Question

write all properties of addition of rational numbers​

Answer 1

Answer:

We will learn the properties of addition of rational numbers i.e. closure property, commutative property, associative property, existence of additive identity property and existence of additive inverse property of addition of rational numbers.

Answer 2

Properties of Addition of Rational Numbers

Closure Property of Addition of Rational Numbers. ...

Commutative Property of Addition of Rational Numbers. ...

Associative Property of Addition of Rational Numbers. ...

Additive Identity Property of Addition of Rational Numbers. ...

Additive Inverse Property of Addition of Rational Numbers.

Examples

(i) Consider the Rational Numbers 5/4 and 1/3

= (5/4+1/3)

= (5*3 +1*4)/12

= (15+4)/12

= 19/12

Therefore, the Sum of Rational Numbers 5/4 and 1/3 i.e. 19/12 is also a Rational Number.

(ii) Consider the Rational Numbers -4/3 and 2/5

= - 4/3+2/5

= (-4*5+2*3)/15

= (-20+6)/15

= -14/15 is also a Rational Number.

Commutative Property of Addition of Rational Numbers

Commutative Property is applicable for the Addition Operation of Rational Numbers. Two Rational Numbers can be added in any order. Let us consider two rational numbers a/b, c/d then we have

(a/b+c/d) = (c/d+a/b)

Examples

(i) 1/3+4/5

= (5+12)/15

= 17/15

and 4/5+1/3

= (12+5)/15

= 17/15

Therefore, (1/3+4/5) = (4/5+1/3).

(ii) -1/2+3/2

= (-1+3)/2

= 2/2

and 3/2+(-1/2)

= (3-1)/2

= 2/2

Therefore, (-1/2+3/2) = (3/2+-1/2)

Associative Property of Addition of Rational Numbers

While adding Three Rational Numbers you can group them in any order. Let us consider three Rational Numbers a/b, c/d, e/f we have

(a/b+c/d)+e/f = a/b+(c/d+e/f)

Example

Consider Three Rational Numbers 1/2, 3/4 and 5/6 then

(1/2+3/4)+5/6 = (2+3)/4+5/6

= 5/4+5/6

= (15+10)/12

25/12 and

1/2+(3/4+5/6) = 1/2+(9+10)/12

= 1/2+19/12

= (6+19)/12

=25 /12

Therefore, (1/2+3/4)+5/6 = 1/2+(3/4+5/6)

Additive Identity Property of Addition of Rational Numbers

0 is a Rational Number and any Rational Number added to 0 results in a Rational Number.

For every Rational Number a/b (a/b+0)=(0+1/b)= a/b and 0 is called the Additive Identity for Rationals.

Example

(i) (4/5+0) = (4/5+0/5) =(4+0)/5 =4/5 and similarly (0+4/5) = (0/5+4/5) = (0+4)/5 = 4/5

Therefore, (4/5+0) = (0+4/5) = 4/5

(ii)(-1/3+0) =(-1/3+0/3) =(-1+0)/3 = -1/3 and similarly (0-1/3) = (0/3-1/3) =(0-1)/3 = -1/3

Therefore, (-1/3+0) =(0+-1/3) = -1/3

Additive Inverse Property of Addition of Rational Numbers

For every Rational Number a/b there exists a Rational Number -a/b such that (a/b+-a/b)=0 and (-a/b+a/b)=0

Thus, (a/b+-a/b) = (-a/b+a/b) = 0

-a/b is called the Additive Inverse of a/b

Example

(4/3+-4/3) = (4+(-4))/3 = 0/3 = 0

Similarly, (-4/3+4/3) = (-4+4)/3 = 0/3 = 0

Thus, 4/3 and -4/3 are additive inverse of each other.