Rational and Irrational Numbers: Rules, Examples and Classification

Rational and Irrational Numbers are classified according to whether they can be written as a ratio of two integers. This topic covers rational numbers, irrational numbers, terminating and recurring decimals, square-root tests, number-line properties and rules for addition, subtraction, multiplication and division. Worked examples help identify the correct number type quickly.

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What Are Rational and Irrational Numbers?

A rational number can be written as p/q, where p and q are integers and q ≠ 0. An irrational number cannot be expressed as the ratio of two integers.

Rational numbers include integers, fractions, terminating decimals and recurring decimals. Examples are 5, -3/7, 0.25 and 0.\overline{6}. Irrational numbers have decimal expansions that are non-terminating and non-repeating, such as √2, π and 0.1010010001... . Every real number is either rational or irrational, and no number belongs to both categories.

Rational and Irrational Numbers Formula & Tricks

Important Formulas

Rational number form
x = p/q, where p, q ∈ ℤ and q ≠ 0

A number is rational if suitable integers p and q can represent it as a fraction.

Terminating decimal conversion
0.375 = 375/1000 = 3/8

A terminating decimal becomes rational by placing it over a power of 10 and simplifying.

Recurring decimal conversion
0.\overline{3} = 1/3

Every recurring decimal can be converted into a fraction, so it is rational.

Square-root test
√n is irrational when n is a natural number that is not a perfect square

For example, √2 and √7 are irrational, while √9 = 3 is rational.

Quick Tricks

Check the decimal pattern

A terminating or repeating decimal is rational. A decimal that neither terminates nor repeats is irrational.

Example: 0.125 is rational because it terminates, and 0.272727... is rational because 27 repeats.
Check perfect squares under roots

For a natural number inside a square root, first check whether it is a perfect square. If it is not, the square root is irrational.

Example: √50 = 5√2, so it is irrational because √2 is irrational. √144 = 12, so it is rational.
Use the operation rules carefully

A rational number plus an irrational number is always irrational, and a non-zero rational number multiplied by an irrational number is always irrational. Two irrational numbers may produce either type.

Example: 2 + √3 is irrational, but √2 + (-√2) = 0 is rational.

Rational and Irrational Numbers Concepts

Rational Numbers and Their Decimal Forms

Rational numbers are numbers expressible as p/q with integers p and q, where q ≠ 0.

Integers, fractions, zero and decimal numbers with a finite or repeating pattern are rational. A decimal terminates when its reduced denominator has only 2 and/or 5 as prime factors. For example, 7/40 is terminating because 40 = 2³ × 5, while 5/12 is recurring because 12 also contains the factor 3.

Example: 0.875 = 875/1000 = 7/8, so 0.875 is rational.

Irrational Numbers and Non-Repeating Decimals

An irrational number cannot be written in the form p/q and has a non-terminating, non-repeating decimal expansion.

Common irrational numbers include √2, √3, √5 and π. A square root of a natural number is irrational when the number is not a perfect square. Simplifying the radical is necessary before classification.

Example: √72 = √(36 × 2) = 6√2; since √2 is irrational, √72 is irrational.

Rules for Arithmetic Operations

The sum, difference, product or quotient of rational numbers remains rational whenever the quotient has a non-zero rational denominator.

Rational ± irrational is irrational. A non-zero rational × irrational is irrational, and irrational ÷ non-zero rational is irrational. Operations involving two irrational numbers require calculation: their result can be rational or irrational. For example, √2 × √8 = √16 = 4, but √2 × √3 = √6 is irrational.

Example: √5 + (-√5) = 0 is rational, whereas √5 + √2 is irrational.

Number Classification on the Real Number Line

Natural numbers, whole numbers, integers and fractions are all rational subsets of the real numbers; irrational numbers form the remaining real numbers.

The inclusion relationship is: natural numbers ⊂ whole numbers ⊂ integers ⊂ rational numbers ⊂ real numbers. Irrational numbers are real numbers outside the rational set. Thus, -4, 0, 3/5 and 2.75 are rational, while √7 and π are irrational.

Example: The number 6 belongs to the natural, whole, integer, rational and real number sets.

Rational and Irrational Numbers Video Lessons

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Practice Rational and Irrational Numbers Questions

Practise published questions related to this topic.

1The Highest Common Factor (HCF) of two numbers is 48 and their Least Common Multiple (LCM) is 3456. If one of the numbers is 768, find the other number.→ 2The LCM of the numbers 3, 7.2 and 0.64 is:→ 3If the ratio of two numbers is 4 : 1, and their HCF is 47, then their LCM is:→ 4Which of the following number is divisible by both 3 and 4?→ 5The ratio of two numbers is 3:4 and their HCF is 7. Their LCM is:→ 6The product of Highest Common Factor and Least Common Multiple of 2 numbers is 6930. One of the numbers is 154. What is the other number?→ 7What is the smallest number that must be added to 4583 so that the resulting number is divisible by 9?→ 8Which of the following number is divisible by both 3 and 11?→ 9Find the square root of 64009.→ 10Which of the following number is NOT divisible by 8?→

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Quick Revision Notes

Rational and Irrational Numbers: Quick Revision

Use these rules to classify numbers and evaluate arithmetic results.

  • A rational number has the form p/q, where p and q are integers and q ≠ 0.
  • Terminating and repeating decimals are rational; non-terminating, non-repeating decimals are irrational.
  • Every integer is rational because n = n/1.
  • √n is irrational for a natural non-perfect-square n; √n is rational when n is a perfect square.
  • Rational ± irrational is irrational.
  • A non-zero rational multiplied by or divided into an irrational number gives an irrational result.
  • The sum, difference or product of two irrational numbers may be rational or irrational.
  • Natural numbers ⊂ whole numbers ⊂ integers ⊂ rational numbers ⊂ real numbers.

Rational and Irrational Numbers FAQs

Is 0 a rational number?

Yes. 0 can be written as 0/1, where both numbers are integers and the denominator is non-zero.

Is every terminating decimal rational?

Yes. For example, 0.48 = 48/100 = 12/25, so it is rational.

Is 0.333... rational or irrational?

It is rational because the digit 3 repeats indefinitely. In fact, 0.333... = 1/3.

How can √98 be classified?

√98 = √(49 × 2) = 7√2. Since √2 is irrational, √98 is irrational.

Is the sum of two irrational numbers always irrational?

No. It may be rational or irrational. √3 + (-√3) = 0 is rational, while √3 + √5 is irrational.

What is the result of multiplying a non-zero rational number by an irrational number?

The result is always irrational. For example, (3/4) × √2 is irrational.

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