mention commutativity, associativity and closure propertie of rational number
Answers
Answer:
For two rational numbers say x and y the results of addition, subtraction and multiplication operations give a rational number. We can say that rational numbers are closed under addition, subtraction and multiplication.
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Closure property
(7/6)+(2/5) = 47/30.
(5/6) – (1/3) = 1/2.
(2/5). (3/7) = 6/35.
Answer:(i) Closure Property :
The sum of any two rational numbers is always a rational number. This is called ‘Closure property of addition’ of rational numbers. Thus, Q is closed under addition
If a/b and c/d are any two rational numbers, then
(a/b) + (c/d) is also a rational number
Example :
2/9 + 4/9 = 6/9 = 2/3 is a rational number
(ii) Commutative Property :
Addition of two rational numbers is commutative.
If a/b and c/d are any two rational numbers, then
(a/b) + (c/d) = (c/d) + (a/b)
Example :
2/9 + 4/9 = 6/9 = 2/3
4/9 + 2/9 = 6/9 = 2/3
So,
2/9 + 4/9 = 4/9 + 2/9
(iii) Associative Property :
Addition of rational numbers is associative.
If a/b, c/d and e/f are any three rational numbers, then
a/b + (c/d + e/f) = (a/b + c/d) + e/f
Example :
2/9 + (4/9 + 1/9) = 2/9 + 5/9 = 7/9
(2/9 + 4/9) + 1/9 = 6/9 + 1/9 = 7/9
So,
2/9 + (4/9 + 1/9) = (2/9 + 4/9) + 1/9
(iv) Additive Identity :
The sum of any rational number and zero is the rational number itself.
If a/b is any rational number, then
a/b + 0 = 0 + a/b = a/b
Zero is the additive identity for rational numbers.
Example :
2/7 + 0 = 0 + 2/7 = 2/7
(v) Additive Inverse :
(-a/b) is the negative or additive inverse of (a/b).
If a/b is a rational number,then there exists a rational number (-a/b) such that
a/b + (-a/b) = (-a/b) + a/b = 0
Example :
Additive inverse of 3/5 is (-3/5).
Additive inverse of (-3/5) is 3/5.
Additive inverse of 0 is 0 itself.
Subtraction
(i) Closure Property :
The difference between any two rational numbers is always a rational number.
Hence Q is closed under subtraction.
If a/b and c/d are any two rational numbers, then
(a/b) - (c/d) is also a rational number.
Example :
5/9 - 2/9 = 3/9 = 1/3 is a rational number.
(ii) Commutative Property :
Subtraction of two rational numbers is not commutative.
If a/b and c/d are any two rational numbers, then
(a/b) - (c/d) ≠ (c/d) - (a/b)
Example :
5/9 - 2/9 = 3/9 = 1/3
2/9 - 5/9 = -3/9 = -1/3
And,
5/9 - 2/9 ≠ 2/9 - 5/9
Therefore, Commutative property is not true for subtraction.
(iii) Associative Property :
Subtraction of rational numbers is not associative.
If a/b, c/d and e/f are any three rational numbers, then
a/b - (c/d - e/f) ≠ (a/b - c/d) - e/f
Example :
2/9 - (4/9 - 1/9) = 2/9 - 3/9 = -1/9
(2/9 - 4/9) - 1/9 = -2/9 - 1/9 = -3/9
And,
2/9 - (4/9 - 1/9) ≠ (2/9 - 4/9) - 1/9
Therefore, Associative property is not true for subtraction.
Multiplication
(i) Closure Property :
The product of two rational numbers is always a rational number. Hence Q is closed under multiplication.
If a/b and c/d are any two rational numbers, then
(a/b) x (c/d) = ac/bd is also a rational number.
Example :
5/9 x 2/9 = 10/81 is a rational number.
(ii) Commutative Property :
Multiplication of rational numbers is commutative.
If a/b and c/d are any two rational numbers, then
(a/b) x (c/d) = (c/d) x (a/b)
Example :
5/9 x 2/9 = 10/81
2/9 x 5/9 = 10/81
So,
5/9 x 2/9 = 2/9 x 5/9
Therefore, Commutative property is true for multiplication.
(iii) Associative Property :
Multiplication of rational numbers is associative.
If a/b, c/d and e/f are any three rational numbers, then
a/b x (c/d x e/f) = (a/b x c/d) x e/f
Example :
2/9 x (4/9 x 1/9) = 2/9 x 4/81 = 8/729
(2/9 x 4/9) x 1/9 = 8/81 x 1/9 = 8/729
So,
2/9 x (4/9 x 1/9) = (2/9 x 4/9) x 1/9
Therefore, Associative property is true for multiplication.
(iv) Multiplicative Identity :
The product of any rational number and 1 is the rational number itself. ‘One’ is the multiplicative identity for rational numbers.
If a/b is any rational number, then
a/b x 1 = 1 x a/b = a/b
Example :
5/7 x 1 = 1 x 5/7 = 5/7
(v) Multiplication by 0 :
Every rational number multiplied with 0 gives 0.
If a/b is any rational number, then
a/b x 0 = 0 x a/b = 0
Example :
5/7 x 0 = 0 x 5/7 = 0