# 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. ## 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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