Difference Between Rational and Irrational Number

Difference Between Rational and Irrational Number

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Vikram
Vikram Singh
Assistant Manager - Content
Updated on Dec 4, 2023 17:03 IST

Want to explore the difference between Rational and Irrational Numbers or look for the solution to whether pi is a rational number or an irrational number? Then, this blog is for you.

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Numbers are an integral part of our lives that are used to quantify, measure, and calculate everything from the amount of time we spend on our smartphones to the distance between two celestial bodies. But all these numbers are different.

In mathematics, numbers are broadly classified into real and complex numbers, and real numbers are further classified as rational or irrational; understanding the difference between them is fundamental to many mathematical concepts and calculations.

One of the most confusing questions asked to check whether you are familiar with rational and irrational numbers is:

"Is pi a rational or irrational number?"

Do you need clarification on the same? Then this article is for you. This article will explore the difference between rational and irrational numbers and why they matter in mathematics.
Let's explore the difference between them.

Table of Content

Difference Between Rational Number and Irrational Number

Parameter

Rational Number

Irrational Number

Definition

A number of the form p/q, where p and q are integers, and q is not equal to 0.

A number that can not be expressed as p/q (i.e., the ratio of two numbers)

Nature

Rational numbers are generally finite or recurring in nature.

Irrational numbers are non-terminating and non-repeating in nature.

Countability

Countable

Uncountable

Arithmetic Operation

When arithmetic operations are performed over two rational numbers, the result is always a rational number.

When arithmetic operations are performed over two irrational numbers, the result may or may not be irrational.

Example

22/7, 3.14, 1

pi, e, 3.1415926……

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What is a Rational Number?

A number that can be expressed as a fraction of two integers (where the denominator is not equal to zero) is called a rational number. In simple terms, any number that can be expressed as p/q, where p and q are integers, and q is not equal to zero, is known as a rational number.

A rational number can also be expressed as a decimal expression, but these decimal expressions must be terminated or repeated.

Example: 22/7, 3.45, 3.14145

Note: Integers, natural numbers, and whole numbers are subsets of the rational number.

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Properties of a Rational Number

If a, b, and c are three rational numbers, then

  • Closure:  Rational numbers are closed under addition, subtraction, multiplication, and division, i.e., a + b, a - b, a * b, and a/b belonging to a rational number.
  • Commutativity: 
    • a + b = b + a 
    • a * b = b * a
  • Associativity
    • a + (b + c) = (a + b) + c
    • a * (b * c) = (a * b) * c
  • Identity: Rational numbers have additive and multiplicative identities, which are 0 and 1, i.e.,
    • a + 0 = 0 + a = a
    • a * 1 = a = 1 * a
  • Inverse: Rational numbers have additive and multiplicative Inverse.
    • Additive inverse of a/b is -a/b
    • Multiplicative Inverse of a/b is b/a.
  • Distributive
    • a * (b + c) = a * b + a * c
    • (a + b) * c = a * c + b * c

Note: One of the most important properties of a rational number is that it is dense in nature, i.e., Between two rational numbers, an irrational number always exists. This property helps to approximate the irrational number with the help of a rational number to the desired degree of accuracy.

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What is an Irrational Number?

A number that can’t be expressed as a fraction of two integers, in simple terms, any real number that is not rational is irrational. Irrational numbers are generally expressed as non-terminating and non-repeating decimals.

Example: sqrt(2), pi, e (euler coefficient)

Properties of an Irrational Number

  • Addition and Subtraction of two irrational numbers may or may not be irrational.

Example: 

  1. sqrt(2) + sqrt(2) = 2sqrt(2)
  2. (2 + sqrt(2)) + (2 - sqrt(2)) = 4
  3. Multiplication of two irrational numbers may or may not be an irrational number.

Example:

sqrt(2) * sqrt(3) = sqrt(6)

sqrt(2) * sqrt(2) = 2

  • Irrational numbers are uncountable, i.e., infinitely many rational numbers that can't be listed in a finite sequence.
  • Similar to rational numbers, irrational numbers can be expressed as decimal numbers, but these decimals are non-terminating and non-repeating.
  • Many irrational numbers (Such as pi, e, etc) are not the solution to any polynomial equation with rational coefficients, i.e., many irrational numbers are transcendental.
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Key Difference Between Rational and Irrational Numbers

  • Rational numbers can be expressed as fractions of two integers, while irrational numbers can not.
  • Rational numbers can be expressed as terminating and repeating decimals, whereas irrational numbers have infinite non-repeating decimal representation.
  • Irrational numbers are uncountable, while rational numbers are countable.
  • Arithmetic operations over a rational number are always rational, whereas when we perform an arithmetic operation over an irrational number may or may not be irrational.

Until now, we clearly understand rational and irrational numbers, so it’s time to discuss the elephant in the room.

Is pi a rational and irrational number?

pi is the ratio of the circumference of a circle and the diameter of a circle. It is a constant value that is equal to 3.14159265359…… which is a non-terminating and non-repeating decimal form. Hence, pi is an irrational number.

We often get confused with the popular approximation of pi, i.e., 22/7, which is a rational number. But keep in mind this is not the actual value of pi. 22/7 is a rational number that is close to the actual value but not exact.

Conclusion

In conclusion, rational and irrational numbers are two different types of real numbers. A rational number can be expressed as a fraction, whereas an irrational number can not be expressed in fraction form. In this article, we have discussed what a rational number is, what an irrational number is, and the difference between them. Later in the article, we also discussed whether pi is a rational number or an irrational number.

Hope you will like the article.
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FAQs of Difference Between Rational Number and Irrational Number

What is Rational Number?

A number that can be expressed as a fraction of two integers (where the denominator is not equal to zero) is called a rational number. In simple terms, any number that can be expressed as p/q, where p and q are integers, and q is not equal to zero, is known as a rational number.

A rational number can also be expressed as a decimal expression, but these decimal expressions must be terminated or repeated.

Example: 22/7, 3.45, 3.14145

Note: Integers, natural numbers, and whole numbers are subsets of the rational number.

What is Irrational Number?

A number that can’t be expressed as a fraction of two integers, in simple terms, any real number that is not rational is irrational. Irrational numbers are generally expressed as non-terminating and non-repeating decimals.

Example: sqrt(2)pie (euler coefficient)

What are the properties of Rational Number?

If a, b, and c are three rational numbers, then

  • Closure:  Rational numbers are closed under addition, subtraction, multiplication, and division, i.e., a + b, a - b, a * b, and a/b belonging to a rational number.
  • Commutativity: 
    • a + b = b + a 
    • a * b = b * a
  • Associativity
    • a + (b + c) = (a + b) + c
    • a * (b * c) = (a * b) * c
  • Identity: Rational numbers have additive and multiplicative identities, which are 0 and 1, i.e.,
    • a + 0 = 0 + a = a
    • a * 1 = a = 1 * a
  • Inverse: Rational numbers have additive and multiplicative Inverse.
    • Additive inverse of a/b is -a/b
    • Multiplicative Inverse of a/b is b/a.
  • Distributive
    • a * (b + c) = a * b + a * c
    • (a + b) * c = a * c + b * c

What are the different properties of Rational Number?

  • Addition and Subtraction of two irrational numbers may or may not be irrational.

Example: 

  1. sqrt(2) + sqrt(2) = 2sqrt(2)
  2. (2 + sqrt(2)) + (2 - sqrt(2)) = 4
  3. Multiplication of two irrational numbers may or may not be an irrational number.

Example:

sqrt(2) * sqrt(3) = sqrt(6)

sqrt(2) * sqrt(2) = 2

  • Irrational numbers are uncountable, i.e., infinitely many rational numbers that can't be listed in a finite sequence.
  • Similar to rational numbers, irrational numbers can be expressed as decimal numbers, but these decimals are non-terminating and non-repeating.
  • Many irrational numbers (Such as pi, e, etc) are not the solution to any polynomial equation with rational coefficients, i.e., many irrational numbers are transcendental.

What is the difference between Rational and Irrational Number?

Parameter

Rational Number

Irrational Number

Definition

A number of the form p/q, where p and q are integers, and q is not equal to 0.

A number that can not be expressed as p/q (i.e., the ratio of two numbers)

Nature

Rational numbers are generally finite or recurring in nature.

Irrational numbers are non-terminating and non-repeating in nature.

Countability

Countable

Uncountable

Arithmetic Operation

When arithmetic operations are performed over two rational numbers, the result is always a rational number.

When arithmetic operations are performed over two irrational numbers, the result may or may not be irrational.

Example

22/7, 3.14, 1

pi, e, 3.1415926……

About the Author
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Vikram Singh
Assistant Manager - Content

Vikram has a Postgraduate degree in Applied Mathematics, with a keen interest in Data Science and Machine Learning. He has experience of 2+ years in content creation in Mathematics, Statistics, Data Science, and Mac... Read Full Bio

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