In this post we will learn about the concept of terminating and non terminating rational number with examples.

At the and of the chapter, problems are also given for practice.

## Checking rational number is terminating or non terminating

To check if the rational number is terminating or non terminating, we have to **first convert the rational number into decimals**.

To learn **rational number to decimal conversion**, click the red link.

After conversion to decimal number, we get two type of numbers;**(a) Terminating decimals**

The **decimals which have finite number of digits** are called terminating decimals.

As the name suggests, these numbers ends after certain digits.

The rational numbers producing terminating decimals are called terminating rational number.**(b) Non terminating decimals**

The decimal which have **unending number of digits** is non terminating decimal.

In non terminating decimal, the decimal digit just go on and on.

Hence, the rational number producing non terminating decimals are called non terminating rational numbers.

I hope you understood the above concept. Given below are some examples for your understanding.

### Shortcut to identify terminating or non terminating rational number

We know that rational number are expressed in the form of fraction \mathtt{\frac{p}{q}} .

Where p & q are the integers.**When the given rational number are factorized in the form of ** \mathtt{\frac{p}{2^{m} \ 5^{n}}} , then the given rational number is terminating rational number.**For example;**

Consider the rational number \mathtt{\frac{7}{20}} **Solution**

The above rational number can be expressed as;

\mathtt{\Longrightarrow \ \frac{7}{20}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7}{4\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7}{2^{2} \times 5}}

Since, we have expressed the rational number in form of \mathtt{\frac{p}{2^{m} \ 5^{n}}} , the given number is a terminating rational number.

We can prove this by dividing numerator by denominator.

\mathtt{\frac{7}{20} \Longrightarrow \ 0.35}

Here 0.35 is a terminating decimal, hence the given rational number is a terminating rational number.**Note:**

If the rational number cannot be represented in the form of above expression then we have to individually check the number using division.

## Examples of Terminating and non terminating rational number

**Example 01**

Check if 50/21 is terminating or non terminating rational number.**Solution**

Factorizing the rational number into smaller components.

\mathtt{\Longrightarrow \ \frac{50}{21}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2\times 5^{2}}{3\times 7}}

Since the number cannot be expressed in form of \mathtt{\frac{p}{2^{m} \ 5^{n}}} , it is a non terminating rational number.

We can prove this by simply dividing numerator by denominator.

\mathtt{\frac{50}{21} =\ 2.3809523\ .\ .\ .}

Here the decimal digit will go on and on.

**Example 02**

Check if 89 / 40 is terminating or non terminating rational number.**Solution**

Factorizing the rational number into smaller components.

\mathtt{\Longrightarrow \ \frac{89}{40}}\\\ \\ \mathtt{\Longrightarrow \ \frac{89}{8\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{89}{2^{3} \times 5}}

Since the number is expressed in form of \mathtt{\frac{p}{2^{m} \ 5^{n}}} , it is a terminating rational number.

**Example 03**

Check if 16 / 125 is terminating or non terminating rational number.

**Solution**

Factorizing the rational number into smaller components.

\mathtt{\Longrightarrow \ \frac{16}{125}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2^{4}}{5\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2^{4}}{5^{2}}}

Since the rational number is expressed in the form of \mathtt{\frac{p}{2^{m} \ 5^{n}}} , it is a terminating rational number.

We can prove this by dividing numerator by denominator.

\mathtt{\frac{16}{125} =0.128} **Hence proved**.

**Example 04**

Check if 1000 / 6 is terminating or non terminating rational number.

**Solution**

Factorizing the rational number into smaller components.

\mathtt{\Longrightarrow \ \frac{1000}{6}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1000}{2\times 3}}

As the rational number cannot be expressed in form of \mathtt{\frac{p}{2^{m} \ 5^{n}}} , we have to divide the number to get the solution.

Dividing numerator by denominator.

\mathtt{\frac{1000}{6} =166.66666\ .\ .\ .\ .}

Here we get the decimal with unending numbers. Hence, the given number is non terminating decimal.

**Example 05**

Check if 343/35 is terminating or non terminating decimal.

**Solution**

First factorize the number in smaller components.

\mathtt{\Longrightarrow \ \frac{343}{35}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7\times 7}{7\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\cancel{7} \times 7}{\cancel{7} \times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7}{5}}

Here the number is expressed in form of \mathtt{\frac{p}{2^{m} \ 5^{n}}} . So it is a terminating rational number.

**Next chapter **:** Convert repeating decimal into rational number**