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Nikita Parmar

Updated on 29th August, 2023 , 5 min read

Rational Numbers: Definition, Properties, Types, Arithmetic Operations, Multiplicative Inverse, and Difference Between Rational and Irrational

Rational Numbers Overview

A rational number is a sort of real number that takes the form p/q and is not equal to zero. A rational number is any fraction with non-zero denominators. Examples of rational numbers are 1/2, 1/5, 3/4, and so on. The number "0" is also a rational number since it may be represented in a variety of ways, including 0/1, 0/2, 0/3, and so on. However, 1/0, 2/0, 3/0, and so on are not reasonable because they offer us unlimited values

What are Rational Numbers?

In mathematics, a rational number is any integer that can be expressed in the form p/q where q = 0. Furthermore, every fraction that has an integer denominator and a numerator that is not equal to zero falls into the category of rational numbers. When a rational number (i.e., a fraction) is divided, the output is in decimal form, which can be either ending or repeating. 

Rational Numbers

Rational Numbers Definition

A rational number is one that can be expressed as p/q, where p and q are integers and q is not equal to zero. So, a rational number can be-

p/q

where q is not zero.

Examples of Rational Numbers

The following table shows examples of rational numbers-

p

q

p/q

 

 

Rational

36

3

36/3 = 12

100

1

1/100 = 0.01

80

10

80/10 = 8

How to Identify Rational Numbers?

The following conditions are needed to see if a number is rational or not-

  1. It is written in the form p/q, where q≠0.
  2. The p/q ratio can be reduced further and given in decimal form.
  3. The set of rational numbers includes positive, negative, and zero numbers.
  4. It is possible to represent it as a fraction.

Properties of Rational Numbers

Because a rational number is a subset of a real number, it will obey all of the real number system's characteristics. The following are some of the most important features of rational numbers-

  1. If we divide or multiply both the numerator and denominator with the same factor, a rational number remains the same.
  2. If we multiply, add, or subtract any two rational numbers, the outcome is always a rational number.
  3. When we add zero to a rational number, we receive the same number.
  4. Addition, subtraction, and multiplication are all close rational numbers.

Rational Numbers

Types of Rational Numbers

A number is rational if it can be written as a fraction with both the denominator and numerator being integers and the denominator being non-zero. The below image explains the number sets in further detail-

  1. Fractions having integer numerators and denominators, such as 1/2, -9/4, 11/999, and so on.
  2. Decimal endings such as 0.9991, 0.2, and so on.
  3. Integers such as 0, -7, 12, and so on.
  4. Non-terminating repeating decimals like 0.666..., 0.2121..., 0.619619..., and so on. 
  5. Positive rational numbers are those that have the same sign in both their numerator and denominator, such as -1/-5, 5/7, -11/-22, and so on. All of them are more than zero.
  6. Negative rational numbers are rational numbers with opposite signs in their numerator and denominator, such as 2/-15, -2/99, and so on. All of them are less than zero.

Read more about the Expanded Form.

Standard Form of Rational Numbers

The standard form of a rational number is defined as having no common factors other than one between the dividend and divisor and thus being positive. For example, 14/38 is a rational number. However, it may be reduced to 7/19 because the divisor and dividend share just one common element. As a result, the rational number 7/19 is in standard form.

Related Articles- Courses after 12th Commerce and Great Mathematicians of India.

Arithmetic Operations on Rational Numbers

Arithmetic operations are the fundamental operations we perform on integers in mathematics. Rational numbers can be added, subtracted, multiplied, and divided as well. These four operations on rational numbers are as follows-

Addition of Rational Numbers (both with the same and different denominators)

Adding two rational numbers is equivalent to adding two fractions. To add two rational numbers, we just match their denominators, and then add them.

Example: 1/3 + 2/7 = (7+6)/21 = 13/21

Subtraction of Rational Numbers (having both the same and distinct denominators)

Similar to addition, we first match the denominators and then subtract. 

Example: 1/3 - 4/6 = 2 - 4/6 = - 2/6 = -1/3

Multiplication of Rational Numbers

To multiply two rational numbers, multiply their numerators and denominators independently, then simplify the resulting fraction.

Example: 1/6 × - 5/7 = (1 × - 5)/(6 ×  7) = - 5/42 

Division of Rational Numbers

The division is accomplished by multiplying the first fraction (dividend) by the reciprocal of the second fraction (divisor)-

p/q ÷ r/s = p/q × s/r

Example: 1/4 ÷ 3/4 = 1/4 × 4/3 = …

Multiplicative Inverse of Rational Numbers

Because the rational number is represented as a fraction in the form p/q, the multiplicative inverse of the rational number is the reciprocal of the provided fraction. For example, if 6/5 is a rational integer, then the multiplicative inverse of 6/5 is 5/6, resulting in (6/5) x (5/6) = 1.

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How to Find Rational Numbers Between Two Rational Numbers?

Between two rational numbers, there exists an unlimited number of rational numbers. Two distinct strategies may be used to easily get the rational numbers between two rational numbers. The following are the two distinct techniques-

Method 1

Determine the equivalent fraction for the given rational numbers and the rational numbers in between. Those figures should be reasonable.

Method 2

Determine the mean value of the two rational numbers given. The needed rational number should be the mean value. Repeat the method using the old and newly obtained rational numbers to find more rational numbers.

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Difference Between Irrational and Rational Numbers

Irrational numbers are those that are not rational numbers. They are symbolized by the symbol Q' (Q dash). The following are the distinctions between rational and irrational numbers-

Parameters

Irrational Numbers

Rational Numbers

Definition

Numbers that cannot be stated as integer fractions.

Numbers that may be stated as fractions of integers, such as p/q when q is greater than zero.

Examples

√5, π (pi)

1/-2, 13/21, 5/2, and so forth.

Other feature

They are never-ending and never repeated.

Example: √5 = 2.236067977499789696... has no repeating patterns of decimals

They might be both terminating and non-terminating in nature.

Example: 2.141141…

Read more about the Difference Between Fractions and Rational Numbers.

Tips and Tricks of Rational Numbers 

The following are some of the tips and tricks of rational numbers-

  1. Fractions and any integer that can be written as a fraction are examples of rational numbers.
  2. Non-terminating decimals with repeated decimal patterns, or recurring decimals, are also rational numbers.
  3. Rational numbers include natural numbers, whole numbers, integers, fractions of integers, and terminating decimals.

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Frequently Asked Questions

Is it true that all integers are rational numbers?

Ans. Yes, all integers are rational numbers since rational numbers comprise all natural numbers, whole numbers, integers, terminating decimals, and fractions of the form p/q where 'p' and 'q' are integers and q should not equal 0.

Is it true that all whole numbers are rational numbers?

Ans. Yes, all whole numbers are rational numbers since rational numbers comprise all natural numbers, whole numbers, integers, terminating decimals, and fractions of the form p/q where 'p' and 'q' are integers and q should not equal 0.

What exactly are rational numbers? Provide examples.

Ans. A rational number is one that has the form p/q, where p and q are integers and q is greater than zero. Some rational number examples are 1/3, 2/4, 1/5, 9/3, and so on.

Is 0 a logical number?

Ans. Yes, 0 is a rational number since it can be written as 0/1, where 0 and 1 are integers and the denominator is greater than zero.

Is Pi (π) a rational number?

Ans. No, the number Pi (π) is not a rational number. It is an irrational number with a value of 3.142857.

Where do Rational Numbers begin?

Ans. Rational numbers do not begin with any particular number. They contain all-natural numbers, whole numbers, and integers, as well as terminating decimals and fractions that may be expressed as p/q when q is greater than zero.

Is the number 0 a rational number?

Ans. Yes, because it is an integer that may be represented in any form, such as 0/1, 0/2, where b is a non-zero integer, and 0 is a rational number. It may be expressed as p/q = 0/1. As a result, we may infer that 0 is a rational number.

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