In this chapter we will discuss important properties of addition of rational numbers with examples.
The following properties are discussed in this chapter;
(a) Closure Property
(b) Commutative Property
(c) Associative Property
(d) Distributive Property
(e) Additive Property
(f) Additive Inverse Property
We will discuss all these property one by one.
Addition Property of rational numbers
Important properties of addition of rational number has been discussed below with examples.
Closure Property
The property states that ” addition of two rational numbers result in another rational number“.
For example;
Let 2/3 and 6/3 are the rational numbers.
The addition of these numbers is expressed as;
\mathtt{\Longrightarrow \ \frac{2}{3} \ +\ \frac{6}{3}}\\\ \\ \mathtt{\Longrightarrow \frac{2+6}{3} \ }\\\ \\ \mathtt{\Longrightarrow \frac{8}{3}}
The final result is 8/3 which is also a rational number.
Example 02
Add \mathtt{\frac{1}{5} \ +\ \frac{2}{7}}
Solution
Adding the given rational numbers.
\mathtt{\Longrightarrow \ \frac{1}{5} \ +\ \frac{2}{7}}\\\ \\ \mathtt{\Longrightarrow \frac{7+10}{35} \ }\\\ \\ \mathtt{\Longrightarrow \frac{17}{35}}
The final result is 17 / 35 which is also a rational number.
Conclusion
Hence, the sum of two rational numbers is always a rational number.
Commutative Property
The Commutative property of rational number states that changing the position of rational number in addition will not change the final result.
So if (a/b) & (c/d) are two rational numbers then;
(a/b) + (c/d) = (c/d) + (a/b)
For example;
Consider the rational numbers \mathtt{\frac{13}{11} \ \&\ \ \frac{5}{11}}
Adding the numbers we get;
\mathtt{\frac{13}{11} \ +\ \frac{5}{11} \ \Longrightarrow \ \frac{18}{11}}
Now interchanging the position of numbers.
\mathtt{\frac{5}{11} \ +\ \frac{13}{11} \ \Longrightarrow \ \frac{18}{11}}
Hence, interchanging the position of rational number in addition will not change the final result.
Associative property
It states that during addition, changing the group of rational number will not affect the end result.
If a/b, c/d & d/e are the three rational numbers. Then according to the additive property.
( a/b + c/d ) + d/e = a/b + ( c/d + d/e )
For Example;
Let 2/3, 5/3 and 7/3 are the given rational numbers.
Then according to associative property.
(2/3 + 5/3) + 7/3 = 2/3 + (5/3 + 7/3)
7/3 + 7/3 = 2/3 + 12/3
14/3 = 14/3
Hence Proved.
Distributive Property
The multiplication of rational number with sum of two rational number is equal to sum of product of rational numbers.
If a/b, c/d & d/e are three rational numbers.
Then according to distributive property.
a/b ( c/d + d/e ) = a/b . c/d + a/b . d/e
Let’s prove the distributive property with the help of example.
Given are three rational numbers 1/5, 4/5 and 3/5.
According to distributive property;
\mathtt{\frac{1}{5} \ \left(\frac{4}{5} +\frac{3}{5}\right) =\ \frac{1}{5} .\frac{4}{5} \ +\ \frac{1}{5} .\frac{3}{5} \ }\\\ \\ \mathtt{\frac{1}{5} \ .\frac{7}{5} \ \ =\frac{4}{25} +\frac{3}{25}}\\\ \\ \mathtt{\frac{7}{25} \ =\ \frac{7}{25}}
Since, LHS = RHS.
We have proved the distributive property of addition of rational number.
Additive Property
The property states that ” when we add any rational number with 0 we get the same rational number “
Hence by adding with number 0, the identity of rational number will be preserved.
a/b + 0 = a/b
Additive Inverse Property
The number which addition with rational number produce 0 result is called additive inverse.
If a/b is the rational number then -a/b is the additive inverse.
a/b + (-a/b) = 0
Hence, for any rational number if we insert negative sign in the front it becomes its additive inverse.
Example 01
Find the additive inverse of 2/3
-2/3 is the additive inverse as it adds up with 2/3 to produce 0.
⟹ 2/3 + (-2/3)
⟹ 0
Example 02
Find additive inverse of -6/7
The additive inverse is;
⟹ – (-6/7)
⟹ 6/7
If we add this inverse with original number, we will get 0.
-6/7 + 6/7 = 0