Verifying Given Properties of Rational Number :
Here we are going to see how to verify the given properties of rational numbers.
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
Closure property for addition :
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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