Rational Number: Introduction

We use basic math frequently in daily life, whether when figuring out your age, determining the temperature, or simply adding the bills. While doing so, we use different types of numbers that we already understand, such as natural numbers, whole numbers, integers, etc. Now, as you move to higher classes, you’ll learn more about numbers, and one crucial topic you’ll learn is Rational numbers. So, what is a rational number, and what are their properties? Read on to find out. 

What are Rational Numbers?

In mathematics, any number can be written as p/q, where q ≠ 0 is considered a rational number. Additionally, every fraction that has an integer denominator and numerator and a denominator that is not zero falls into the category of rational numbers. The outcome of dividing a rational number, or fraction, will be a decimal number, either a terminating decimal or a repeating decimal. It is hard to tell fractions from rational numbers when the numbers are expressed in the p/q form. Unlike the numerator and denominator of rational numbers, which are made up of integers, fractions are composed of whole numbers. Examine the following conditions to determine whether a number is rational or not.

• A number that is expressed in the form of p/q, where q ≠ 0.
• Further simplification and representation of the ratio p/q in decimal form are possible.

Rational Numbers Examples

As mentioned above, a number is rational if it can be written as a fraction with both integers in the numerator and denominator. Some examples of rational numbers include:

• 0.3 or 3/10
• 0.141414… or 14/99
• -0.7 or -7/10
• 1/2
• 2/3

Types of Rational Numbers

There are two types of rational numbers:

1. Positive rational numbers – If a rational number’s numerator and denominator are positive, the number is said to be positive. A rational number is said to be positive if both its numerator and denominator are positive integers or both are negative integers. In other words, if both the denominator and numerator have the same sign, it is defined as positive rational numbers. 

Examples: 2/9 , 7/9, 5/7, -3/-5.

The top and bottom can both be multiplied by -1 even if the numerator and denominator of the rational numbers are negative. Since the sum of two negative signs equals a positive sign, the numerator and denominator are once more both positive integers.

2. Negative rational numbers – If a rational number’s numerator or the denominator is negative, the number is said to be negative. 

Examples: –2/7, –0.5 are some examples of negative rational numbers. Here, 10.5 can also be written as -1/2.

If the denominator includes a negative integer, the numerator and denominator can both be multiplied by -1 to bring the minus sign to the top. 

Here’s a table that could sum up the characteristics of positive and negative rational numbers. 

Properties of Rational Numbers

The subset of real numbers that adheres to all of the real number system’s features is known as a rational number. The following are a few of the crucial properties of rational numbers:

• Any two rational numbers can be multiplied, added to, subtracted from, or divided to always produce a rational number only.
• When the numerator and the denominator are the same integer, the rational number remains constant.
• Any rational number will provide the same number as the result when we add zero to it.
• When doing subtraction, addition, and multiplication, rational numbers are closed.

Rational Numbers Calculations

Since you are already proficient in adding, subtracting, multiplying, and dividing both numbers and fractions, let’s examine these fundamental operations on rational numbers right now.

• When adding rational numbers with the same denominator, just add the numerators only by keeping the same denominator. However, to add two rational numbers with different denominators, the LCM of the two denominators is subtracted, and the two rational rational numbers are then converted to their equivalent forms. Then by keeping the LCM as the denominator, you can add two rational numbers with different denominators.
• When we subtract two rational numbers, we add the additive inverse of the rational number that is being subtracted from the other rational number.
• When multiplying, Multiply the numerators and denominators of the two rational numbers independently, and then express the result as product of numerators/product of denominators. When multiplying a rational number by a positive integer, the denominator is left unaltered while the numerator is multiplied by the integer.
• By multiplying the rational number by the reciprocal of the other non-zero rational number, we can divide one rational number by the other.

How to Find a Rational Number Between Two Rational Numbers?

Between any two rational numbers, there are an endless number of rational numbers. There are two simple ways for finding the rational numbers that lie between two rational numbers. Let’s now examine the two distinct approaches.

Approach 1

Find the rational numbers between the given rational numbers as well as the equivalent fraction for each. Those figures must correspond to the necessary rational figures.

Approach 2:

For the two given rational numbers, determine their respective means. The needed rational number should be the mean value. Repeat the method with the old and the new rational numbers to find further rational numbers.

We hope this article on Rational numbers has been helpful to you. Stay tuned to Embibe for such informative articles.