Rational and Irrational Numbers

Definition of Rational and Irrational Numbers

• Rational Numbers can be written as a ratio that compares two numbers or quantities, giving a simple fraction or mixed fraction p/q. These include integers or decimals terminating (finite) or recurring (repeating patterns). The denominator ‘q’ is a natural number, i.e., non-zero.
• Irrational Numbers cannot be written as a ratio. In simple words, irrational numbers include numbers that cannot be simplified further to fraction with natural numbers and integers. The decimal of irrational numbers, if expanded, give neither finite nor recurring decimals. These include surds and unique numbers like π (pi). Surds are non-perfect squares or cubes that cannot be simplified and remove the square or cube roots. ‘pi’ is the most common irrational number.

Below is the figure representing the Venn diagram of rational and irrational numbers.

Figure 1 (heading: Rational and Irrational Numbers, filename: Rational and Irrational Numbers)

[ Sample link: https://cdn1.byjus.com/wp-content/uploads/2020/08/Rational-and-Irrational-Numbers.png ]

We cannot make a list of rational and irrational numbers since both have an infinite range. Some examples are given below.

Examples of Rational and Irrational Numbers

Examples of Rational Numbers

Common examples of rational numbers are:

1. 6; it can be written as 6/1 where 6 and 1 are integers
2. 0.125; it can be written as 1/8 or 125/1000
3. √81; it can be simplified further to 9 or 9/1
4. 5.232323…, or 0.111; these are recurring decimals as they are repeated in patterns

Examples of Irrational Numbers

Common examples of irrational numbers are:

1. 1/0; denominator is zero
2. π; its value is 3.142, non-terminating and non-recurring
3. √99; its value is 9.94987.. and it cannot be simplified further

Rational Numbers vs Irrational Numbers

While discussing about rational and irrational numbers, we need to compare to find the how the both terms mathematically differ from each other.

Below are the differences between rational and irrational numbers in a table.

Difference Between Rational and Irrational Numbers

Rational Numbers	Irrational Numbers
Can be written as ratio form, i.e., p/q	Cannot be written as ratio (p/q) form
Include decimals that are terminating (finite) or recurring (repeated in patterns)	Include decimals that are non-terminating (infinite) or non-recurring (not repeated in patterns)
Include perfect squares such as 1, 4, 9, 16, 25, 36.	Include surds (numbers that cannot be simplified to remove a square root or cube root etc.) such as √2, √3, √5
The numerator and denominator are whole numbers; the denominator is not 0. E.g., 3/4, 1/9	It is impossible to write the numbers in fractional form. E.g., π, √7

Now, let us learn about identifying rational and irrational numbers through some examples.

Solved Examples

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