RATIONAL NUMBERS :
RATIONAL NUMBERS Rational numbers are any numbers that can be expressed in the form of a/b , where a and b are integers, and b ≠ 0. They can always be expressed by using terminating decimals or repeating decimals. Example : 2/3, 6/7,1
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UNDER ADDITION UNDER SUBTRACTION UNDER MULTIPLICATION UNDER DIVISION RATIONAL NUMBERS ARE CLOSED UNDER ADDITION. FOR ANY TWO RATIONAL NUMBERS A AND B, A+B IS ALSO A RATIONAL NUMBER. FOR EXAMPLE: 8+3=11. RATIONAL NUMBERS ARE CLOSED UNDER SUBTRACTION. FOR ANY TWO RATIONAL NUMBERS A AND B, A-B IS ALSO A RATIONAL NUMBER. FOR EXAMPLE: 25-11=14. RATIONAL NUMBERS ARE CLOSED UNDER MULTIPLICATION. FOR ANY TWO RATIONAL NUMBERS A AND B, A*B IS ALSO A RATIONAL NUMBER. FOR EXAMPLE: 4*2=8. RATIONAL NUMBERS ARE NOT CLOSED UNDER DIVISION. FOR ANY RATIONAL NUMBER A , A/0 IS NOT DEFINED. FOR EXAMPLE: 6/0 IS NOT DEFINED. CLOSURE PROPERTY
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COMMUTATIVE PROPERTY UNDER ADDITION UNDER SUBTRACTION RATIONAL NUMBERS CAN BE ADDED IN ANY ORDER.THEREFORE, ADDITION IS COMMUTATIVE FOR RATIONAL NUMBERS. FOR EXAMPLE: -3/8 + 1/7 LCM = 56 -3/8 = -21/56, 1/7 = 8/56 = -21 + 8/56 = -13/56 1/7 + -3/8 LCM = 56 -3/8 = -21/56, 1/7 = 8/56 = 8 + (-21)/56 = -13/56. THEREFORE , -13/56 = -13/56. RATIONAL NUMBERS CANNOT BE SUBTRACTED IN ANY ORDER.THEREFORE, SUBTRACTION IS NOT COMMUTATIVE FOR RATIONAL NUMBERS. FOR EXAMPLE: 2/3 – 5/4 LCM = 12 2/3 = 8/12, 5/4 = 15/12 = 8 – 15/12 = -7/12 5/4 – 2/3 LCM = 12 2/3 = 8/12, 5/4 = 15/12 = 15 – 8/12 = 7/12 THEREFORE,-7/12 IS NOT EQUAL TO 7/12
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COMMUTATIVE PROPERTY UNDER MULTIPLICATION UNDER DIVISION RATIONAL NUMBERS CAN BE MULTIPLIEDIN ANY ORDER. THEREFORE IT IS SAID THAT MULTIPLICATION IS COMMUTATIVE FOR RATIONAL NUMBERS. FOR EXAMPLE: 7/3 * 6/5 = 7*6 / 3*5 = 42/15 6/5 * 7/3 = 6*7 / 5*3 = 42/15 THEREFORE, 42/15 = 42/15. RATIONAL NUMBERS CANNOT BE DIVIDED IN ANY ORDER. THEREFORE IT IS SAID THAT DIVISION IS NOT COMMUTATIVE FOR RATIONAL NUMBERS. FOR EXAMPLE: 2/3 / ¾ = 2/3 * 4/3 = 8/9 ¾ / 2/3 = ¾ * 3/2 = 9/8 THEREFORE, 8/9 IS NOT EQUAL TO 9/8.
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ASSOCIATIVE PROPERTY UNDER ADDITION UNDER SUBTRACTION UNDER MULTIPLICATION UNDER DIVISION Addition is associative for rational numbers. For any three rational numbers a, b and c, a + (b + c) = (a + b) + c. Subtraction is not associative for rational numbers. 2/3-(5/3-8/3) =(2-5-8)/3 =-11/3 (5-2-8)/3 =-5. Multiplication is associative for rational numbers. That is for any rational numbers a, b and c a* (b*c) = (a*b) * c. Division is not associative for Rational numbers. For example: ½ /(-1/3 /2/5) =-3/5 (1/2 /-1/3)/2/5 =-15/4.
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SOME MORE PROPERTIES DISTRIBUTI--VE LAW THE ROLE OF ZERO (0) ADDITIVE INVERSE THE ROLE OF ONE (1) For all rational numbers a, b and c, a ( b+c ) = ab + ac and a (b-c) = ab – ac. For Example : 4(2+6) = 4(8) = 32 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. Therefore, for any rational number a, a+0 = 0+a = a For Example - 2+0 = 0+2 = 2 -5+0 = 0+(-5) = -5 . 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).It is called the reciprocal. 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.
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