square + 1 square)
root 2 square value (should be equal to 2) plus 1
i.e root (2+1) = root 3
now OC value, what we get is root 3,
so what we can consider that
OC value is now root 3
that you write over here
Ohhh I'm so sorry, you can write
over here that OC value is root 3
now you can consider OC as a
radius of circle and the same thing as previous
one you can write by considering OC as
the radius and we draw an arc
and wherever this arc will touch over the
number line, that point (Q) will show as root 3
and so on you can proceed this same method to
calculate next higher or many more value
and it will help to solve or to locate 
 higher (Irrational number)
over the number line, hope you understand if not
please comments me and let me know
in which point you have confusion, and I will definitely
help you to solve your problem :)
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