Question

if rational numbers satisfies closures commutive and associative properties under addition and subtraction explain with ​

Answer 1

Answer:

What are the properties of rational numbers?

The word rational has evolved from the word ratio. In general, rational numbers are those numbers that can be expressed in the form of p/q, in which both p and q are integers and q≠0. The properties of rational numbers are:

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

Answer 2

Answer:

closure property of rational numbers:

it satisfies addition , subtraction and multiplication but not under division

Step-by-step explanation:

examples: addition = 2+3=5(rational numbers)

                subtraction= 2-3= -1(rational number)

                 multiplication= 2*3= 6(rational number)

                 division=20/3=6.666and so on (not a rational number)

plzzzzzzz plzzzz mark me the brainliest if it helps plzzzzzzzzzzz