Write the closure,commutative,associative and distributive properties of rational number. In short answer
Answers
Answer:
Subtraction and division is not commutative with respect to rational numbers.
For eg.
2
1
−
3
1
=
3
1
−
2
1
.
For eg. 5÷3
=3÷5
So, option B and D is correct.
Answer:
Closure property
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. For example:
(7/6)+(2/5) = 47/30
(5/6) – (1/3) = 1/2
(2/5). (3/7) = 6/35
Do you know why division is not under closure property?
The division is not under closure property because division by zero is not defined. We can also say that except ‘0’ all numbers are closed under division.
Commutative Property
For rational numbers, addition and multiplication are commutative.
Commutative law of addition: a+b = b+a
Commutative law of multiplication: a×b = b×a
Associative Property
Rational numbers follow the associative property for addition and multiplication.
Suppose x, y and z are rational, then for addition: x+(y+z)=(x+y)+z
For multiplication: x(yz)=(xy)z.
Example: 1/2 + (1/4 + 2/3) = (1/2 + 1/4) + 2/3
⇒ 17/12 = 17/12
And in case of multiplication;
1/2 x (1/4 x 2/3) = (1/2 x 1/4) x 2/3
⇒ 2/24 = 2/24
⇒1/12 = 1/12
Distributive Property
The distributive property states, if a, b and c are three rational numbers, then;
a x (b+c) = (a x b) + (a x c)
Example: 1/2 x (1/2 + 1/4) = (1/2 x 1/2) + (1/2 x 1/4)
LHS = 1/2 x (1/2 + 1/4) = 3/8
RHS = (1/2 x 1/2) + (1/2 x 1/4) = 3/8
Hence, proved
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