# How to Identify Rational and Irrational Numbers?

In this post blog, we teach you the definition of rational and irrational numbers and how to identify them.

**Related Topics**

**Step-by-step guide to rational and irrational numbers**

Different types of numbers depend on their properties. For example, rational and irrational numbers are as follows:

**Rational numbers**

A rational number is a number in the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q\) is not equal to \(0\). The set of rational numbers is denoted by \(Q\).

**Types of rational numbers**

The different types of rational numbers are given below:

- Integers like \(-2, 0, 3\), etc. are rational numbers.
- Fractions whose numerator and denominator are integers such as \(\frac{7}{3}, \frac{5}{6}\), etc. are rational numbers.
- Terminating decimals like \(0.35, 0.7116\), etc., are rational numbers.
- Non-terminating decimals with some repeating patterns (after the decimal point) such as \(0.333…\), etc., are rational numbers. These are generally known as non-terminating repeating decimals.

**How to identify rational numbers?**

Rational numbers can be easily identified with the help of the following properties.

- All integers, whole numbers, natural numbers, and fractions with integers are rational numbers.
- If the decimal form of the number is terminating or repeating, such as \(5.6\) or \(3.151515\), we know that they are rational numbers.
- If the decimal numbers seem never-ending or non-repeating, they are called irrational numbers.
- Another way to identify rational numbers is to see if the number can be expressed as \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q\) is not equal to \(0\).

**Irrational numbers**

Irrational numbers are the set of real numbers that cannot be expressed as fractions, \(\frac{p}{q}\) where \(p\) and \(q\) are integers. The denominator \(q\) is not equal to zero \((q≠0)\). The set of irrational numbers is represented by \(Q´\).

**Examples of irrational numbers**

The specific irrational numbers that are commonly used:

- \(π\) (pi) is an irrational number. \(π=3⋅14159265….\). The decimal value never stops at any point.
- \(\sqrt{2}\) is an irrational number.
- Euler’s number \(e\) is an irrational number. \(e=2⋅718281….\)
- Golden ratio, \(φ 1.61803398874989….\)

**How to identify an irrational number?**

We know that irrational numbers are just real numbers that cannot be expressed as \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q≠0\). For example, \(\sqrt{5}\) and \(\sqrt{3}\), etc. are irrational numbers. On the other hand, numbers that can be represented as \(\frac{p}{q}\), such that \(p\) and \(q\) are integers and \(q≠0\), are rational numbers.

**Rational vs irrational numbers**

The difference between rational and irrational numbers can be understood from the figure below.

Determine the rational numbers among the following. \(\sqrt{16},\:\sqrt{3},\:-\frac{4}{5},\:\pi ,\:2.3137134623730860…..\)

**Solution:** \(\sqrt{16},-\frac{4}{5}\) are rational numbers.

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