Rational Numbers
Introduction of Rational numbers
A number r is called a
rational number, if it can be written in the form p⁄q, where p and q are
integers and q≠0. i.e. 0.3796, 4.1333...,4⁄5 etc.
Properties of rational numbers
(1)Closure
(2)Commutativity
(3)Associativity
(4)Additiveidentity
(5) Multiplicative identity
(6) Additive inverse
(7) Multiplicative inverse or Reciprocal
(8) Distributivity
Closure property consists numbers
Closure property consists
three numbers following below :
(i) Whole numbers : In whole numbers there have four operation
consists for closure property :-
a. Addition
In addition opertaion, 0 + 3 = 3, 4 + 5 = 9, is a whole number, so we can say
that whole numbers are closed in under addition. i.e., a + b
b. Subtraction
In Subtraction opertaion, 4 - 3 = 1 is a whole number, but 4 - 5 = -1, is not a
whole number, so we can say that whole numbers are not closed in under
Subtraction. i.e., a - b
c. Multiplication
In multiplication opertaion, 3 x 2 = 6, is a whole number, so we can say that
whole numbers are closed in under multiplication. i.e., a x b
d. Division
In division opertaion, 4 ÷ 3 = 4/3, is not a whole number, so we can say that
whole numbers are not closed in under division. i.e., a ÷ b
(ii) Rational numbers : In rational numbers there have four
operation consists for closure property :-
a. Addition
In addition opertaion, 1/2 + 3/5 = (5 + 6)/10 = 11/10, 4/5 + (-5/4) = (16 -
25)/20 = -(9/20), are a rational numbers, so we can say that rational numbers
are closed in under addition.
b. Subtraction
In Subtraction opertaion, 4/3 - 3/2 = (8 - 9)/6 = -(1/6) is a rational number,
so we can say that rational numbers are closed in under Subtraction.
c. Multiplication
In multiplication opertaion, 3/2 x 2/5 = 6/10, is a rational number, so we can
say that rational numbers are closed in under multiplication.
d. Division
In division opertaion, 4/3 ÷ 0 = ∞, is not a rational number, so we can say
that rational numbers are not closed in under division.
(iii) Integer numbers : In integer numbers there have four
operation consists for closure property :-
a. Addition
In addition opertaion, 4 + (-5) = -1, is an integer number, so we can say that
integer numbers are closed in under addition.
b. Subtraction
In Subtraction opertaion, -4 - 3 = -7 is an integer number, so we can say that
integer numbers are closed in under Subtraction.
c. Multiplication
In multiplication opertaion, (-3) x 2 = -6, is an integer number, so we can say
that integer numbers are closed in under multiplication.
d. Division
In division opertaion, 4 ÷ 5 = 4/5, is not an integer number, so we can say
that integer numbers are not closed in under division.
Commutativity property consists numbers
Commutativity property
consists three numbers following below :
(i) Whole numbers : In whole numbers there have four operation
consists for Commutativity property :-
a. Addition
In addition opertaion, 4 + 3 = 3 + 4, is a whole number, so we can say that
whole numbers are commutative in under addition. i.e., a + b = b + a.
b. Subtraction
In Subtraction opertaion, 4 - 3 ≠ 3 - 4, is not a whole number, so we can say
that whole numbers are not commutative in under Subtraction. i.e., a - b ≠ b -
a
c. Multiplication
In multiplication opertaion, 3 x 2 = 2 x 3, is a whole number, so we can say
that whole numbers are commutative in under multiplication. i.e., a x b = b x a
d. Division
In division opertaion, 4 ÷ 3 ≠ 3 ÷ 4, is not a whole number, so we can say that
whole numbers are not commutative in under division. i.e., a ÷ b ≠ b ÷ a
(ii) Rational numbers : In rational numbers there have four
operation consists for Commutativity property :-
a. Addition
In addition opertaion, 1/2 + 3/5 = 3/5 + 1/2, are a rational numbers, so we can
say that rational numbers are commutative in under addition.
b. Subtraction
In Subtraction opertaion, 4/3 - 3/2 ≠ 3/2 - 4/3 is not a rational number, so we
can say that rational numbers are not commutative in under Subtraction.
c. Multiplication
In multiplication opertaion, 3/2 x 2/5 = 2/5 x 3/2, is a rational number, so we
can say that rational numbers are commutative in under multiplication.
d. Division
In division opertaion, 4/3 ÷ 0 ≠ 0 ÷ 4/3 , is not a rational number, so we can
say that rational numbers are not commutative in under division.
(iii) Integer numbers : In integer numbers there have four
operation consists for commutativity property :-
a. Addition
In addition opertaion, 4 + (-5) = (-5) + 4, is an integer number, so we can say
that integer numbers are commutative in under addition.
b. Subtraction
In Subtraction opertaion, (-4) - 3 ≠ 3 - (-4) is not an integer number, so we
can say that integer numbers are not commutative in under Subtraction.
c. Multiplication
In multiplication opertaion, (-3) x 2 = 2 x (-3), is an integer number, so we can
say that integer numbers are commutative in under multiplication.
d. Division
In division opertaion, 4 ÷ 2 ≠ 2 ÷ 4, is not an integer number, so we can say
that integer numbers are not commutative in under division.
Associativity property consists numbers
Associativity property
consists three numbers following below :
(i) Whole numbers : In whole numbers there have four operation
consists for Associativity property :-
a. Addition
In addition opertaion, (4 + 3) + 2 = 4 + (3 + 2), is a whole number, so we can
say that whole numbers are associative in under addition. i.e., (a + b) + c = a
+ (b + c).
b. Subtraction
In Subtraction opertaion, (4 - 3) - 1 ≠ 4 - (3 - 1), is not a whole number, so
we can say that whole numbers are not associative in under Subtraction. i.e.,
(a - b) - c ≠ a - (b - c)
c. Multiplication
In multiplication opertaion, (3 x 2) x 4 = 3 x (2 x 4), is a whole number, so
we can say that whole numbers are associative in under multiplication. i.e., (a
x b) x c = a x (b x c)
d. Division
In division opertaion, (8 ÷ 4) ÷ 2 ≠ 8 ÷ (4 ÷ 2), is not a whole number, so we
can say that whole numbers are not associative in under division. i.e., (a ÷ b)
÷ c ≠ a ÷ (b ÷ c)
(ii)
Rational numbers : In rational numbers there have four operation consists for Associativity
property :-
a. Addition
In addition opertaion, (1/2 + 1/3) + 1/4 = 1/2 + (1/3 + 1/4), are a rational
numbers, so we can say that rational numbers are associative in under addition.
b. Subtraction
In Subtraction opertaion, (1/2 - 1/3) - 1/4 ≠ 1/2 - (1/3 - 1/4) is not a
rational number, so we can say that rational numbers are not associative in
under Subtraction.
c. Multiplication
In multiplication opertaion, (1/2 x 1/3) x 1/4 = 1/2 x (1/3 x 1/4), is a
rational number, so we can say that rational numbers are associative in under
multiplication.
d. Division
In division opertaion, (1/2 ÷ 1/3) ÷ 1/4 ≠ 1/2 ÷ (1/3 ÷ 1/4), is not a rational
number, so we can say that rational numbers are not associative in under
division.
(iii) Integer numbers : In integer numbers there have four
operation consists for Associativity property :-
a. Addition
In addition opertaion, [(–2) + 3)] + (– 4) = (–2) + [3 + (– 4)], is an integer
number, so we can say that integer numbers are associative in under addition.
b. Subtraction
In Subtraction opertaion,(5 – 7) – 3 ≠ 5 – (7 – 3) is not an integer number, so
we can say that integer numbers are not associative in under Subtraction.
c. Multiplication
In multiplication opertaion, [(– 4) × (– 8)] × (–5) = (– 4) × [(– 8) × (–5)],
is an integer number, so we can say that integer numbers are associative in
under multiplication.
d. Division
In division opertaion, [(–10) ÷ 2] ÷ (–5) ≠ (–10) ÷ [2 ÷ (– 5)], is not an
integer number, so we can say that integer numbers are not associative in under
division.
The role of zero (0)
Zero is called the identity
for the addition of rational numbers. It is the additive identity for integers
and whole numbers as well.
In general,
a + 0 = 0 + a = a, where a is a whole number. e.g., 2 + 0 = 0 + 2 = 2.
b + 0 = 0 + b = b, where b is an integer. e.g., (-2) + 0 = 0 + (-2) = -2.
c + 0 = 0 + c = c, where c is a rational number. e.g., 1/2 + 0 = 0 + 1/2 = 1/2.
The role of one (1)
1 is the multiplicative
identity for rational numbers.
In general,
a x 1 = 1 x a = a, where a is a whole number. e.g., 2 x 1 = 1 x 2 = 2.
b x 1 = 1 x b = b, where b is an integer. e.g., (-2) x 1 = 1 x (-2) = -2.
c x 1 = 1 x c = c, where c is a rational number. e.g., 1/2 x 1 = 1 x 1/2 = 1/2.
Negative of a number (additive inverse)
Here, -(a/b) is the additive
inverse of a/b and a/b is the additive inverse of -(a/b) or vice-versa.
In general, for an integer a, we have, a + (–a) = (–a) + a = 0; so, a is the
negative of –a and –a is the negative of a. e.g., 2 + (-2) = (-2) + 2 = 0.
In general, for a rational number a/b, we have a/b + (-a/b) = (-a/b) + a/b = 0.
e.g., 1/2 + (-1/2) = (-1/2) + 1/2 = 0
Reciprocal (multiplicative inverse)
p/q is the multiplicative
inverse or reciprocal of another rational numbers a/b if a/b x p/q = 1. And zero
(0) has no any reciprocal values.
In general,
e.g., 4/5 x 5/4 = 1 is a reciprocal values.
Distributivity of multiplication
For all rational numbers a,
b and c, a(b + c) = ab + ac and a(b – c) = ab – ac.
For addition, e.g., 1/2 x (1/3 + 1/4) = (1/2 x 1/3) + (1/2 x 1/4) = 1/6 + 1/8 =
7/24.
For Subtraction, e.g., 1/2 x (1/3 - 1/4) = (1/2 x 1/3) - (1/2 x 1/4) = 1/6 -
1/8 = 1/24.
Rational Numbers between Two Rational Numbers
Between any two given
rational numbers there are countless rational numbers. The idea of mean helps
us to find rational numbers between two rational numbers.
If a and b are two rational numbers, then (a + b)/2 is a rational number
between a and b such that a < (a + b)/2 < b.
Representation of number line
We have learnt to represent
natural numbers, whole numbers, integers and rational numbers on a number line
Natural number
The line extends
indefinitely only to the right side of 1.
Whole number
The line extends
indefinitely to the right, but from 0. There are no numbers to the left of 0.
Integer number
The line extends
indefinitely on both sides. Do you see any numbers between –1, 0; 0, 1 etc.
Rational number
The line extends
indefinitely on both sides. But you can now see numbers between –1, 0; 0, 1
etc.
Closure Property :-
1. Addition : a + b
2. Substraction : a - b
3. Multiplication : a x b
4. Division : a ÷ b
Commutative Property :-
1. Addition : a + b = b + a
2. Substraction : a - b = b -
a
3. Multiplication : a x b = b
x a
4. Division : a ÷ b = b ÷ a
Associative Property :-
1. Addition : (a + b) + c = a
+ (b + c)
2. Substraction : (a - b) - c
= a - (b - c)
3. Multiplication : (a x b) x
c = a x (b x c)
4. Division : (a ÷ b) ÷ c = a
÷ (b ÷ c)
Additive identity :-
1. a + 0 = 0 + a = a, where a
is a whole number
2. b + 0 = 0 + b = b, where b
is an integer
3. c + 0 = 0 + c = c, where c
is a rational number
Multiplicative identity :-
1. a x 1 = 1 x a = a, where a
is a whole number
2. b x 1 = 1 x b = b, where b
is an integer
3. c x 1 = 1 x c = c, where c
is a rational number
Additive & Multiplicative inverse :-
1. a + (– a) = (– a) + a = 0;
so, a is the negative of – a and – a is the negative of a.
2. a/b + (-a/b) = (-a/b) +
a/b = 0
3. Reciprocal : a/b x c/d = 1
Distributive Property :-
1. Addition : a(b + c) = ab +
ac
2. Substraction : a(b – c) =
ab – ac
Example-1 :- Find 3/7 + (-6/11) + (-8/21) + 5/22 .
Solution :-
3/7 + (-6/11) + (-8/21) + 5/22
L.C.M of 7, 11, 21, 22 is 462
Then, (198 - 252 - 176 + 105 )/462
= (303 - 428)/462
= -125/462
Example-2 :- Find -4/5 x 3/7 x 15/16 x (-14/9)
Solution :-
-4/5 x 3/7 x 15/16 x (-14/9)
= [(-4/5) x 3/7] x [15/16 x (-14/9)]
= (-12/35) x (-35/24)
= -12/24
= -1/2
Example-3 :- Write the additive inverse of the following:
(i) -7/19 (ii) 21/112
Solution :-
(i) -7/19
Example-1 :- Find 3/7 + (-6/11) + (-8/21) + 5/22 .
Solution :-
3/7 + (-6/11) + (-8/21) + 5/22
L.C.M of 7, 11, 21, 22 is 462
Then, (198 - 252 - 176 + 105 )/462
= (303 - 428)/462
= -125/462
Example-2 :- Find -4/5 x 3/7 x 15/16 x (-14/9)
Solution :-
-4/5 x 3/7 x 15/16 x (-14/9)
= [(-4/5) x 3/7] x [15/16 x (-14/9)]
= (-12/35) x (-35/24)
= -12/24
= -1/2
Example-3 :- Write the additive inverse of the following:
(i) -7/19 (ii) 21/112
Solution :-
(i) -7/19
Here, 7/19 is the additive inverse of -7/19
because -7/19 + 7/19 = 0
(ii) 21/112
Here, -21/112 is the additive inverse of
21/112 because -21/112 + 21/112 = 0
Example-4 :- Verify that – (– x) is the same as x for
(i) x = 13/17 (ii) x = -21/31
Solution :-
(i) x = 13/17
The additive inverse of x = 13/17 is -x =
-13/17
Then -13/17 + 13/17 = 0
Now, - (- x) = - (- 13/17) = 13/17
(i) x = -21/31
The additive inverse of x = -21/31 is -x =
21/31
Then -21/31 + 21/31 = 0
Now, - (- x) = - [- (-21/31] = -21/31
Example-5 :- Find 2/5 x (-3/7) - 1/14 - 3/7 x 3/5 .
Solution :-
2/5 x (-3/7) - 1/14 - 3/7 x 3/5
= 2/5 x (-3/7) - 3/7 x 3/5 - 1/14 [by commutativity]
= 2/5 x (-3/7) + (-3/7) x 3/5 - 1/14
= (-3/7)[2/5 + 3/5] - 1/14 [by distributivity]
= -3/7 x 1 - 1/14
= -3/7 - 1/14
= (-6 - 1)/14
= -7/14
= -1/2
Example-6 :- Write any 3 rational numbers between –2 and
0.
Solution :-
In these numbers, we can multiply and divide
by 10 in -2 and 0.
Then, -2 x 10/10 = -20/10 and 0 x 10/10 = 0
/10
Since, the numbers in between -20/10 and
0/10 are -19/10, -18/10, -17/10,.....-1/10.
So, we can take any 3 rational numbers in
between them numbers.
Example-7 :- : Find any ten rational numbers between -5/6
and 5/8 .
Solution :-
Firstly, we convert same denominator of the
both values.
Then, In -5/6 multiply and divide by 4. I
will get (-5 x 4)/(6 x 4) = -20/24.
Also, In 5/8 multiply and divide by 3. I
will get (5 x 3)/(8 x 3) = 15/24.
So, we can find that ten rational number in
-20/24, -19/24, -18/24,......15/24.
Example-8 :- Find a rational number between 1/4 and 1/2.
Solution :-
Firstly, We find the mean of the given
rational numbers. i.e., (a + b)/2
since, a = 1/4 and b = 1/2
(1/4 + 1/2)/2 = (1 + 2)/8 = 3/8
So, 3/8 lies in between 1/4 and 1/2.
Example-9 :- Find three
rational numbers between 1/4 and 1/2.
Solution :-
Firstly, We find the mean of the given
rational numbers. i.e., (a + b)/2
since, a = 1/4 and b = 1/2
(1/4 + 1/2)/2 = (1 + 2)/8 = 3/8
So, 3/8 lies in between 1/4 and 1/2. i.e.,
1/4 < 3/8 < 1/2
Secondly, we again find the mean in between
1/4 and 3/8.
since, a = 1/4 and b = 3/8
(1/4 + 3/8)/2 = (2 + 3)/16 = 5/16
So, 5/16 lies in between 1/4 and 3/8.
Finally, we again find the mean in between
3/8 and 1/2
(3/8 + 1/2)/2 = (3 + 4)/16 = 7/16
So, 7/16 lies in between 3/8 and 1/2.
Now, 5/16, 3/8, 7/16 are be three rational
numbers in between 1/4 and 1/2.
Here, 7/19 is the additive inverse of -7/19
because -7/19 + 7/19 = 0
(ii) 21/112
Here, -21/112 is the additive inverse of
21/112 because -21/112 + 21/112 = 0
Example-4 :- Verify that – (– x) is the same as x for
(i) x = 13/17 (ii) x = -21/31
Solution :-
(i) x = 13/17
The additive inverse of x = 13/17 is -x =
-13/17
Then -13/17 + 13/17 = 0
Now, - (- x) = - (- 13/17) = 13/17
(i) x = -21/31
The additive inverse of x = -21/31 is -x =
21/31
Then -21/31 + 21/31 = 0
Now, - (- x) = - [- (-21/31] = -21/31
Example-5 :- Find 2/5 x (-3/7) - 1/14 - 3/7 x 3/5 .
Solution :-
2/5 x (-3/7) - 1/14 - 3/7 x 3/5
= 2/5 x (-3/7) - 3/7 x 3/5 - 1/14 [by commutativity]
= 2/5 x (-3/7) + (-3/7) x 3/5 - 1/14
= (-3/7)[2/5 + 3/5] - 1/14 [by distributivity]
= -3/7 x 1 - 1/14
= -3/7 - 1/14
= (-6 - 1)/14
= -7/14
= -1/2
Example-6 :- Write any 3 rational numbers between –2 and 0.
Solution :-
In these numbers, we can multiply and divide
by 10 in -2 and 0.
Then, -2 x 10/10 = -20/10 and 0 x 10/10 = 0
/10
Since, the numbers in between -20/10 and
0/10 are -19/10, -18/10, -17/10,.....-1/10.
So, we can take any 3 rational numbers in
between them numbers.
Example-7 :- : Find any ten rational numbers between -5/6 and 5/8 .
Solution :-
Firstly, we convert same denominator of the
both values.
Then, In -5/6 multiply and divide by 4. I
will get (-5 x 4)/(6 x 4) = -20/24.
Also, In 5/8 multiply and divide by 3. I
will get (5 x 3)/(8 x 3) = 15/24.
So, we can find that ten rational number in
-20/24, -19/24, -18/24,......15/24.
Example-8 :- Find a rational number between 1/4 and 1/2.
Solution :-
Firstly, We find the mean of the given
rational numbers. i.e., (a + b)/2
since, a = 1/4 and b = 1/2
(1/4 + 1/2)/2 = (1 + 2)/8 = 3/8
So, 3/8 lies in between 1/4 and 1/2.
Example-9 :- Find three rational numbers between 1/4 and 1/2.
Solution :-
Firstly, We find the mean of the given
rational numbers. i.e., (a + b)/2
since, a = 1/4 and b = 1/2
(1/4 + 1/2)/2 = (1 + 2)/8 = 3/8
So, 3/8 lies in between 1/4 and 1/2. i.e.,
1/4 < 3/8 < 1/2
Secondly, we again find the mean in between
1/4 and 3/8.
since, a = 1/4 and b = 3/8
Example-1 :- Find 3/7 + (-6/11) + (-8/21) + 5/22 .
Solution :-
3/7 + (-6/11) + (-8/21) + 5/22
L.C.M of 7, 11, 21, 22 is 462
Then, (198 - 252 - 176 + 105 )/462
= (303 - 428)/462
= -125/462
Example-2 :- Find -4/5 x 3/7 x 15/16 x (-14/9)
Solution :-
-4/5 x 3/7 x 15/16 x (-14/9)
= [(-4/5) x 3/7] x [15/16 x (-14/9)]
= (-12/35) x (-35/24)
= -12/24
= -1/2
Example-3 :- Write the additive inverse of the following:
(i) -7/19 (ii) 21/112
Solution :-
(i) -7/19
Here, 7/19 is the additive inverse of -7/19
because -7/19 + 7/19 = 0
(ii) 21/112
Here, -21/112 is the additive inverse of
21/112 because -21/112 + 21/112 = 0
Example-4 :- Verify that – (– x) is the same as x for
(i) x = 13/17 (ii) x =
-21/31
Solution :-
(i) x = 13/17
The additive inverse of x = 13/17 is -x =
-13/17
Then
-13/17 + 13/17 = 0
Now, - (- x) = - (- 13/17) = 13/17
(i) x = -21/31
The additive inverse of x = -21/31 is -x =
21/31
Then -21/31 + 21/31 = 0
Now, - (- x) = - [- (-21/31] = -21/31
Example-5 :- Find 2/5 x (-3/7) - 1/14 - 3/7 x 3/5 .
Solution :-
2/5 x (-3/7) - 1/14 - 3/7 x 3/5
= 2/5 x (-3/7) - 3/7 x 3/5 - 1/14 [by commutativity]
= 2/5 x (-3/7) + (-3/7) x 3/5 - 1/14
= (-3/7)[2/5 + 3/5] - 1/14 [by distributivity]
= -3/7 x 1 - 1/14
= -3/7 - 1/14
= (-6 - 1)/14
= -7/14
= -1/2
Example-6 :- Write any 3 rational numbers between –2 and
0.
Solution :-
In these numbers, we can multiply and divide
by 10 in -2 and 0.
Then, -2 x 10/10 = -20/10 and 0 x 10/10 = 0
/10
Since, the numbers in between -20/10 and
0/10 are -19/10, -18/10, -17/10,.....-1/10.
So, we can take any 3 rational numbers in
between them numbers.
Example-7 :- : Find any ten rational numbers between -5/6
and 5/8 .
Solution :-
Firstly, we convert same denominator of the
both values.
Then, In -5/6 multiply and divide by 4. I
will get (-5 x 4)/(6 x 4) = -20/24.
Also, In 5/8 multiply and divide by 3. I
will get (5 x 3)/(8 x 3) = 15/24.
So, we can find that ten rational number in
-20/24, -19/24, -18/24,......15/24.
Example-8 :- Find a rational number between 1/4 and 1/2.
Solution :-
Firstly, We find the mean of the given
rational numbers. i.e., (a + b)/2
since, a = 1/4 and b = 1/2
(1/4 + 1/2)/2 = (1 + 2)/8 = 3/8
So, 3/8 lies in between 1/4 and 1/2.
Example-9 :- Find three
rational numbers between 1/4 and 1/2.
Solution :-
Firstly, We find the mean of the given
rational numbers. i.e., (a + b)/2
since, a = 1/4 and b = 1/2
(1/4 + 1/2)/2 = (1 + 2)/8 = 3/8
So, 3/8 lies in between 1/4 and 1/2. i.e.,
1/4 < 3/8 < 1/2
Secondly, we again find the mean in between
1/4 and 3/8.
since, a = 1/4 and b = 3/8
(1/4 + 3/8)/2 = (2 + 3)/16 = 5/16
So, 5/16 lies in between 1/4 and 3/8.
Finally, we again find the mean in between
3/8 and 1/2
(3/8 + 1/2)/2 = (3 + 4)/16 = 7/16
So, 7/16 lies in between 3/8 and 1/2.
Now, 5/16, 3/8, 7/16 are be three rational
numbers in between 1/4 and 1/2.
(1/4 + 3/8)/2 = (2 + 3)/16 = 5/16
So, 5/16 lies in between 1/4 and 3/8.
Finally, we again find the mean in between
3/8 and 1/2
(3/8 + 1/2)/2 = (3 + 4)/16 = 7/16
So, 7/16 lies in between 3/8 and 1/2.
Now, 5/16, 3/8, 7/16 are be three rational
numbers in between 1/4 and 1/2.
Example-1 :- Find 3/7 +
(-6/11) + (-8/21) + 5/22 .
Solution :-
3/7 + (-6/11) + (-8/21)
+ 5/22
L.C.M of 7, 11, 21, 22
is 462
Then, (198 - 252 - 176 +
105 )/462
= (303 - 428)/462
= -125/462
Example-2 :- Find -4/5 x
3/7 x 15/16 x (-14/9)
Solution :-
-4/5 x 3/7 x 15/16 x
(-14/9)
= [(-4/5) x 3/7] x [15/16
x (-14/9)]
= (-12/35) x (-35/24)
= -12/24
= -1/2
Example-3 :- Write the
additive inverse of the following:
(i) -7/19 (ii) 21/112
Solution :-
(i) -7/19
Here, 7/19 is the
additive inverse of -7/19 because -7/19 + 7/19 = 0
(ii) 21/112
Here, -21/112 is the
additive inverse of 21/112 because -21/112 + 21/112 = 0
Example-4 :- Verify that
– (– x) is the same as x for
(i) x = 13/17 (ii) x = -21/31
Solution :-
(i) x = 13/17
The additive inverse
of x = 13/17 is -x = -13/17
Then -13/17 + 13/17 =
0
Now, - (- x) = - (-
13/17) = 13/17
(i) x = -21/31
The additive inverse
of x = -21/31 is -x = 21/31
Then -21/31 + 21/31 =
0
Now, - (- x) = - [-
(-21/31] = -21/31
Example-5 :- Find 2/5 x
(-3/7) - 1/14 - 3/7 x 3/5 .
Solution :-
2/5 x (-3/7) - 1/14 -
3/7 x 3/5
= 2/5 x (-3/7) - 3/7 x
3/5 - 1/14 [by commutativity]
= 2/5 x (-3/7) + (-3/7) x
3/5 - 1/14
= (-3/7)[2/5 + 3/5] -
1/14 [by distributivity]
= -3/7 x 1 - 1/14
= -3/7 - 1/14
= (-6 - 1)/14
= -7/14
= -1/2
Example-6 :- Write any 3
rational numbers between –2 and 0.
Solution :-
In these numbers, we
can multiply and divide by 10 in -2 and 0.
Then, -2 x 10/10 =
-20/10 and 0 x 10/10 = 0 /10
Since, the numbers in
between -20/10 and 0/10 are -19/10, -18/10, -17/10,.....-1/10.
So, we can take any 3
rational numbers in between them numbers.
Example-7 :- : Find any
ten rational numbers between -5/6 and 5/8 .
Solution :-
Firstly, we convert
same denominator of the both values.
Then, In -5/6 multiply
and divide by 4. I will get (-5 x 4)/(6 x 4) = -20/24.
Also, In 5/8 multiply
and divide by 3. I will get (5 x 3)/(8 x 3) = 15/24.
So, we can find that
ten rational number in -20/24, -19/24, -18/24,......15/24.
Example-8 :- Find a
rational number between 1/4 and 1/2.
Solution :-
Firstly, We find the
mean of the given rational numbers. i.e., (a + b)/2
since, a = 1/4 and b =
1/2
(1/4 + 1/2)/2 = (1 +
2)/8 = 3/8
So, 3/8 lies in between
1/4 and 1/2.
Example-9 :- Find three rational numbers between 1/4 and 1/2.
Solution :-
Firstly, We find the
mean of the given rational numbers. i.e., (a + b)/2
since, a = 1/4 and b =
1/2
(1/4 + 1/2)/2 = (1 +
2)/8 = 3/8
So, 3/8 lies in between
1/4 and 1/2. i.e., 1/4 < 3/8 < 1/2
Secondly, we again find
the mean in between 1/4 and 3/8.
since, a = 1/4 and b =
3/8
(1/4 + 3/8)/2 = (2 +
3)/16 = 5/16
So, 5/16 lies in
between 1/4 and 3/8.
Finally, we again find
the mean in between 3/8 and 1/2
(3/8 + 1/2)/2 = (3 +
4)/16 = 7/16
So, 7/16 lies in
between 3/8 and 1/2.
Now, 5/16, 3/8, 7/16
are be three rational numbers in between 1/4 and 1/2.
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