By Anuradha

1. Rational numbers can be
multiplied in any order. Therefore, it is
said that multiplication is
commutative for rational numbers.
For Example –
Since, L.H.S = R.H.S.
Therefore, it is proved that rational
numbers can be multiplied in any
order.
Rational numbers can not be divided
in any order.Therefore,division is not
commutative for rational numbers.
For Example –
Since, L.H.S. is not equal to R.H.S.
Therefore, it is proved that rational
numbers can not be divided in any
order.
L.H.S. R.H.S.
-7/3*6/5 =
-42/15
6/5*(7/3) =
-42/15
L.H.S. R.H.S.
(-5/4) / 3/7
= -5/4*7/3
= -35/12
3/7 / (-5/4)
= 3/7*4/-5
= -12/35

2. Associative property
Addition is associative for rational numbers.
That is for any three rational numbers a, b and c, a + (b + c) =
(a + b) + c.
For Example
Since, -9/10 = -9/10
Hence, L.H.S. = R.H.S.
Therefore, the property
has been proved.
Subtraction is not associative for rational numbers.
For Example -
Since, 19/30 is not equal to
29/30
Hence, L.H.S. is not equal
to R.H.S. Therefore, the
property has been proved.
L.H.S. R.H.S.
-2/3+[3/5+(-5/6)]
= -2/3+(-7/30)
= -27/30
= -9/10
[-2/3+3/5]+(-5/6)
=-1/15+(-5/6)
=-27/30
=-9/10
-2/3-[-4/5-1/2]
= -2/3 + 13/10
=-20 +39 /30
= 19/30
[2/3-(-4/5)]-1/2
= 22/15 – ½
= 44 – 15/30
= 29/30

3. Multiplication is associative for rational numbers. That is
for any rational numbers a, b and c
a* (b*c) = (a*b) * c
For Example –
Since, -5/21 = -5/21
Hence, L.H.S. = R.H.S
Division is not associative
for Rational numbers.
For Example –
Since,
Hence, L.H.S. Is Not
equal to R.H.S.
L.H.S. R.H.S.
-2/3* (5/4*2/7)
= -2/3 * 10/28
= -2/3 * 5/14
= -10/42
= -5/21
(-2/3*5/4) * 2/7
= -10/12 * 2/7
= -5/6 * 2/7
= -10/42
= -5/21
L.H.S. R.H.S.
½ / (-1/3 / 2/5)
= ½ / -5/6
= -6/10
= -3/5
[½ / (-1/2)] / 2/5
= -1 / 2/5
= -5/2
= -5/2

4. Distributive Law
Distributivity of multiplication over addition and
subtraction :
For all rational numbers a, b and c,
a (b+c) = ab + ac
a (b-c) = ab – ac
For Example –
Since, L.H.S. = R.H.S.
Hence, distributive law is proved
L.H.S. R.H.S.
4 (2+6)
= 4 (8)
= 32
4*2 + 4*6
= 8 + 24
= 32

5. DISTRIBUTIVE PROPERTY
 Distributivity of multiplication over
addition for rational number :
 For all rational numbers
 a, b and c, a(b + c) = ab + ac

6. DISTRIBUTIVE PROPERTY

7. DISTRIBUTIVE PROPERTY
Distributivity of multiplication
over subtraction for rational
number:
 For any three rational numbers
a, b and c,
 a (b – c) = ab – ac

8. DISTRIBUTIVE PROPERTY

9. 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.
Therefore,
for any rational number a,
a+0 = 0+a = a
For Example -
2+0 = 0+2 = 2
-5+0 = 0+(-5) = -5

10. The role of one (1)
 1 is the multiplicative identity for rational
numbers.
Therefore,
a*1 = 1*a = a
for any rational number a.
For Example -
2*1 = 2
1*-10 = -10

11. RATIONAL NUMBERS
Property Addition Multiplication Subtraction Division
1. Commutative Property x + y = y+ x x × y = y × x x – y ≠ y – x x ÷ y ≠ y ÷ x
2. Associative Property x + (y + z) = (x + y) +z x × (y × z) = (x × y) × z (x – y) – z ≠ x – (y – z) (x ÷ y) ÷ z ≠ x ÷ (y ÷ z)
3. Identity Property x + 0 = x =0 + x x × 1 = x = 1 × x x – 0 = x ≠ 0 – x x ÷ 1 = x ≠ 1 ÷ x
4. Closure Property x + y ∈ Q x × y ∈ Q x – y ∈ Q x ÷ y ∉ Q
5. Distributive Property x × (y + z) = x × y + x× z
x × (y − z) = x × y − x × z

12. WORK SHEET
Q1) Verify that –(-x) is the same as x for x = 5/6
A1) The additive inverse
of x = 5/6 = -x = -5/6
Since, 5/6 + (-5/6) = 0
Hence, -(-x) = x.
Q2) Find any four rational numbers between -5/6 and 5/8
A2) Convert the given numbers to rational numbers with same
denominators :
-5*4-6*4 = -20/24 5*3/8*3 = 15/24
Thus, we have -19/24; -18/24; ........13/24; 14/24
Any four rational numbers can be chosen.
L.H.S. R.H.S.
-(-5/6)
= +5/6
= 5/6
5/6
= + 5/6
= 5/6

13. Qn.Find ten rational numbers between and .
Ans. And can be represented as respectively.
Therefore, ten rational numbers between and are
Qn.Represent on the number line.
Ans. can be represented on the number line as follows.