Question

Prepare a chart explaining which properties (closure, commutative and associative) are satisfied by which type of numbers.

Answer 1

Step-by-step explanation:

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