A whole number or the quotient of any whole numbers, excluding zero as a denominator A whole number or the quotient of any whole numbers, excluding zero as a denominator Examples - 5/8; -3/14; 7/-15; -6/-11 Examples - 5/8; -3/14; 7/-15; -6/-11 Natural numbers – They are counting numbers. Natural numbers – They are counting numbers. Integers - Natural numbers, their negative and 0 form the system of integers. Integers - Natural numbers, their negative and 0 form the system of integers. Fractional numbers – the positive integer which are in the form of p/q where q is not equal to 0 are known as fractional numbers. Fractional numbers – the positive integer which are in the form of p/q where q is not equal to 0 are known as fractional numbers.

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. 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 = 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.

Commutative Property 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 = = /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

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

Associative property Addition is associative for rational numbers. Addition is associative for rational numbers. That is for any three rational numbers a, b and c, a + (b + c) = (a + b) + c. That is for any three rational numbers a, b and c, a + (b + c) = (a + b) + c. For Example For Example Since, -9/10 = -9/10 Hence, L.H.S. = R.H.S. Therefore, the property has been proved. 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. L.H.S. R.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-[-4/5-1/2] = -2/3 + 13/10 = /30 = 19/30 [2/3-(-4/5)]-1/2 = 22/15 – ½ = 44 – 15/30 = 29/30 = 29/30

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 Rational numbers. For Example – Since, Hence, L.H.S. Is Not equal to R.H.S. equal to R.H.S. L.H.S. L.H.S. R.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. L.H.S. R.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

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

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. 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 Therefore, for any rational number a, a+0 = 0+a = a For Example = 0+2 = 2 For Example = 0+2 = = 0+(-5) = = 0+(-5) = -5 The role of one (1) The role of one (1) 1 is the multiplicative identity for rational numbers. 1 is the multiplicative identity for rational numbers. Therefore, a*1 = 1*a = a for any rational number a. Therefore, a*1 = 1*a = a for any rational number a. For Example - 2*1 = 2 For Example - 2*1 = 2 1*-10 = -10 1*-10 = -10

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

Some Problems On Rational Numbers Q 1 ) Verify that –(-x) is the same as x for x = 5/6 A 1 ) The additive inverse of x = 5/6 = -x = -5/6 of x = 5/6 = -x = -5/6 Since, 5/6 + (-5/6) = 0 Since, 5/6 + (-5/6) = 0 Hence, -(-x) = x. Hence, -(-x) = x. Q 2 ) Find any four rational numbers between -5/6 and 5/8 A 2 ) Convert the given numbers to rational numbers with same denominators : -5*4-6*4 = -20/24 5*3/8*3 = 15/24 -5*4-6*4 = -20/24 5*3/8*3 = 15/24 Thus, we have -19/24; -18/24; /24; 14/24 Thus, we have -19/24; -18/24; /24; 14/24 Any four rational numbers can be chosen. Any four rational numbers can be chosen. L.H.S. L.H.S. R.H.S. R.H.S. -(-5/6) = +5/6 = 5/6 5/6 = + 5/6 = 5/6