In this article, we will learn about irrational and rational numbers. Rational numbers are numbers that can be expressed in the form of a fraction, i.e., p/q, where q is not equal to zero. The decimal expansion of a rational number can either terminate after a particular set of digits (3/6= 0.5), or the digits start repeating themselves after a finite sequence of digits (⅓ = 0.3333). This statement is true for all bases, including 10, binary, hexadecimal, etc.

Irrational numbers are numbers that cannot be expressed in the form of a ratio or fraction, i.e., p/q. Any real number that is not rational can be classified as irrational. The decimal expansion of irrational numbers is non-recurring and non-terminating.


Rational Numbers

Research has shown that the concept of fractional numbers dates back to prehistoric times. Ancient Egyptians wrote texts that involved converting fractions to their special notation. Greek and Indian Mathematicians studied the theory of rational numbers while they were researching the number theory. The texts Sthananga Sutra and Euclid’s Elements mention the use of rational numbers. There are several texts that also speak of the decimal place value notations that are another critical aspect associated with rational numbers.

Irrational Numbers

In 800 – 500 BC, the Indian Sulba Sutras made use of irrational numbers. Pythagorean Hippasus of Metapontum has been attributed to giving the existential proofs of irrational numbers, specifically the square root of two. In the 19th Century, irrationals were separated into algebraic and transcendental parts, prompting a much deeper study of these numbers. As research progressed further, several great mathematicians came together to give us the concept of irrational numbers leading to real numbers as we know today.


The best way to learn how to identify rational and irrational numbers is by looking at examples.

Rational numbers

1. A number such as 5 can be written in the form of 5/1, where both 5 and 1 are integers.

2. Any decimal that is terminating, such as 0.5, 0.25. Fractions such as ½, ¼ have terminating decimal expansions and hence form rational numbers.

3. Square root of 81 is 9, or the square root of 4 is 2, and hence, they are also rational numbers.

4. 0.33333 is a non-terminating and repeating decimal; hence, this also falls under the category of rational numbers.

Irrational Numbers

1. 8/0 is an irrational number. It cannot be expressed in the form of p/q as the denominator is 0.

2. The decimal expansion of pi is 3.14159… which is non-recurring and non-terminating.

3. Square root of 2 is an irrational number as it cannot be simplified to a form that falls under the category of rational numbers.

Thus, when given a number, you can do all the checks as mentioned. By the process of elimination, you can identify if a number is rational or irrational.


The basics of Mathematics start with identifying the various categories of numbers; hence, ensuring that a child has a strong foundation of this topic is imperative. The best way to do so is by availing the proper guidance. Cuemath is a fantastic online educational platform that provides quality education to students. The certified tutors use a vast plethora of resources such as online worksheets, interactive puzzles, visual simulations, and apps to deliver an impactful lecture. Students are also encouraged to work at their own pace, enabling them to master a topic confidently. Hopefully, this article gave you an insight into how to identify numbers, and I wish you all the best!