Rational Numbers

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Rational Numbers Maths Assignment Made By : Class - Viii A Aditya Sinha

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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 Natural numbers – They are counting numbers. 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. Rational Numbers

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Properties Of Rational 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. 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.

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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 = -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

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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: 

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

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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

Distributive Law: 

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

The Role Of Zero (0): 

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 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

Additive Inverse: 

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: 

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 Since, 5/6 + (-5/6) = 0 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 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

Acknowledgement: 

Acknowledgement I want to acknowledge Purnima mam, our mathematics teacher for her valuable contribution as she gave and explained me this assignment. Bibliography Sources of information : Class 8 th N.C.E.R.T. textbook. Internet (Google)