Hi, I'm Rupal Gupta.
I welcome you all to the "Mathematics Classes".
Today we will revise the
representation of the various numbers
on the number line.
Some important points
While drawing our number line.
How to count the number of integers
between two integers.
What are the fractions and
how to represent it on number line.
How many rational numbers are there
between two rational numbers and
how to find them.
Let's revise the different type of numbers.
We have learned about the Natural number
in the earlier classes.
It starts from the 1 and extends indefinitely
only to the right side of the 1.
These numbers
will start from 1 and extend
only to the right side up to infinity.
The whole numbers is also extends indefinitely
or up to infinity only to the right side.
but, it starts from zero (0).
Like Natural number it also extend on the
right side
but will start from zero (0).
The integers are the numbers that are not
in fraction
and extends on the both sides of zero.
On the left side of Zero (0)
you can see the negative integers like -1,-2
and on the right side
you can see the positive integers like
positive 1 , positive 2 and so on.
For Rational numbers, also the line extends,
on the both side,
as you can see here all the positive intergers,
as well as all the negative integers,
but now we can also see the numbers
in between the integers
like 1/2 in between 0 and 1
and 3/2 or 1.5 in between 1 and 2.
Similarly we can show these
number on the negative side as well.
The position of -1/2 will lie in between -1
and 0.
Position of -3/2 will lie in between -2 and
-1.
How to position them, we will learn in this
lecture.
We have already learned about the number line.
While, drawing number line
we need to keep some important point in mind.
0 will always be the mid point.
right side of 0 will show all the positive
numbers.
while left side will show the negative numbers.
All the consecutive numbers marked on the
number line
will be equally spaced from each other.
they must be in order like
1,2,3 and -1,-2 and -3.
If we have taken 1 cm between two consecutive
numbers
then, for the rest of the numbers we need
to
maintain the same distance.
you can see this number line
and as comparison with first number line
the spacing between two number is bit larger
but, consistent
because of equal spacing,
it is a valid number line.
Similarly this number line is also valid becasue
every points on the line is equally spaced.
could you predict whether this line is valid
or not?
Correct, its invalid,
Since the space between these points are not
equal
so, it is not a valid number line.
We always keep all the number at equal space
and can not change their order.
Let us represent the fraction number on number
line.
There are two types of fractions namely
Proper Fraction and Improper Fraction.
Let us first study about the Proper fraction
We know that proper fractions are those
in which numerator is less than the denominator
as shown here.
they always lie between only zero and one.
some of the examples given
are: 1/2 , 4/5 and 9/11
we can see the numerator is less than the
denominator.
like 1 is less than 2 and
similarly here 9 is less than 11,
so, these numbers will always be
greater than zero and less than one.
They are always less than one and greater
than Zero (0).
So, proper fractions always exist
between zero and one on number line.
In the negative numbers consider
only the values not the sign.
In -1/2 and -9/11
while comparing the numerator and
denominator we will only compare the values
like 1 and 2 and here 9 and 11.
this number is also less than 1
and
will lie in between the -1 and 0,
since they are negative numbers.
Improper fractions are those
in which numerator of the fraction
will be greater than its denominator.
like 12/5 , 12 is greater than 5 so it is
a improper fraction.
These numbers are always greater than one
because the numerator is greater than the
denominator.
We will first convert the improper fraction
into mixed fraction,
then we would be able to know the numbers,
where the position of fraction is lying.
The from of mixed fraction is given here as
Q R/D,
where Q is a whole number and R/D is a fraction.
so number lies between Q and Q+1.
let us learn how to convert the fraction into
mixed fraction.
To write in the mixed fraction,
we will divide the numerator by denominator.
We know that when we divide a divident by
a devisor
we get some quotient and remainder.
Now for 3/2 ,
if we divide 3 by 2,
it will go by 1 and there would be 1 as remainder.
so we can write 3/2 as a 1 whole number plus
1/2
that means this number lies
between itself that is one and the next numer
that is 2.
So, 3/2 lie between 1 and 2.
Similarly in 12/5, when we divide 12 by 5,
we get quotient 2 and remainder is also 2.
We can write 12/5 in mixed fraction like Q
R/D
then 12/5 will become 2 2/5 and this number
tell us that its position lie between 2 and
3.
Whatever, be the whole number here then
the position of fraction will lie
between itself and the next number.
Here, whole number is 2 and fraction part
is 2/5
So, it is like 2 point something
that is greater than 2 but less than 3.
similarly , in 32/9 we get Q as 3 and remainder
5,
so can also write is as 3 5/9.
and this number will lie between 3 and 4.
To represent the fraction on the number line,
we need to follow certain rules as given here:
suppose our fraction is in the from of n/d
where n is numerator
and d is denominator.
denominator will tell us the
number of equal division between two
required consecutive integers.
like if our fraction is proper
then we will divide the line segment
between 0 and 1 in d-equal parts.
if its improper and in Q R/D form then
the division will be in between Q and Q+1.
numerator n will tell us how many such part,
we have to include.
for a proper fraction,
we will count n part starting from zero
and for improper fraction,
we will count nth part starting from Q.
the fraction may be positive or negative.
positive means Right side of 0
and negative means left side of 0.
Lets understand this with a example.
Suppose our number is 1/2,
Here we can see that 1 is less than 2
so, the fraction would be less than 1.
on the number line
it will lie between 0 and 1.
This 2 tell us the no. of equal division
between 0 and 1.
Now, if we divide from 0 to 1 in two parts
then this 1 inform us to count the number
of part
(first part) starting from 0.
so, this is our 1/2.
Similarly, we can see on the number line
that, for every postive number,
we have one corresponding negative number
at equal distance from 0.
The distance at which we get 1/2 ,
the same distance but in opposite direction
we will get -1/2.
So, easily we can find
the corresponding positive fraction
or integer number,
as they lies at equal distance from 0.
can you tell us
what would be the number at these divisions?
it will start from 0/2
then 1/2 and finally 2/2.
2/2 is nothing but 1.
Let us represent 2/3 and
here the numerator 2 is less than denominator
3
hence this fraction is less than 1
and would lie in between 0 and 1.
The denominator 3
tells us about the number of division
between 0 and 1.
so we will divide it in three parts.
the numerator 2 tells us to count 2 parts
starting from 0 that is second part from 0.
so here
first,
second,
and
this is our 2/3.
and its corresponding value negative faction
will be, -2/3 here.
the distance between 0 and 2/3
is same as 0 and -2/3
so,
if we know the positive side distance
we need not to divide on the left side.
we can simply measure this distance OA
and it would be equal to OB
to mark our point -2/3.
All these in between division would be 0/3
,
1/3
as it is first part,
2/3
and
finally 3/3
as up to here we have counted all the part
which is this complete line segment from 0
to 1.
hence this point would be 3/3 or 1.
Similarly, for 4/7 as it is also less than
1
so, we will divide the 0 and 1 in 7 parts.
as the denominator is 7.
After dividing it the numerator is 4
so we will count the 4 part starting from
0.
1 ,
2 ,
3
and
4
this is our 4/7.
Similarly we can measure the OA distance
and, if we go to the left side from 0
by the same distance OA
we will reach the Point B Which is -4/7.
This time we have not divided
on the negative side but
If we divide the negative side
that is from 0 to -1
then we will get
the same value at same point.
after counting it from 0
considering the 4 part in left side.
On the positive side
the rest of the division is shown here.
like 0/7 to 7/7.
For 9/11, which is also less than 1.
the 11 will be the number of division
in between 0 and 1.
9 will tell us to count 9 part starting
from 0 and that is
1,
2,
3,
4,
5,
6,
7,
8,
9
this is our number.
Similarly we can also find
its negative point on the left side.
Let us represent the 3/2
on the number line.
Here, 3 is greater than 2
so, the fration is greater than 1.
Without writing it
in the form of mixed fraction
we have to divide every unit in two parts
and due to numerator 3,
we will consider 3 parts starting from 0.
Now after dividing the each unit
and counting the three parts
starting from zero
we get 3/2 here, i.e 1.5
but it is always useful
if we have considered it
in the form of mixed fraction
becuase it help in finding the ranges
between which the number lies.
like 3/2 can be written as 1 1/2,
1 is a whole number
and the fraction will lie
between 1 and 2,
so we need to divide only this line segment
between 1 and 2.
Suppose our next number is 31/2
and without converting it into mixed fraction,
then we have to mark every unit
that is from 0 to 1 , 1 to 2 and so on.
after, dividing it in two parts,
we need to consider 31 parts,
which is very lengthy procedure.
to avoid it we will simply convert it
in mixed fraction and we get 15 1/2.
so, the position of number is between 15 and
16.
we will only divide this line segment in two
parts
and our first part counting from 15
would be the required number.
to represent 30/9,
let us convert it into mixed fraction
we get 3 3/9
3 is a whole number
and telling us the number 30/9 is lying
between 3 and 4.
the fraction 3/9 would provide
the same information as earlier.
9 is the number of division
and we need to consider 3 parts
counting from
this time we will count it from 3.
so, our point will be here 3 3/9.
Also this number can be written as 10/3,
and in mixed fraction 3 1/3
According to this data
we need to divide the line in between three
parts
and have to consider only the first part
we will get the same answer as earlier.
Our next number is 19/4
and writing it in mixed fraction
we get 4 3/4.
all these three number, tell us,
about the position of fraction 19/4
the whole number 4 indicates that
the position of number is between itself
and the next number that is 5.
the rest fraction denominator 4
tell us to divide the distance in to 4 parts
and finally this 3, is to count the number
of part
starting from 4.
if we count 1 , 2 , 3,
so, this 3rd part
is our number 19/4.
We all know that to represent
any integer parts is easy on number line.
We have represented here -6, -2, 2 and 4.
Let us count how many numbers
are there in between -6 and 4 ,
we get -5, -4, -3, -2, -1 , 0, 1, 2, 3
these are all the numbers between them
a total 9 numbers.
Since the number was between -6 to 4 so
on the positive side, up to 4 we have 3 numbers.
on the left side we have 5 negative numbers
and including 0 our total count is 9.
Similarly the numbers
between 1 and 5 are 3 that is 2, 3, 4.
-2 ,-1, 0 ,1 these are all the four numbers
between -3 and 2.
Similarly, on the negative side,
we can also find the numbers
between any two negative quatity .
like here
We have learned
that there would always be a countable
or definite natural numbers or integers between
two natural numbers or integers.
Let us consider two rational number
-1/2 and 7/6.
we have to tell how many rational numbers
are there in between these two numbers.
First we make the denomnator equal
by taking the L.C.M
here L.C.M is 6
so, the number becomes -3/6 and 7/6
so, we get total 9 numbers as : -2/6 ,-1/6
till 6/6
but what if we multiply each term by 10
we get -30/60 to 70/60
and the number between -30/60 and 70/60
are:- -29/60, -28/60 till 69/60
You can easily count this number
as up to 70 there would be 69 positive numbers,
on the left side there would be 29 negative
numbers
and including 0 it will be total 99 numbers.
again, if we multiply by 100
we will end up with 999 numbers
and multiplying by 1 lakhs
then we will be having nearly
9.9 Million numbers.
If We can keep multiplying then
our count in between the numbers
will also keep increasing.
We have find out that we get countless
or infinite rational numbers between any
two given rational numbers.
Since there are infinite number of rational
numbers
then what would happen if we need only few
rational number between two numbers.
consider -5/6 and 5/8 and
we have to find out 10 rational numbers
in between them.
We will multiply the numerator and denominator
by a suitable number to make our denominator equal.
Here, we have multiplied 4 to -5/6 and 3 to
5/8
to make their denominator equal i.e 24.
If we count the number between them
then, we get 19 on negative side
and 14 on positive side
and including 0, we get 34 numbers.
We can take any ten numbers.
According to our need we choose our multiplier.
e.g here we need to find ten rational number
between -4/3 and 5/6.
if we take 6 as L.C.M
and multiply only -4/3 by 2 both numerator
and denominator
then we get -8/6 and 5/6.
In between we have -7/6 , -6/6 ,
0 ,
1/6 , till 4/6
total 7 +1 + 4 equals 12 numbers.
We can take any ten.
What if we need more numbers like 20 numbers?
In that case we will choose different multiplier,
earlier, we have multiplied by 2,
now this time it is 4,
we will get 12, as denominator
and to make the denominator as 12 for 5/6,
we will multiply it by 2 to both 5 and 6.
so finally we get our number as
-16/12
and 10/12.
On the negative side,
we will get 15 numbers.
and on the positive side,
we will get 9 numbers as
and including 0 ,
we have total 25 numbers.
earlier, we had only 12 numbers
and now we have 25 numbers.
we can take any 20 numbers from here.
There is one more method.
We had studied it in Class 7th,
how to find mean.
Mean or Average of two numbers
will always lie between them.
Let us understand it with an example.
We have two numbers 1 and 2,
to find the mean,
we need to add 1 and 2
and divide the result by 2
means (1+2)/2,
we will get 3/2 or 1.5
and 1.5 is always greater than 1 but less
than 2.
Mean will always lies between these numbers
as, it tells about the mid point of these
numbers.
We can utilize the idea of mean
to find the rational numbers between two numbers.
If a and b are two rational numbers,
then (a+b) by 2 is a rational number
between a and b
such that,
a plus b
by 2
will lie between a and b.
We can
always find the mid points of two points
using the same approach.
then, again average (mean) can be taken
between two points we can perform this
continuously again and again.
hence we can say that
there are countless numbers of rational numbers
between any two given rational numbers.
Let us see this in action
Let us find 1 rational number between 1/4
and 1/2.
We will use the average method
and this can be written as
1/4 plus 1/2 divided by 2
here L.C.M is 4
so 1 +2 is 3
and 3/4
multiplied by 1/2 will be 3/8.
The same thing has been shown here
on the number line.
initially 1/2 is the mid point of 0 and 1.
1/4, can also be finded out by dividing it
in to four part and taking the first part.
In the mid of 1/4 and 1/2 we get 3/8
this is the mean of these two number.
and using our representation
we can also find the 3/8 at the same place
by dividing the whole line in to 8 parts
and marking the 3rd point.
Just like 1/4 and 1/2,
we find out the mid point as 3/8,
similarly a mid point can be found in
between 1/4 and 3/8
then, again in between these and so on...
That's why from there,
we can also conclude the same
that there are infinite rational numbers
between two numbers.
Now, Let us find three rational numbers
between 1/3 and 1/2.
Firstly, we will calculate the mean of 1/3
and 1/2
the calculation is shown here...
and the mean of these two number
is 5/12
5/12 is the exact mid point of the
1/3 and 1/2.
As it is a mid point
so, it will always lie between 1/3 and 1/2.
We can also find the same fraction 5/12
if we divide the whole line
into
12 parts
and
our 5th part will be the 5/12,
which is also at the same position.
Now, we can take the mean of 1/3 and 5/12
to find out the mid point
we will calculate like this as shown here
L.C.M is 12.
and after performing basic addition and multiplication
we will get 3/8.
3/8 is the mid point of 1/3 and 5/12.
which lie between 1/3 and 5/12.
we can get the same fraction
if we
divide the line into 8 parts
and
3rd point would be the exact same position.
but you need not to show by dividing the line
if the question is related to
demonstrate the idea of mean.
Only show the calculations of mean.
again we will find one more number
between 5/12 and 1/2,
which is shown here.
and the mean is 11/24.
and dividing the line into 24 parts
and selecting the 11 part
counting from zero
would give us the exact same number.
All these points are in between the 1/3 and
1/2.
so, our three points which lying
between them are:
3/8,
5/12,
and 11/24.
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