The integers which are in the form of p/q where q The integers which are in the form of p/q where q is not equal to 0 are known as rational numbers. is not equal to 0 are known as rational numbers.

Examples - 5/8; -3/14; 7/-15; -6/-11Examples - 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 Integers - Natural numbers, their negative and 0 form the system of integers.form the system of integers.

Fractional numbers – the positive integer which are Fractional numbers – the positive integer which are in the form of p/q where q is not equal to 0 are in the form of p/q where q is not equal to 0 are known as fractional numbers. known as fractional numbers.

Closure PropertyClosure Property Rational numbers are closed under addition. That is, for any two 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 a and b, a+b s also a rational number. For Example - 8 + 3 = 11 ( a rational number. )For Example - 8 + 3 = 11 ( a rational number. )Rational numbers are closed under subtraction. That is, for any Rational numbers are closed under subtraction. That is, for any two rational numbers a and b, a – b is also a rational number,two rational numbers a and b, a – b is also a rational number,For Example - 25 – 11 = 14 ( a rational number. )For Example - 25 – 11 = 14 ( a rational number. )Rational numbers are closed under multiplication. That is, for any Rational numbers are closed under multiplication. That is, for any two rational numbers a and b, a * b is also a rational number.two rational numbers a and b, a * b is also a rational number.For Example - 4 * 2 = 8 (a rational number. )For Example - 4 * 2 = 8 (a rational number. )Rational numbers are not closed under division. That is, for any Rational numbers are not closed under division. That is, for any rational number a, a/0 is not defined.rational number a, a/0 is not defined.For Example - 6/0 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

= -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

Rational numbers can be Rational numbers can be multiplied in any order. Therefore, it is multiplied in any order. Therefore, it is said that multiplication is said that multiplication is commutative for rational numbers.commutative for rational numbers.For Example – For Example – Since, L.H.S = R.H.S.Since, L.H.S = R.H.S.Therefore, it is proved that rational Therefore, it is proved that rational numbers can be multiplied in any numbers can be multiplied in any order.order.

Rational numbers can not be divided Rational numbers can not be divided in any order.Therefore,division is not in any order.Therefore,division is not commutative for rational numbers.commutative for rational numbers.For Example – For Example – Since, L.H.S. is not equal to R.H.S.Since, L.H.S. is not equal to R.H.S.Therefore, it is proved that rational Therefore, it is proved that rational numbers can not be divided in any numbers can not be divided in any order.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 propertyAssociative 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 That is for any three rational numbers a, b and c, a + (b

+ c) = (a + b) + c.+ c) = (a + b) + c. For ExampleFor ExampleSince, -9/10 = -9/10Since, -9/10 = -9/10Hence, L.H.S. = R.H.S.Hence, L.H.S. = R.H.S.Therefore, the propertyTherefore, the property has been proved.has been proved.

Subtraction is not associative for rational numbers. Subtraction is not associative for rational numbers. For Example -For Example -Since, 19/30 is not equal toSince, 19/30 is not equal to29/3029/30Hence, L.H.S. is not equal Hence, L.H.S. is not equal to R.H.S. Therefore, the to R.H.S. Therefore, the property has been proved.property has been proved.

L.H.S.L.H.S. R.H.S.R.H.S.

-2/3+[3/5+(-5/6)]-2/3+[3/5+(-5/6)]

= -2/3+(-7/30)= -2/3+(-7/30)

= -27/30= -27/30

= -9/10= -9/10

[-2/3+3/5]+(-[-2/3+3/5]+(-5/6)5/6)

=-1/15+(-5/6)=-1/15+(-5/6)

=-27/30=-27/30

=-9/10=-9/10

-2/3-[-4/5-1/2]-2/3-[-4/5-1/2]

= -2/3 + = -2/3 + 13/1013/10

=-20 +39 /30=-20 +39 /30

= 19/30= 19/30

[2/3-(-4/5)]-1/2[2/3-(-4/5)]-1/2

= 22/15 – ½= 22/15 – ½

= 44 – 15/30= 44 – 15/30

= 29/30= 29/30

Multiplication is associative for rational numbers. That is Multiplication is associative for rational numbers. That is for any rational numbers a, b and cfor any rational numbers a, b and ca* (b*c) = (a*b) * c a* (b*c) = (a*b) * c For Example –For Example –Since, -5/21 = -5/21Since, -5/21 = -5/21Hence, L.H.S. = R.H.SHence, L.H.S. = R.H.S

Division is not associativeDivision is not associative for Rational numbers.for Rational numbers.For Example – For Example – Since,Since,Hence, L.H.S. Is NotHence, 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* (5/4*2/7)

= -2/3 * 10/28= -2/3 * 10/28

= -2/3 * 5/14= -2/3 * 5/14

= -10/42= -10/42

= -5/21= -5/21

(-2/3*5/4) * 2/7(-2/3*5/4) * 2/7

= -10/12 * 2/7= -10/12 * 2/7

= -5/6 * 2/7= -5/6 * 2/7

= -10/42= -10/42

= -5/21= -5/21

L.H.S.L.H.S. R.H.S.R.H.S.

½ / (-1/3 / 2/5)½ / (-1/3 / 2/5)

= ½ / -5/6= ½ / -5/6

= -6/10= -6/10

= -3/5= -3/5

[½ / (-1/2)] / 2/5[½ / (-1/2)] / 2/5

= -1 / 2/5= -1 / 2/5

= -5/2= -5/2

= -5/2= -5/2

Distributive LawDistributive Law

Distributivity of multiplication over Distributivity of multiplication over addition and subtraction :addition and subtraction :

For all rational numbers a, b and c,For all rational numbers a, b and c,

a (b+c) = ab + aca (b+c) = ab + ac

a (b-c) = ab – aca (b-c) = ab – ac

For Example – For Example –

Since, L.H.S. = R.H.S.Since, L.H.S. = R.H.S.

Hence, distributive law is proved 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)= 4 (8)

= 32= 32

4*2 + 4*2 + 4*64*6

= 8 = 24= 8 = 24

= 32= 32

The Role Of Zero (0)The Role Of Zero (0)

Zero is called the identity for the addition of rational Zero is called the identity for the addition of rational numbers. It is the additive identity for integers and numbers. It is the additive identity for integers and whole numbers as well.whole numbers as well.

Therefore, for any rational number a, a+0 = 0+a = aTherefore, for any rational number a, a+0 = 0+a = a For Example - 2+0 = 0+2 = 2 For Example - 2+0 = 0+2 = 2 -5+0 = 0+(-5) = -5 -5+0 = 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 = 2For Example - 2*1 = 2 1*-10 = -10 1*-10 = -10

WORK SHEETWORK SHEET

QQ11)) Verify that –(-x) is the same as x for x = 5/6Verify that –(-x) is the same as x for x = 5/6AA11) The additive inverse) The additive inverse of x = 5/6 = -x = -5/6of x = 5/6 = -x = -5/6 Since, 5/6 + (-5/6) = 0Since, 5/6 + (-5/6) = 0 Hence, -(-x) = x.Hence, -(-x) = x.

QQ22) Find any four rational numbers between -5/6 and 5/8) Find any four rational numbers between -5/6 and 5/8 AA22) Convert the given numbers to rational numbers with ) Convert the given numbers to rational numbers with

same denominators :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; ........13/24; 14/24Thus, we have -19/24; -18/24; ........13/24; 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

5/65/6

= + 5/6= + 5/6

= 5/6= 5/6

Qn.Find ten rational numbers between and .

Ans. And can be represented as respectively.

Therefore, ten rational numbers between and are

Qn.Represent on the number line.

Ans. can be represented on the number line as follows.