Rational numbers are numbers that can be written as fractions p/q, where p and q are integers and q is not equal to 0. Rational numbers have important properties: 1) They are closed under addition and multiplication, meaning the sum or product of two rational numbers is also rational. 2) Operations like addition and multiplication are commutative and associative, following standard order of operations rules. 3) They follow the distributive property, where multiplying a number times the sum of two other numbers equals the sum of multiplying each number individually. 4) Each rational number has an additive inverse (its negative) and a multiplicative inverse (its reciprocal), such that adding/multiplying a number by its inverse results

1. Rational Numbers
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2. Definition of Rational Numbers
 The integers which are in the form of p/q where q ≠ 0 are known as Rational Numbers.
 Examples : 5/8; -3/14; 7/-15; -6/-11

3. Properties of Rational Numbers
1) Closure Property
2) Associative Property
3) Distributive Law
4) Additive Inverse
5) Multiplicative Inverse

4. Closure Property
 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 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. )

5. Commutative Property
Addition
 Rational numbers can be added in any order. Therefore, addition is
commutative for rational numbers.
 For Example :- -3/8 + 1/7 = 1/7 + (-3/8)
(-21 + 8 )/56 = (8 – 21)/56
-13/56 = -13/56

6. Subtraction
 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
- 3/7 – 1/7 ≠ 3/7 – (- 1/7 )

7. Multiplication
 Rational numbers can be multiplied in any order. Therefore, it is said that
multiplication is commutative for rational numbers.
 For Example : -7/3*6/5 = 6/5 * (-7/3)
-14/5 = -14/5

8. Associative Property
Addition
 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 : 2 + (5 + 3) = (2 + 5) + 3
2 + 8 = 7 + 3
10 = 10

9. Multiplication
 Multiplication is associative for rational numbers. That is for any rational
numbers a, b and c :
a* (b*c) = (a*b) * c
 For Example : 2 * (5*3) = (2*5) *3
2*15 = 10*3
30 = 30

10. Distributive Law
 For all rational numbers a, b and c,
a (b+c) = ab + ac
a (b-c) = ab – ac
For Example : 2(5+3) = 2*5 + 2*3
2*8 = 10 + 6
16 = 16
2(5-3) = 2*5 - 2*3
2*2 = 10 - 6
4 = 4

11. 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).

12. Multiplicative Inverse
 Multiplicative inverse is also known as reciprocal number. For any rational
number a/b,
a/b * b/a = 1

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