Properties of Rational Numbers

Rational numbers can be expressed as a fraction p/q, where p and q are integers. However, q is not equal to zero (q ≠ 0). The set of rational numbers includes whole numbers, natural numbers, integers, and decimals (terminating and repeating decimals). Examples of rational numbers are:

•  1/2 
• -3/4
•  0.5 (1/2)
•  0.121212 (repeating decimal) 
•  46 (which can be written as 46/1)
   

Rational numbers follow several properties, such as: 

• Closure Property 
• Commutative Property
• Associative Property
• Distributive Property
• Multiplicative Property
• Additive Property

These six properties help us solve arithmetic operations accurately and efficiently. 
 

According to the closure property, if we add, subtract, or multiply two rational numbers, the results will also be rational numbers. Now, let’s discuss how the closure property is applied to the four arithmetic operations, such as addition, subtraction, multiplication, and division.   

For addition: If a and b are two rational numbers, (a + b) = rational number. 
For example, add 1/4 and 2/3. 
First, we need to find a common denominator since the denominators are different. The least common denominator of 4 and 3 is 12. Convert the unlike denominators to a common denominator. 
1/4 = 1 × 3 / 4 × 3 = 3/12
2/3 = 2 × 4 / 3 × 4 = 8/12 
Now, add the fractions:
3/12 + 8/12 = 11/12 
Thus, 1/4 + 2/3 = 11/12 
11/12 is a rational number, and it is closed under addition. 

For subtraction: If a and b are two rational numbers, (a - b) = rational number. 
For instance, subtract 1/4 and 2/3.
Since the denominators are different, we must first find a common denominator. The LCD of 4 and 3 is 12. 
Convert the unlike denominators to a common denominator.
1/4 = 1 × 3 / 4 × 3 = 3/12
2/3 = 2 × 4 / 3 × 4 = 8/12 
Next, subtract the fractions:
3/12 - 8/12 = -5/12
Therefore, 1/4 - 2/3 = -5/12 
-5/12 is a rational number, and it is closed under subtraction. 

For multiplication: If a and b are two rational numbers, (a × b) = rational number. 
For example, multiply 1/4 and 2/3. 
First, multiply the numerators together.
    1 × 2 = 2
Next, the denominators:
    4 × 3 = 12
Finally, write the fraction as: 
  2/12 
The fraction 2/12 can be simplified to 1/6.
Hence, 1/4 × 2/3 = 1/6.
1/6 is a rational number, and it is closed under multiplication. 

For division: If we divide any number by zero, the result will be undefined. 
 a ÷ 0 = undefined. 
On the other hand, if we divide any number by another non-zero rational number, then the result will be a rational number. Rational numbers are closed under division except when they are divided by zero. For instance, divide 1/4 and 2/3.
When we divide two fractions, we take the reciprocal of the second fraction and multiply it by the first fraction. 
Here, 2/3 will become 3/2.
1/4 ÷ 2/3 = 1/4 × 3/2 = 3/8
The fraction 3/8 is a rational number. 
 

According to the commutative property, if we add or multiply two rational numbers in any order, the result will be the same. However, if we subtract or divide any two rational numbers, and the order of the numbers is changed, the result will also change. 

For addition: If a and b are two rational numbers, a + b = b + a.
For example, add 1/2 and 2/3.
First, we need to find a common denominator since the denominators are different. The least common denominator of 2 and 3 is 6. Convert the unlike denominators to a common denominator. 
1/2 = 1 × 3 / 2 × 3 = 3/6
2/3 = 2 × 2 / 3 × 2 = 4/6 
Now, add the fractions:
4/6 + 3/6 = 7/6
3/6 + 4/6 = 7/6
Thus, 1/2 + 2/3 = 7/6
The addition of rational numbers is commutative. 

For subtraction: If a and b are two rational numbers, a - b ≠ b - a.
For instance, subtract 1/2 and 2/3.
Since the denominators are different, we must first find a common denominator. The LCD of 2 and 3 is 6. 
Convert the unlike denominators to a common denominator.
1/2 = 1 × 3 / 2 × 3 = 3/6
2/3 = 2 × 2 / 3 × 2 = 4/6 
Next, subtract the fractions: 
3/6 - 4/6 = -1/6 
4/6 - 3/6 = 1/6 
Thus, 3/6 - 4/6 ≠ 4/6 - 3/6 
Subtraction is not commutative for rational numbers. 

For multiplication: If a and b are two rational numbers, a ×  b = b × a.
For instance, multiply 1/2 and 2/3.
First, multiply the numerators together.
    1 × 2 = 2
Next, the denominators:
    2 × 3 = 6
Finally, write the fraction as: 
  2/6  
The fraction 2/6 can be simplified to 1/3.
1/2 × 2/3 = 2/3 × 1/2 
The multiplication of rational numbers is commutative. 

For division: If a and b are two rational numbers, a ÷ b ≠ b ÷ a.
For example, divide 1/2 and 2/3.
When we divide two fractions, we take the reciprocal of the second fraction and multiply it by the first fraction. 
1/2 ÷ 2/3 = 1/2 × 3/2 = 3/4 
2/3 ÷ 1/2 = 2/3 × 2/1 = 4/3 
Therefore, 1/2 ÷ 2/3 ≠ 2/3 ÷ 1/2 
The division of rational numbers is not commutative. 
 

According to the associative property of rational numbers, if we add or multiply three rational numbers, the result will be the same even if we change the order of the numbers. However, if we subtract or divide three rational numbers, the result will vary if the order of the numbers is changed.  

For addition: If a, b, and c are three rational numbers, (a + b) + c = a + (b + c).
For example, add 1/2, 1/4, and 1/6. 
(1/2 + 1/4) + 1/6 = 1/2 + (1/4 + 1/6)
Here, we start with the first two numbers: 
 1/2 + 1/4 
To add these two numbers, we need to find a common denominator. 
The LCM of 2 and 4 is 4. 
1/2 = 1 × 2 / 2 × 2 = 2/4 
1/4 = 1/4; the denominator is already 4. 
Now, add the two fractions: 
2/4 + 1/4 = 3/4 
Hence, the expression becomes 3/4 + 1/6.
 Next, find the LCD of 4 and 6. 
The LCD of 4 and 6 = 12.
3/4 = 3 × 3 / 4 × 3 = 9/12 
1/6 = 1 × 2 / 6 × 2 = 2/12 
Add the two fractions: 
9/12 + 2/12 = 11/12 

Next, we can solve the right side, 1/2 + (1/4 + 1/6)
     = 1/4 + 1/6 
The LCD of 4 and 6 is 12. 
1/4 = 1 × 3 / 4 × 3 = 3/12  
1/6 = 1 × 2 / 6 × 2 = 2/12 
Next, add the two fractions: 
3/12 + 2/12 = 5/12 
Then, add 1/2 + 5/12 
The LCM of 2 and 12 is 12. 
1/2 = 1 × 6 / 2 × 6 = 6/12  
5/12 = 5/12 
Next, add the two fractions: 
6/12 + 5/12 = 11/12 
Here, both sides are equal, so the addition of three rational numbers is associative. 
(1/2 + 1/4) + 1/6 = 1/2 + (1/4 + 1/6) = 11/12 

For subtraction: If a, b, and c are three rational numbers, (a - b) - c ≠ a - (b - c).
The subtraction of three rational numbers is not associative.
For example, (1/2 - 1/4) - 1/6 ≠ 1/2 - (1/4 - 1/6) 
First, solve the left side: 
(1/2 - 1/4) - 1/6 
= 1/4 - 1/6 
The LCD of 4 and 6 is 12.
1/4 = 3/12 
1/6 = 2/12 
Now, subtract these two fractions: 
3/12 - 2/12 = 1/12 
(1/2 - 1/4) - 1/6 = 1/12 

Next, the right side: 
1/2 - (1/4 - 1/6)  
= 1/2 - 1/12
LCD of 2 and 12 is 12.
1/2 = 6/12 
1/12 = 1/12 
Subtract the two fractions: 
6/12 - 1/12 = 5/12 

Hence, 1/12 ≠ 5/12        
(1/2 - 1/4) - 1/6 ≠ 1/2 - (1/4 - 1/6) 

For multiplication: If a, b, and c are three rational numbers, (a × b) × c = a × (b × c).
The multiplication of three rational numbers is associative. 
For instance, (1/2 × 1/4) × 1/6 = 1/2 × (1/4 × 1/6)

Left side: 
(1/2 × 1/4) × 1/6 
= 1/8 × 1/6 = 1/48 

Right side: 
1/2 × (1/4 × 1/6)
= 1/2 × 1/24 = 1/48 

Both sides are equal, so multiplication is associative. 

For division: If a, b, and c are three rational numbers, (a ÷ b) ÷ c ≠ a ÷ (b ÷ c).
For example, (1/2 ÷ 1/4) ÷ 1/6 ≠ 1/2 ÷ (1/4 ÷ 1/6)

First, we can solve the left side: 
(1/2 ÷ 1/4) ÷ 1/6 
= 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2 
Then, 2 ÷ 1/6
2 ÷ 1/6 = 2 × 6 = 12 

Next, we can solve the right side: 
1/2 ÷ (1/4 ÷ 1/6)
= 1/4 ÷ 1/6 = 1/4 ÷ 6/1 
= 6/4 can be simplified to 3/2
Then, 1/2 ÷ 3/2
1/2 ÷ 2/3 = 1/3 
Hence, (1/2 ÷ 1/4) ÷ 1/6 ≠ 1/2 ÷ (1/4 ÷ 1/6) 
12 ≠ 1/3  
Therefore, the division of three rational numbers is not associative. 
 

This property is also known as the distributive property of multiplication over addition or subtraction. If three rational numbers are added, such as a, b, and c, it can be solved as a × (b + c) = ab + ac. 
When three rational numbers are subtracted, it can be solved as a × (b - c) = ab - ac.

For addition: If a, b, and c are three rational numbers, a × (b + c) = ab + ac. 
For instance, 2/3 × (1/2 + 1/4)
According to the distributive property of rational numbers:
2/3 × (1/2 + 1/4) = (2/3 × 1/2) + ( 2/3 × 1/4)
Next, multiply the fractions: 
(2/3 × 1/2) + ( 2/3 × 1/4)
= 2/6 + 2/12 
Here, we can simplify the fractions.
2/6 = 1/3 
2/12 = 1/6 
Hence, the expression becomes: 
1/3 + 1/6 
For that, first find the LCD of 3 and 6.
The LCD of 3 and 6 is 6. 
1/3 = 1 × 2 / 3 × 2 = 2/6
1/6 = 1/6 
Now, add the two fractions: 
2/6 + 1/6 = 3/6 
3/6 can be simplified to 1/2.

For subtraction: If a, b, and c are three rational numbers, a × (b - c) = ab - ac.
For instance, 2/3 × (1/2 - 1/4)
According to the distributive property of rational numbers:
2/3 × (1/2 + 1/4) = (2/3 × 1/2) - ( 2/3 × 1/4)
Next, multiply the fraction:
(2/3 × 1/2) - ( 2/3 × 1/4) 
= 2/6 - 2/12 
The simplified fractions of 2/6 and 2/12 are:
= 1/3 - 1/6 
Next, find the LCD of 3 and 6.
6 is the LCD of 3 and 6. 
1/3 = 1 × 2 / 3 × 2 = 2/6
1/6 = 1/6 
Now, subtract the two fractions:
2/6 - 1/6 = 1/6
 

The multiplicative identity property and multiplicative inverse property are the two important multiplicative properties of rational numbers. 

Multiplicative identity property: According to this property, if we multiply any rational number by 1, the product will be the rational number itself. So, the multiplicative identity for rational numbers is 1. 
If a/b is a rational number, then if we multiply a/b by 1:
a/b × 1 = 1 × a/b = a/b. 
For example, 3/5 × 1 = 1 × 3/5 = 3/5.

Multiplicative inverse property: If a/b is a rational number (where a and b ≠ 0), and its multiplicative inverse is b/a. Every rational number, except 0, has a reciprocal, and their product is 1. The reciprocal of a rational number is its multiplicative inverse. 
  a/b × b/a = 1
For example, 3/2 × 2/3 = 1 

On the other hand, if we multiply any rational number by 0, the product will be 0. 
For instance, 3/5 × 0 = 0  
 

The two additive properties of rational numbers are the additive identity property and the additive inverse property. 

Additive identity property: According to the additive identity property, if we add zero to any rational number, the sum will be the same rational number itself. If a/b is a rational number, 
a/b + 0 = 0 + a/b = a/b. 
So, 0 is the additive identity for rational numbers. 
For example, 4/3 + 0 = 0 + 4/3 = 4/3. 

Additive inverse property: According to this property, for a rational number a/b, there is an inverse number (-a/b); then, if we add these two numbers, a/b + (-a/b) = (-a/b) + a/b = 0. 
For instance, 2/3 is a rational number, and -2/3 is the additive inverse. 
If we add: 
    2/3 + -2/3 = 0
 

Understanding the properties of rational numbers helps us perform accurate calculations in our daily lives. We use rational numbers when dealing with measurements, money, shopping, and cooking. Here are some real-world applications of the properties of rational numbers listed below:  

• We use the closure property when we add, subtract, or multiply rational numbers, and it ensures we get a rational number as a result. For example, if we run 1/2 kilometer in the morning and 3/4 kilometer in the evening. So, we can add the rational numbers to find the total kilometers we cover:
  1/2 + 3/4 = 5/4 kilometers 

• When we go shopping, we can use the commutative property to calculate the total cost of various items we buy. In the commutative property, the order does not affect the result. For instance, if we buy a water bottle for $15 and a lunch box for $20: 
  Total cost = 15 + 20 = 35 = 20 + 15
  The total cost is $35.

• In banking and billing, the identity property helps us maintain values unchanged. For example, if we win a lottery of $25,000, and it did not cost any money, the final amount remains: 
  25,000 + 0 = 25,000

• The properties of rational numbers help us make better financial decisions, find the total cost of bulk items, and help us solve mathematical problems easily and accurately.