# Rational and irrational numbers

Last updated: September 17, 2021

## What are rational and irrational numbers?

Numbers are one of the mathematical objects that are used mainly to count other objects and measure quantities. Numbers are classified into natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, etc. Every natural number is a whole number; every whole number is an integer; every integer is a rational number. The real number is a set of all rational and irrational numbers.

Here, we will discuss rational numbers and irrational numbers.

### E1.1A: Identify and use rational numbers

Rational numbers

A rational number is defined as a number that can be represented in the form of $$\frac{p}{q}$$, where $$q\neq 0$$. Also, both $$p$$ and $$q$$ are integers. Some of the rational numbers are given below:

The number $$\frac{1}{2}$$ is rational. Its decimal form is $$0.5$$. So, the finite entries after the decimal point classify it as a rational number.

The number $$\frac{1}{3}$$ is again a rational number. The decimal value is $$0.333…$$, where $$3$$ repeats after the decimal point.

The number $$5$$ is again rational. It can be written in the form of $$\frac{5}{1}$$.

So, from the above examples, it is clear terminating decimals and non-terminating repeating decimals are rational numbers and can be changed into $$\frac{p}{q}$$ form.

In real-world scenarios, if a whole pizza is sliced into eight pieces, then the fractional representation of the one-piece as compared to the entire pizza is written as $$\frac{1}{8}$$. Also, the decimal representation of $$\frac{1}{8}=0.125$$ qualifies it as a rational number.

### Worked examples

Example 1: Find the number $$\frac{1}{4}$$. Is it rational or not?

Step 1: Change the fraction into the form of decimal.

$$\frac{1}{4} = 0.25$$

Step 2: Check the numbers after the decimal point.

The numbers after the decimal point terminate at $$5$$. So, the given number $$\frac{1}{4}$$ is a rational number.

Example 2:

Write the number $$0.6$$ in $$\frac{p}{q}$$ form, where $$q\neq 0$$.

Step 1: Multiply the given number by $$\frac{10}{10}$$.

$$0.6 \times \frac{10}{10}=\frac{6}{10}$$

Step 2: Change the fraction into its simplest form.

$$\frac{6}{10}=\frac{3}{5}$$.

Thus, $$\frac{p}{q}$$ form of the number $$0.6$$ is $$\frac{3}{5}$$.

## E1.1B: Identify and use irrational numbers

Irrational numbers

An irrational number is defined as a number that cannot be represented in the form of $$\frac{p}{q}$$, where $$q\neq 0$$. So, in other words, irrational numbers are the opposite of rational numbers. If we remove rational numbers from the set of real numbers, we will only have irrational numbers in that set.

For example, the square root of the number $$2$$ is an irrational number, as the numbers after the decimal point are non-terminating. It is represented as $$\sqrt{2}$$.

The number $$0.030030003 \cdots$$ is a rational number because the sequence is repeating.

The number $$\pi$$ equals $$3.1415926 \cdots$$ is an irrational number.

So, from the above examples, it is clear that every non-terminating and non-repeating decimal number is an irrational number.

### Worked examples

Example 1: Write the value of the Euler number and find whether it is irrational or not.

Step 1: Write the value of the Euler number $$e$$.

$$e=2.718281828…$$

Step 2: Check the value is irrational or not.

The numbers after the decimal point do not terminate, and also, no repeating value is there. So, the given number $$e=2.7181828…$$ is an irrational number.

Example 2: Write the value of the Golden ratio $$\phi$$ and then identify whether it is an irrational number or not.

Step 1: Write the value of the Golden ratio $$\phi$$.

$$\phi=1.61803398874 \cdots$$

Step 2: Check if the value is irrational.

The numbers after the decimal point do not terminate, and there is no repetition in the values. So, the given number $$\phi=1.61803398874 \cdots$$ is an irrational number.