White all properties of rational number with 4 examples
Answers
Answer:
Closure Property
Commutative Property
Associative Property
Distributive Property
Identity Property
Inverse Property
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
For example:
Commutative law example
Subtraction is not commutative property i.e. a-b ≠ b-a. This can be understood clearly with the following example:
Commutative law - subtraction LHS
Whereas
Commutative law - subtraction RHS
The division is also not commutative i.e. a/b ≠ b/a, since,
Commutative law - Division LHS
Whereas,
Commutative law - Division RHS
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
Identity and Inverse Properties of Rational Numbers
Identity Property: 0 is an additive identity and 1 is a multiplicative identity for rational numbers.
Examples:
1/2 + 0 = 1/2 [Additive Identity]
1/2 x 1 = 1/2 [Multiplicative Identity]
Inverse Property: For a rational number x/y, the additive inverse is -x/y and y/x is the multiplicative inverse.
Examples:
The additive inverse of 1/3 is -1/3. Hence, 1/3 + (-1/3) = 0
The multiplicative inverse of 1/3 is 3. Hence, 1/3 x 3 = 1