Types of Decimal Numbers(With Examples)

In mathematics, decimals are numbers that consist of two parts – a whole part and a fractional part. These two parts are separated by a symbol ‘.’ known as a decimal point.

Numbers like $32.7$, $28.328$, $21.5555…$, $98.252525…$, and $67.4532567…$ are all decimal numbers. Did you notice the number of digits after the decimal point and the pattern followed by these digits?

Based on the number of digits after the decimal point and the pattern followed by these digits, the decimal numbers are divided into three types – terminating, non-terminating but recurring, and non-terminating and non-recurring.

Let’s understand what these different types of decimal numbers are.

Types of Decimal Numbers

Based on whether there are a countable or an uncountable number of digits after the decimal point, a decimal number is divided into two categories

• Terminating Decimal Numbers
• Non-Terminating Decimal Numbers

Terminating Decimal Numbers

A decimal number that has a countable number of digits after the decimal point is called a terminating decimal. For example, $8.6$, $12.58$, $13.654$, etc. are terminating decimal numbers.

In the case of $8.6$, the fractional/decimal part is terminating after the $1$ digit. Similarly, in the case of $12.58$, the fractional/decimal part is terminating after $2$ digits, and in the case of $13.654$, the fractional/decimal part is terminating after $3$ digits.

Note: The number of digits after the decimal point is countable or finite, i.e., you can count.

Conversion of Terminating Decimal Numbers to Fractions

The following steps are used to convert a terminating decimal to a fraction.

Step 1: Write the decimal number as a numerator but without a decimal point. e.g., for $8.6$ the numerator will be $86$, and similarly, for $12.58$ the numerator will be $1258$.

Step 2: Count the number of digits in decimal places and write the denominator as a power of $10$, with the number of decimal places as power. For example, in the case of $8.6$, the number of digits after the decimal point is $1$, therefore denominator will be $10^{1} = 10$. And similarly, in the case of $12.58$, the number of digits after the decimal point is $2$, therefore the denominator will be $10^{2} = 100$.

Step 3: Reduce the fraction, if required.

Terminating Decimal Examples

Ex 1: Convert $12.6$ to a fraction.

The numerator will be $126$ (Decimal number without decimal point).

Number of digits after decimal places = $1$.

Therefore, the denominator will be $10^{1} = 10$.

And fraction $= \frac {126}{10} = \frac {63}{5}$. (Reducing it to the lowest form)

Ex 2: Convert $14.258$ to a fraction.

The numerator is $14258$ (Decimal number without decimal point).

Number of digits after decimal places = $3$.

Therefore, the denominator is $10^{3} = 1000$.

And fraction $= \frac {14258}{1000} = \frac {14258}{1000} = \frac {7129}{500}$.

Ex 3: Convert $0.005$ to a fraction.

The numerator is $5$ (Decimal number without decimal point).

Number of digits after decimal places = $3$.

Therefore, the denominator is $10^{3} = 1000$.

And fraction $= \frac {5}{1000} = \frac {1}{200}$.

Note: The fractions in Ex 1 and Ex 2 are improper fractions and hence can be converted to mixed fractions.

$\frac {63}{5} \frac 12 \frac {3}{5}$ and $\frac {7129}{500} = 14 \frac{129}{500}$.

Non Terminating Decimal Numbers

A decimal number that has an uncountable number of digits after the decimal point is called a non-terminating decimal. In the case of non-terminating decimal numbers, the digits after the decimal point keep on appearing.

Non Terminating Decimal Examples

For example, $2.3333…$, $11.121212…$, $5.6765675….$, etc. are non terminating decimal numbers.

Notice that in the case of $2.3333…$ and $11.121212…$, the digits after the decimal point are repeating. For $2.3333…$, one digit $3$ is repeating (or recurring). Similarly, in the case of  $11.121212…$, two digits viz., $1$ and $2$ are repeating (or recurring) in a fixed pattern ($1$ followed by $2$). Such types of non terminating decimal numbers are called non terminating but recurring decimal numbers.

Now, notice the decimal number $5.6765675….$. Here the digits after the decimal are not terminating and not recurring (repeating). Such types of non-terminating decimal numbers are called non terminating and non recurring decimal numbers.

Non Terminating But Recurring Decimal Numbers

In the case of non-terminating but recurring decimal or non-terminating repeating decimal, the decimal places will continue forever and never come to an end but since the name says repeating or recurring, it signifies that the repetition of the decimal values forms a specific pattern that can be easily identified. 

Non Terminating But Recurring Decimal Examples

For example, $\frac {1}{3} = 0.333333…$is a non-terminating recurring decimal expansion. The repetition of the decimal can also be indicated by showing a bar on top of the numbers that are repeating i.e., $0.33333…$ can also be represented as $0.\overline {3}$. Similarly, $\frac {1}{7} = 0.142857142857142857…$ which can also be written as $0.\overline{142857}$ is also a non-terminating repeating decimal expansion as the block of decimals $142857$ is repeating after every $6$ digit. You can convert a non-terminating recurring decimal to a rational number.

Converting Non Terminating But Recurring Decimals To Rational Numbers

As seen above, a non terminating recurring decimal can be converted into a rational number. A rational number is defined as the ratio of two integers $p$ and $q$ and is represented as \frac {p}{q}$ where $q \ne 0$.

Steps to Convert Non Terminating But Recurring Decimals To Rational Numbers

A non terminating but recurring decimal number can be converted to its rational number equivalent as

Step 1: Assume the repeating decimal to be equal to some variable $x$

Step 2: Write the number without using a bar and equal to $x$ 

Step 3: Determine the number of digits having a bar on their heads or the number of digits before the bar for mixed recurring decimal

Step 4: If the repeating number is the same digit after the decimal such as $0.3333…$ then multiply by $10$, if the repetition of the digits is in pairs of two numbers such as $0.5656…$ then multiply by $100$ and so on

Step 5: Subtract the equation formed by step $2$ and step $4$

Step 6: Find the value of $x$ in the simplest form

Examples

Ex 1: Convert $0.3333… \left( 0.\overline{3} \right)$ to a rational number.

Let $x = 0.3333… $ —————————————- (1)

Since only $1$ digit is repeated multiply both sides by $10$

$10 \times x = 10 \times 0.3333… $ 

$=> 10x = 3.3333$ —————————————- (2)

Subtract $\left( 1 \right)$ from $\left( 2 \right)$

$9x = 3 => x = \frac {3}{9} => x = \frac {1}{3}$

Therefore, $0.3333… \left( 0.\overline{3} \right) = \frac {1}{3}$.

Ex 2: Convert $0.272727… \left( 0.\overline{27} \right)$ to a rational number.

Let $x = 0.272727… $ —————————————- (1)

Since $2$ digits are repeated multiply both sides by $100$

$100 \times x = 100 \times 0.272727… $ 

$=> 100x = 27.272727$ —————————————- (2)

Subtract $\left( 1 \right)$ from $\left( 2 \right)$

$99x = 27 => x = \frac {27}{99} => x = \frac {3}{11}$

Therefore, $0.272727… \left( 0.\overline{27} \right) = \frac {3}{11}$.

Ex 3: Convert $0.4333333… \left( 0.4\overline{3} \right)$ to a rational number.

Here there are two types of digits after the decimal point – one that is not repeating $\left( 4 \right)$ and the other that is repeating $\left( 3 \right)$. We’ll treat these two separately.

Let $x = 0.4333333… $ —————————————- (1)

First, consider non-repeating digit(s).

Since only $1$ digit is not repeating, so multiply both sides by $10$.

$10 \times x = 10 \times 0.4333333… $ 

$ => 10x = 4.333333…$ —————————————- (2)

Now consider repeating digit(s).

Since only $1$ digit is repeating, so multiply both sides of an equation $2$ by $10$.

$10 \times 10x = 10 \times 4.333333…$

$=> 100x = 43.33333…$ —————————————- (3)

Subtract $\left( 2 \right)$ from $\left( 3 \right)$

$90x = 39 => x = \frac {39}{90} => x = \frac {13}{30}$

Therefore, $0.4333333… \left( 0.4\overline{3} \right) = \frac {13}{30}$

Ex 4: Convert $0.12666666… \left( 0.12\overline{6} \right)$ to a rational number.

Here there are two types of digits after the decimal point – one that is not repeating $\left( 12 \right)$ and the other that is repeating $\left( 6 \right)$. We’ll treat these two separately.

Let $x = 0.12666666… $ —————————————- (1)

First, consider non-repeating digit(s).

Since $2$ digits are not repeating, so multiply both sides by $100$.

$100 \times x = 100 \times 0.12666666… $ 

$=> 100x = 12.666666…$ —————————————- (2)

Now consider repeating digits.

Since, only $1$ digit is repeating, so multiply equation $\left( 2 \right)$ by $10$.

$10 \times 100x = 10 \times 12.666666…$ 

$=> 1000x = 126.666666…$ —————————————- (3)

Subtract equation $\left( 2 \right)$ from $\left( 3 \right)$

$900x = 114 => x = \frac {114}{900} => x = \frac{57}{450}$

Therefore, $0.12666666… \left( 0.12\overline{6} \right) = \frac{57}{450}$.

Non-Terminating And Non Recurring Decimal Numbers

In the case of non-terminating and non-recurring decimal or non terminating non-repeating decimal, the decimal places will continue forever and never come to an end and do not repeat also.

The non terminating and non recurring decimals cannot be converted to $\frac{p}{q}$ form and hence these numbers are not rational numbers. Such numbers are called irrational numbers. 

One of the most common irrational numbers is $\pi$.

$\pi = 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944…$.

Practice Problems

1. Write true or false.
   • Terminating decimal numbers are called rational numbers.
   • Terminating decimal numbers are called irrational numbers.
   • Non Terminating but recurring decimal numbers are called rational numbers.
   • Non Terminating but recurring decimal numbers are called irrational numbers.
   • Non Terminating and recurring decimal numbers are called rational numbers.
   • Non Terminating and recurring decimal numbers are called irrational numbers.
2. Convert the following decimal numbers to $\frac {p}{q}$ form.
   • $0.1111… = 0.\overline{1}$.
   • $0.121212… = 0.\overline{12}$.
   • $0.53333… = 0.5\overline{3}$.
   • $0.724444… = 0.72\overline{4}$.
   • $0.6323232… = 0.6\overline{32}$.

FAQs

Conclusion

Decimal numbers which consist of two parts – a whole part and a fractional/decimal part are of three types 

• Terminating decimal numbers
• Non Terminating but Recurring decimal numbers
• Non Terminating and Non Recurring decimal numbers

The first two types viz. terminating and non-terminating but recurring decimal numbers are called rational numbers as they can be expressed in the form $\frac {p}{q}$, whereas the third type of non-terminating and non-recurring decimals are called irrational numbers and cannot be expressed in the form $\frac {p}{q}$.

Recommended Reading

• Decimal Fraction – Definition, Conversion & Operations (With Examples)
• Decimal Place Value Chart (Definition & Examples)
• What Are Decimals – Definition & Types