Rational numbers and Irrational numbers are real numbers with unlike characteristics. Rational numbers are the numbers which can be represented in the A/B form where B ≠ 0. Irrational numbers are the numbers that cannot be represented in A / B form. In this article, we’ll learn the concepts of rational numbers and irrational numbers and explore the difference between them.

What is Rational number?

Standard form of a rational number is achieved when there are no common factors, except one, between the dividend and divisor. Additionally, the divisor must be positive. Consider the rational number 12/36. By simplifying it, we get 1/3, as the only common factor between the divisor and dividend is one. Therefore, we can confidently state that the rational number 1/3 is in standard form.

Basically, it is number in a form of m/n it can be written as follows: 14/2 , 6/2

Rational Number Definition

Rational numbers are special kinds of numbers that can be written as fractions. These numbers can be positive, negative, or zero. We write them as m/n, where n is not zero (n ≠ 0)

The word “rational” comes from “ratio,” which means comparing two or more things. In simpler terms, rational numbers are like fractions – they show the relationship between two whole numbers.

Rational Number Examples

Some examples of rational numbers are,

How to Identify Rational Numbers?

To identify if a number is rational or not, check the below conditions.

1. It is represented in the form of p/q, where q ≠ 0
2. The ratio p/q can be further simplified and represented in decimal form.

The set of rational numerals: Include positive, negative numbers, and zero and Can be expressed as a fraction

Types of Rational Numbers

A number is rational if we can write it as a fraction, where both denominator and numerator are integers and the denominator is a non-zero number.

The below diagram helps us to understand more about the number sets.

Positive and Negative Rational Numbers

Rational numbers are represented as m/n, where both m and n are integers, and n is a non-zero integer. Rational numbers can be either positive or negative, based on the sign of the fraction.

If the rational number is positive, both m and n are positive integers.

If the rational number takes the form -(m/n), it means either m or n (or both) are negative. This implies that -(m/n) is equivalent to both (-m)/n and m/(-n).

Arithmetic operations on Rational numbers

This term involves performing basic mathematical calculations such as addition, subtraction, multiplication, and division. Let’s explore each operation:

Addition (or Sum) of Rational Numbers

1. To add two rational numbers (fractions), find a common denominator.
2. Add or subtract the numerators while keeping the common denominator unchanged.
3. Simplify the result if possible.

• Example: 3/4 + 1/6 = 11/12
• Explanation: 3/4 + 1/6 = 9/12 + 2/12 = 11/12

Subtraction (or Difference) of Rational Numbers

1. Similar to addition, find a common denominator.
2. Subtract the numerators while maintaining the common denominator.
3. Simplify the result if necessary.

• Example: 5/8 – 1/4 = 3/8
• Explanation: 5/8 – 1/4 = 5/8 – 2/8 = 3/8

Multiplication of Rational Numbers

1. Multiply the numerators together to get the new numerator.
2. Multiply the denominators together to get the new denominator.
3. Simplify the result if possible.

• Example: 2/3 X 4/5 = 8/15
• Explanation: 2/3 X 4/5 = (2 X 4) / (3 X 5 ) =8/15

Division of Rational Numbers

1. Invert (flip) the second fraction (divisor).
2. Multiply the first fraction (dividend) by the inverted second fraction.
3. Simplify the result if necessary.

• Example: 3/4 ÷ 2/3 = 9/8
• Explanation: 3/4 ÷ 2/3 = 3/4 X 3/2 =9/8

Read More About

• Whole Number
• Real Number
• Prime Number
• Complex Number
• Rational Numbers

What are Irrational Numbers?

Irrational numbers belong to the realm of real numbers, but they possess a unique characteristic: they cannot be neatly expressed as simple fractions. In contrast to rational numbers, which can be written as p/q (where both p and q are integers and q≠0), irrational numbers defy such ratio representation.

To further grasp the concept, think of irrational numbers as a contradiction to rationality. They resist being confined within the bounds of simple fractions.

A common way to express irrational numbers is through the notation R\Q, where the backward slash (\) symbol signifies ‘set minus.’ In simpler terms, it denotes the set of real numbers excluding the set of rational numbers. Another representation is R – Q, emphasizing the distinction between the set of all real numbers and the set of rational numbers.

The calculations based on these numbers are a bit complicated. For example, √5, √11, √21, etc., are irrational.

Irrational Number Definition

An irrational number is a type of real number that cannot be expressed as a simple fraction (ratio) of two integers. In other words, it’s a number that cannot be written in the form a/b, where “a” and “b” are integers and “b” is not equal to zero.

Irrational numbers have non-repeating, non-terminating decimal expansions. Examples of irrational numbers include the square root of non-perfect squares (like √2, √3), pi (π), and the mathematical constant “e”.

Examples of Irrational Numbers

Square Root of 2 (√2)

• The square root of 2 is irrational. Its decimal representation goes on forever without repeating, and it cannot be expressed as a fraction.
• Approximate decimal representation: 1.4142…

Pi (π)

• Pi is the ratio of the circumference of a circle to its diameter. It is an irrational number.
• Approximate decimal representation: 3.141…

Euler’s Number (e)

• Euler’s number is an irrational constant that is the base of natural logarithms.
• Approximate decimal representation: 2.718281828459045…

How to Identify Irrational Numbers?

The real numbers which cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0 are known as irrational numbers. For example, √2 and √ 3 etc. are irrational. Whereas any number which can be represented in the form of p/q, such that, p and q are integers and q ≠ 0 is known as a rational number.

There are 4 different aspects by which we can conclude the irrational number:

Square Roots of Prime Numbers

If a number is the square root of a non-perfect square (a number that is not the square of an integer), then it is irrational.

Example: √2, √3, and √5 are irrational.

Note: √prime number always gives an irrational number.

Non-Terminating, Non-Repeating Decimals

If the decimal representation of a number goes on forever without repeating, it might be irrational. Rational numbers have decimals that either terminate (like 1/4 = 0.25) or repeat (like 1/3 = 0.333…).

Example: 1/4 = 0.4444…….

Mathematical Constants

Certain mathematical constants, like pi (π) and Euler’s number (e), are known to be irrational.

Note: Pi (π) is an irrational number because it is neither terminating nor repeating decimal.

Arithmetic Operations on Irrational Number

Various Operations on Irrational Numbers are,

Addition of Irrational Numbers

Add the irrational numbers just as you would with rational numbers.

Example: √2 + √3

Subtraction of Irrational Numbers

Subtract the irrational numbers just as you would with rational numbers.

Example: √5 – √2

Multiplication of Irrational Numbers

Multiply the irrational numbers by multiplying the radicals.

Example: √2 × √3 = √6

Division of Irrational Numbers

Divide the irrational numbers by rationalizing the denominator (multiply both the numerator and denominator by the conjugate).

Example: √3 / √2 = (√3 / √2) × (√2 / √2) = √6 / 2

Properties of Rational and Irrational Number

Some of the properties of rational and irrational numbers are listed below:

Properties of Rational Numbers

Various Properties of Rational Number are,

Expressible as Fractions

Rational numbers are those you can write as simple fractions. For example, 3/4 and -2 are rational because they can be expressed as ratios of integers.

Terminating or Repeating Decimals

When you write a rational number as a decimal, it either stops after a certain number of digits (like 0.25 for 1/4) or repeats a pattern (like 0.333… for 1/3).

Closure under Addition and Subtraction

If you add or subtract two rational numbers, the result is always a rational number.

Closure under Multiplication and Division

When you multiply or divide two rational numbers, the answer is always rational, as long as you don’t divide by zero.

Additive Inverse

Every rational number has a friend that, when added together, gives zero. For example, if you have 3, its additive inverse is -3 because 3 + (-3) equals 0.

Multiplicative Inverse (excluding 0)

Every non-zero rational number has a buddy that, when multiplied, gives 1. For instance, the multiplicative inverse of 2 is 1/2 because 2 × (1/2) equals 1.

Properties of Irrational Numbers

Various Properties of Irrational Number are,

Non-Expressible as Fractions

Irrational numbers are the rebels that cannot be written as fractions. Examples include the square root of 2 (√2) or pi (π).

Non-Terminating, Non-Repeating Decimals

When you write an irrational number as a decimal, it goes on forever without any repeating pattern.

Closure under Addition and Subtraction

If you add or subtract two irrational numbers, the result can be either irrational or rational.

Closure under Multiplication and Division

When you multiply or divide two irrational numbers, the answer can be either irrational or rational.

No Additive Inverse

Unlike rational numbers, irrational numbers don’t have a friend that, when added, equals zero.

No Multiplicative Inverse

Irrational numbers don’t have a buddy that, when multiplied, equals 1 within the set of irrational numbers.

Learn More,

• Rational Numbers
• Irrational Numbers

How to Classify Rational and Irrational Numbers?

According to the definition of rational and irrational numbers can be classified as the numbers which can be written in p/q form are rational whereas the number which cannot be written in p/q form are irrational. Below diagram shows us the diagram for rational and irrational numbers.

Rational number and Irrational number Classification

Difference Between Rational and Irrational Numbers

The difference between rational and irrational number is added in the table below

Read More

• Is pi rational or irrational
• Is 2 rational number
• Is square root of 2 is rational number
• Is square root of 22 is rational number

Examples on Rational and Irrational Numbers

Example 1: Which of the following numbers is a rational number?

1. √12
2. 5/2
3. π
4. -3/7

Solution:

Option (2), (4) are correct

Example 2: Identify the number that is not a rational number?

1. 3/4
2. √5
3. 0.75
4. 1/3

Solution:

Option (2) is correct

Example 3: Check which of the following is irrational or rational: 1/2, 13, -4, √3, and π.

Solution:

Rational numbers are numbers that can be expressed in the form of p/q, where q is not equal to 0.1/2, 13, and -4 are rational numbers as they can be expressed as p/q.√3, and π are irrational numbers as they can not be expressed as p/q.

Example 4: Check if a mixed fraction, 3(5/6) is a rational number or an irrational number.

Solution:

Simplest form of 3(5/6) is 23/6

Numerator = 23, which is an integer

Denominator = 6, is an integer and not equal to zero.

So, 23/6 is a rational number.

Practice Problems on Rational and Irrational Numbers

1. Simplify 3/7 × 28/15 ÷ 14/5

2. Simplify 3/7 + (-2)/21 × (-5)/6

3. Find (2/3) × (-5/7) + (7/3) + (2/3) × ((-2)/7)

4. Prove √144 is not irrational number.

Rational and Irrational Numbers – FAQs

What are Rational and Irrational Numbers?

A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, where p is the numerator and q is the denominator, and q is not equal to zero whereas irrational number is a number that cannot be expressed as the quotient of two integers. In other words, its decimal representation goes on forever without repeating, and it cannot be written as a fraction.

What are some Rational and Irrational Number Examples?

Some rational and irrational number examples are 10 / 2 and √2 respectively.

What are Difference Between Rational and Irrational Numbers?

The main difference between the rational and irrational numbers is that rational numbers can be represented in p/q form and irrational numbers cannot be represented in p/q form.

Can Square Root of a Non-Perfect Square be Rational?

Yes, the square root of a non-perfect square can be rational. For example, √4 is rational because it equals 2. However, the square root of certain non-perfect squares, like √2 or √5, is irrational.

How can you Prove a Number is Irrational?

One common method to prove a number is irrational is proof by contradiction. Assume the number is rational, express it as a fraction, and then derive a contradiction. This contradiction shows that the original assumption of the number being rational must be false.

Are all Square Roots of Integers Irrational?

No, not all square roots of integers are irrational. The square root of a perfect square (e.g., √4, √9, √16) is rational, as it results in a whole number. However, the square root of a non-perfect square (e.g., √2, √5) is typically irrational.

Can an Irrational Number be Raised to a Rational Power to Yield a Rational Result?

Yes, it is possible for an irrational number to be raised to a rational power and yield a rational result. For example, \( (\sqrt{2})^2 = 2 \) is rational.

Can Sum or Product of a Rational and an Irrational Number be Rational?

Yes, it is possible. The sum or product of a rational and an irrational number can be either rational or irrational. It depends on the specific numbers involved.