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.

How to Identify Rational and Irrational Numbers?

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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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