Math, asked by choudharynancy410, 2 months ago


White all properties of rational number with 4 examples

Answers

Answered by rishikatripurarirt
0

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

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