0In 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
