# Terminating and non terminating rational numbers

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