# VERIFYING GIVEN PROPERTIES OF RATIONAL NUMBERS EXAMPLES

## About "Verifying Given Properties of Rational Number"

Verifying Given Properties of Rational Number :

Here we are going to see how to verify the given properties of rational numbers.

## Verifying Given Properties of Rational Numbers Examples - Practice questions

Question 1 :

Verify the closure property for addition and multiplication of the rational numbers −5/7 and 8/9

Solution :

Let a = -5/7 and b = 8/9

a + b  =  b + a

Closure property for multiplication :

a x b  =  b x a

(i)  a + b  =  (-5/7) + (8/9)

=  (-45 + 56)/63

=  11/63  ----(1)

b + a  =  (8/9) + (-5/7)

=  (56 - 45)/63

=  11/63 ----(2)

(1)  =  (2)

(ii)  a x b  =  (-5/7) x (8/9)

=  -40/63 -----(1)

b x a  =  (8/9) x (-5/7)

=  -40/63 ----(2)

(1)  =  (2)

Question 2 :

Verify the associative property for addition and multiplication of the rational numbers

−10/11, 5/6, -4/3

Solution :

Addition and multiplication are associative for rational numbers.That is, for any three rational numbers a,b and c,

(i) (a + b) + c = a + (b + c) and

(ii) (a × b) × c = a × (b × c)

a = -10/11, b = 5/6 and c = -4/3

(i)

L.H.S :

a + b  =  (-10/11) + (5/6)  =  (-60 + 55)/66  =  -5/66

(a + b) + c  =  (-5/66) + (-4/3) =  (-5-88)/66

=  93/66  =  31/22  ---(1)

R.H.S :

b + c  =  (5/6) + (-4/3)  =  (5 - 8)/6  =  -3/6  =  -1/2

a + (b + c)  =  (-10/11) + (-1/2) =  (-20-11)/22

=  31/22  ---(2)

(1)  =  (2)

Hence associative property for addition is true.

(ii) (a × b) × c = a × (b × c)

L.H.S :

a x b  =  (-10/11) x (5/6)  =  -25/33

(a x b) x c  =  (-25/33) x (-4/3)  =  100/99  -----(1)

R.H.S :

(b × c)  =  (5/6) x (-4/3)  =  -10/9

a x (b x c)  =  (-10/11) x (-10/9)  =  100/99 ----(2)

(1)  =  (2)

Hence associative property for multiplication is true.

Question 2 :

Check the commutative property for addition and multiplication of the rational numbers −10/11 and −8/33

Solution :

(i)  Commutative property for addition :

a + b  = b + a

(ii)  Commutative property for multiplication :

a x b  =  b x a

(i) a = -10/11 and b = -8/33

a + b  =  (-10/11) + (-8/33)

=  (-30 - 8)/33

=  -38/33  ----(1)

b + a  =  (-8/33) + (-10/11)

=  (-8 - 30)/33

=  -38/33  ----(2)

(1)  =  (2)

Hence proved.

(ii)

a x b  =  (-10/11) x (-8/33)

=  80/363-----(1)

b x a  =  (-8/33) x (-10/11)

=  80/363 ------(2)

(1)  =  (2) After having gone through the stuff given above, we hope that the students would have understood, "Verifying Given Properties of Rational Numbers Examples"

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