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Rational Numbers

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Rational Numbers PPT for Class VIII, Properties of Rational Numbers: Commutative, Distributive, Associative, etc.

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Rational Numbers

1. 1. Rational Numbers Class VIII
2. 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. 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. 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. 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. 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. 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. 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
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