Operations on Rational Numbers

A number that can be written as a fraction is a rational number. Any fraction where the denominator is not zero qualifies as a rational number. Rational numbers are expressed in the form p/q where q ≠ 0. There are various types of numbers that we can represent as rational numbers:

 

Fraction form of a rational number: A rational number consists of two integers and can be written in the form p/q.

 

Decimal form of a rational number: A rational number can be written in the form of a decimal number if the value is terminated or has recurring digits after the decimal point. For example, 0.33333 is a rational number.

 

Standard form of a rational number: A rational number is expressed as p/q, where p and q are integers with no common factor other than 1. For example, 3/9 is a rational number, but is not in the standard form as both the numerator and denominator have a common factor of 3, and can be further simplified to 1/3. Therefore, the standard form is 1/3.

 

To identify whether a number is rational or not, here are a few properties that we can use to identify the rational numbers:

 

• All natural numbers, whole numbers, fractions, and integers are rational numbers.
  
   
• Any decimal number that terminates is a rational number.
  
   
• A decimal number that keeps recurring is also a rational number.
  
   
• A rational number should be expressed as p/q, where p and q are integers, and q ≠ 0. 
   

When we solve for rational numbers, the operations we use are usually addition, subtraction, division, and multiplication. We know that a rational number is expressed in the form p/q. Here, we will explain each operation of rational numbers in detail:

We add rational numbers similar to how we add fractions. When adding rational numbers, there are usually two cases:

 

• Adding rational numbers with like denominators
  
   
• Adding rational numbers with different denominators 

When we add two rational numbers with common denominators, we simply need to add the numerators and keep the same denominator. When rational numbers have different denominators, certain steps need to be followed:

 

Step 1: Since the denominators are different, we need to find the least common denominator.


Step 2: Find the rational number equivalent with the common denominator.


Step 3: Since the denominators are the same, we just need to add the numerators and use the same denominator.
 

The method of subtracting rational numbers is similar to the method of addition of rational numbers.

 

Step 1: Find the LCM of the denominators. 


Step 2: Rewrite the numbers using a common denominator.


Step 3: Subtract the numbers.

Multiplying rational numbers is similar to multiplying fractions. Here are a few steps to multiply any two rational numbers:

 

Step 1: We first multiply the numerators


Step 2: Multiply the denominators


Step 3: Simplify the resulting number to its lowest form. 
 

When we want to divide any two rational numbers, we see how many parts of the divisors are in the dividend. Below are the steps used to divide rational numbers:

 

Step 1: The reciprocal of the divisor should be taken (the second rational number)


Step 2: Multiply it by the dividend.


Step 3: The solution is the product of these two numbers.
 

Here are some properties that we can apply to operations on rational numbers:


 

We use rational numbers in various fields by researchers and engineers. Here are a few real-life applications of operations on Rational Numbers:

 

• Financial transactions: We use rational numbers in variable fields, including research and engineering. 
  
   
• Construction: Workers use measurements for cutting wood and mixing cement, which often involves fractions.
  
   
• Navigation and travel: Fuel efficiency is measured in miles per gallon or kilometers per liter. Similar time and distance calculations involve adding or subtracting fractions, such as estimating travel time.