Finding: Rational Numbers between two Rational Numbers Video Lecture - Class 7

rational numbers between two rational
numbers let us try to list the natural
numbers between four and seven so as you
can see five and six are the only two
natural numbers between four and seven
now can you try to find out the natural
numbers between seven and eight yes you
are right there is no natural number
between seven and eight
now let us look at the integers that lie
between minus 3 & 2 they are -2 minus 1
0 & 1 in the first case there are two
natural numbers between 4 & 7 in the
second case there was no natural number
between 7 & 8 in the third case the
count of integers between minus 3 & 2 is
4 this tells us that there is finite
number of natural numbers between any
two natural numbers this is also true
for integers we always have finite
number of integers between any two
integers now let us look at rational
numbers how many rational numbers are
there between 3 by 8 and 7 by 8 you
probably will think that the answer is 4
by 8 5 by 8 and 6 by 8 now is it correct
to state that there are only 3 rational
numbers between 3 by 8 and 7 by 8 no
this is not the correct statement let us
understand the reason
3x8 can be written as 30 by 80 and 7 by
8 can also be written as 70 by 80 the
numbers 31 by a t32 by 80 33 by 80 and
so on till 69 by 80 fall between 3 by 8
and 7 by 8
now as done above 3 by 8 and 7 by 8 can
also be written as 300 by 800 and 700 by
800 respectively therefore you can
observe unlike natural numbers and
integers there are in finite number of
rational numbers between any two given
rational numbers let us solve some
examples find any two rational numbers
between 3 & 5 here 3 can be written as
30 by 10 and 5 as 50 by 10 thus we have
31 by 10 32 by 10 33 by 10 and so on
till 40 9 by 10 between 30 by 10 and 50
by 10 or we can see between 3 & 5 any
two rational numbers between 3 & 5 are
31 by 10 and 40 by 10 let us take an
example of alternative method find a
rational number between 2 by 7 and 3 by
4 to find a rational number between two
rational numbers by alternative method
we will find the mean of the two given
rational numbers
so for this first we will find the sum
of 2 by 7 and 3 by 4 the sum is 29 by 28
now we divide the sum by 2 the result is
29 by 56 so we can say 29 by 56 lies
between 2 by 7 and 3 by 4 symbolically
we can write for any two rational
numbers P and Q such that P is less than
Q then P is less than P plus Q by 2
which is less than Q or we can say that
P plus Q by 2 lies between P and Q the
rational number P plus Q by 2 is called
the mean of P and Q let us consider
another example find any two rational
numbers between 3 by 4 and 5 by 3 as the
denominators of these two numbers are
different we will convert them as
rational numbers having the same
denominator the LCM of these
denominators that is 4 and 3 is 12
therefore we will convert the
denominators of both the fractions to 12
as you can see that the equivalent
rational numbers have numerators 9 and
20 respectively
now the rational numbers between them
will be 10 by 12 11 by 12 and so on till
19 by 12 therefore we can take any two
rational numbers out of them so for
instance we can say that 11 by 12 and 19
by 12 are two rational numbers
between nine by 12 and 20 by 12