Rational numbers are defined as integers in the form of p/q where q is not zero and can be added, subtracted, and multiplied with closure, but division is not closed. Addition and multiplication are commutative and associative for rational numbers, while subtraction and division are neither commutative nor associative. The document also discusses properties like the additive inverse and the reciprocal of rational numbers.

1. Rational Numbers
Class VIII

2. The integers which are in the form of
p/q where q is not equal to 0 are
known as Rational Numbers.
Examples : 5/8; -3/14; 7/-15; -6/-11

3.  Rational numbers are closed under addition. That is, for any two rational
numbers a and b, a+b s also a rational number.
For Example - 8 + 3 = 11 ( a rational number. )
 Rational numbers are closed under subtraction. That is, for any two rational
numbers a and b, a – b is also a rational number.
For Example - 25 – 11 = 14 ( a rational number. )
 Rational numbers are closed under multiplication. That is, for any two
rational numbers a and b, a * b is also a rational number.
For Example - 4 * 2 = 8 (a rational number. )
 Rational numbers are not closed under division. That is, for any rational
number a, a/0 is not defined.
For Example - 6/0 is not defined.

4. • Rational numbers can be added in any order.
Therefore, addition is commutative for rational
numbers. For Example :-
• Subtraction is not commutative for rational
numbers. For Example -
Since, -7 is unequal to 7
Hence, L.H.S. Is unequal to R.H.S.
Therefore, it is proved that Subtraction is not commutative for rational numbers.
L.H.S. R.H.S.
- 3/8 + 1/7
L.C.M. = 56
= -21+8
= -13
1 /7 +(-3/8)
L.C.M. = 56
= 8+(-21)
= -13
L.H.S. R.H.S.
2/3 – 5/4
L.C.M. = 12
= 8 – 15
= -7
5/4 – 2/3
L.C.M. = 12
= 15 – 8
= 7

5. • 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

6.  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
 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.
ASSOCIATIVE PROPERTY
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
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

7. DISTRIBUTIVE LAW
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

8. Additive Inverse
 Additive inverse is also known as negative of a
number.
For any rational number a/b, a/b+(-a/b)= (-a/b)+a/b = 0
Therefore, -a/b is the additive inverse of a/b and a/b is
the
Additive Inverse of (-a/b).
Reciprocal
 Rational number c/d is called the reciprocal or
Multiplicative Inverse of another rational number a/b
if a/b * c/d = 1