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Rational numbers and its properties

Namrata P.
02 Jul 0 0

In Mathematics a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational Number.

Properties of addition of rational numbers:

1) closure property

2)commutative property

3)associative property

4)existence of additive identity property

5) existence of additive inverse property of addition of rational numbers.

 

Closure property of addition of rational numbers: The sum of two rational numbers is always a rational number. 

If a/b and c/d are any two rational numbers, then (a/b + c/d) is also a rational number. 


For example:

(i) Consider the rational numbers 1/3 and 3/4 Then, 

(1/3 + 3/4) 

= (4 + 9)/12

= 13/12, is a rational number 

(ii) Consider the rational numbers -5/12 and -1/4 Then, 

(-5/12 + -1/4) 

= {-5 + (-3)}/12

= -8/12 

= -2/3, is a rational number

(iii) Consider the rational numbers -2/3 and 4/5 Then, 

(-2/3 + 4/5) 

= (-10 + 12)/15 

= 2/15, is a rational number



Commutative property of addition of rational numbers:

Two rational numbers can be added in any order. 

Thus for any two rational numbers a/b and c/d, we have

(a/b + c/d) = (c/d + a/b) 

For example: 


(i) (1/2 + 3/4) 

= (2 + 3)/4

=5/4 

and (3/4 + 1/2) 

= (3 + 2)/4

= 5/4

Therefore, (1/2 + 3/4) = (3/4 + 1/2) 

(ii) (3/8 + -5/6) 


= {9 + (-20)}/24 

= -11/24

and (-5/6 + 3/8) 

= {-20 + 9}/24

= -11/24

Therefore, (3/8 + -5/6) = (-5/6 + 3/8) 

(iii) (-1/2 + -2/3) 

= {(-3) + (-4)}/6 

= -7/6

and (-2/3 + -1/2) 

= {(-4) + (-3)}/6

= -7/6

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


Associative property of addition of rational numbers:

While adding three rational numbers, they can be grouped in any order. 

Thus, for any three rational numbers a/b, c/d and e/f, we have 

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

For example:

Consider three rationals -2/3, 5/7 and 1/6 Then, 

{(-2/3 + 5/7) + 1/6} = {(-14 + 15)/21 + 1/6} = (1/21 + 1/6) = (2 + 7)/42

= 9/42 = 3/14

and {(-2/3 + (5/7 + 1/6)} = {-2/3 + (30 + 7)/42} = (-2/3 + 37/42)

= (-28 + 37)/42 = 9/42 = 3/14

Therefore, {(-2/3 + 5/7) + 1/6} = {-2/3 + (5/7 + 1/6)} 


Existence of additive identity property of addition of rational numbers:

0 is a rational number such that the sum of any rational number and 0 is the rational number itself. 

Thus, (a/b + 0) = (0 + a/b) = a/b, for every rational number a/b

0 is called the additive identity for rationals. 


For example: 

(i) (3/5 + 0) = (3/5 + 0/5) = (3 + 0)/5 = 3/5 and similarly, (0 + 3/5) = 3/5

Therefore, (3/5 + 0) = (0 + 3/5) = 3/5

(ii) (-2/3 + 0) = (-2/3 + 0/3) = (-2 + 0)/3 = -2/3 and similarly, (0 + -2/3)

= -2/3

Therefore, (-2/3 + 0) = (0 + -2/3) = -2/3



Existence of 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) = {a + (-a)}/b = 0/b = 0 and similarly, (-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


For example:

(4/7 + -4/7) = {4 + (-4)}/7 = 0/7 = 0 and similarly, (-4/7 + 4/7) = 0

Thus, 4/7 and -4/7 are additive inverses of each other

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