Terminating and Non-Terminating Decimals(Meaning, Conversions & Examples)

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Decimal numbers are one of the ways to represent both whole numbers and fractions. Decimal numbers consist of two parts – the whole part and the decimal (fractional) part separated by a decimal point ‘$.$’. 

The decimals are categorized into two broad categories depending on whether it has a finite number of decimal places or an infinite number of decimal places.

The decimals with a finite number of decimal places are called terminating decimals and the ones with an infinite number of decimal places are called non-terminating decimals.

The non-terminating decimals are further divided into two types – recurring decimals or non-recurring decimals depending on whether the digits after the decimal point repeat in a fixed pattern or not.

What is a Terminating Decimal? 

Terminating decimals are the numbers that have a fixed or a finite number of digits after the decimal point. Decimals are used to express the whole number and fraction together that are separated from each other by inserting a “$.$” i.e. a decimal point. For example, $3.6$, $3$ is the whole number and $6$ is the decimal fraction.

The numbers that are terminating decimal numbers are rational numbers and hence can be expressed in the form $\frac {p}{q}$, where both $p$ and $q$ are integers and $q \ne 0$. For example, the number $3.6$ can be written as $\frac {18}{5}$ which is an improper fraction where both $18$ and $5$ are integers, and obviously, $5 \ne 0$. The fraction $\frac {18}{5}$ can also be written in the form $3 \frac {3}{5}$ which is a mixed fraction. 

Note: The terminating decimals are rational numbers and can be written in the form $\frac {p}{q}$ which be either of the following two types.

• proper fractions
• improper fractions which can be expressed as mixed numbers or mixed fractions

Note: All natural numbers, whole numbers, and integers are terminating decimal numbers.  For example, the numbers $34$, $97$ or $-56$ can also be written as $34.0$, $97.0$ or $-56.0$.

What is Non Terminating Decimal?

Non terminating decimals are the numbers that have an infinite number or an uncountable number of digits after the decimal point. These decimals are also used to express the whole number and fraction together that are separated from each other by inserting a “$.$” i.e. a decimal point. For example, $7.454545…$, or $2.78654324573067…$. In the case of non terminating decimal numbers, the digits do not terminate after the decimal point.

The non terminating decimals are further divided into two types.

• Non terminating but recurring decimal numbers
• Non terminating and non recurring decimal numbers

What is Recurring Decimal?

The recurring decimals or non terminating but recurring decimal numbers are the numbers that consist of two parts a whole part and the fractional part separated by a decimal point ‘$.$’ and the number of decimal places are infinite but repeats in a fixed pattern. 

All non terminating but repeating decimal numbers are rational numbers and can be expressed in the form $\frac {p}{q}$.

Examples of non-terminating but repeating decimals are $5.222222…$, $8.12121212…$, $-56.453453453…$.

In the case of $5.222222…$, the digit $2$ is repeating and it appears an infinite number of times in the number or never terminates. The number $5.222222…$ can also be written as $5. \overline{2}$.

In the case of $8.121212121212…$, the digits $12$ are repeating and they appear an infinite number of times in the number or never terminate. The number $8.121212121212…$ can also be written as $8. \overline{12}$.

Similarly, in the case of $-56.453453453…$, the digits $453$ are repeating and they appear an infinite number of times in the number or never terminate. The number $-56.453453453…$ can also be written as $-56. \overline{453}$.

What is Non Recurring Decimal?

The non recurring decimals or non terminating and non recurring decimal numbers are the numbers that consist of two parts a while part and the fractional part separated by a decimal point ‘$.$’ and the number of decimal places are infinite and never repeats.

All non terminating and non recurring decimal numbers are irrational numbers, i.e., these are the numbers that are not rational numbers and cannot be expressed in the form $\frac {p}{q}$.

Examples of non terminating and non recurring decimal numbers are $5.6754246089…$, $-3.8975630986…$.

Note: $\pi = 3.1415926535 8979323846 2643383279 5028841971$ is the most widely used non terminating and non recurring decimal number. 

Converting Terminating Decimals to Fractions

As discussed above the terminating decimals are rational numbers and hence can be converted into the form $\frac {p}{q}$. 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 $7.8$ the numerator will be $78$, and similarly, for $14.76$ the numerator will be $1476$.

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 $7.8$, the number of digits after the decimal point is $1$, therefore denominator will be $10$. And similarly, in the case of $14.76$, the number of digits after the decimal point is $2$, therefore the denominator will be $100$.

Step 3: Reduce the fraction, if required.

Examples

Ex 1: Convert $15.8$ to a fraction.

The numerator will be  $158$ (Decimal number without decimal point).

Number of digits after decimal places = $1$.

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

And fraction is $\frac {158}{10} = \frac {79}{5}$ . (Reducing it to the lowest form)

It can also be written as $15 \frac {4}{5}$ as a mixed fraction.

Ex 2: Convert $2.548$ to a fraction.

The numerator is  $2548$ (Decimal number without decimal point).

Number of digits after decimal places = $3$.

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

And fraction is $\frac {2548}{1000} = \frac {637}{250}$. (Reducing it to the lowest form) 

It can also be expressed as a mixed fraction as $2 \frac {137}{250}$.

Converting Non Terminating But Recurring Decimals to Fractions

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 decimal such as $0.333333…$  then multiply by $10$, if repetition of the digits is in pairs of two numbers such as  $0.252525…$ 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 $0.222222… \left(= 0. \overline{2} \right)$ to a rational number

Let $x = 0.222222…$ —————————————- (1)

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

$10 \times x = 10 \times 0.222222…$ 

$=> 10x = 2.222222…$ ————————————– (2)

Subtract (1) from (2)

$9x = 2 => x = \frac {2}{9}$

Therefore, $0.222222… \left(= 0. \overline{2} \right) = \frac {2}{9}$

Ex 2: Convert $1.343434… \left(= 1. \overline{34} \right)$ to a rational number

Let $x = 1.343434…$ —————————————- (1)

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

$100 \times x = 100 \times 1.343434…$ 

$=> 100x = 134.343434…$ ————————————– (2)

Subtract (1) from (2)

$99x = 133 => x = \frac {133}{99}$

Therefore, $1.343434… \left(= 1. \overline{34} \right) = \frac {133}{99} = 1 \frac {34}{99}$

Ex 3: Convert $6.27454545… = 6.27 \overline{45}$ to a rational number

Let $x = $6.27454545…$ —————————– (1)

Since $2$ numbers $2$ and $7$ are not repeated, therefore, multiply both sides by $100$

$100 \times x = $100 \times 6.27454545…$ —————————– (1) 

$=> 100x = 627.454545…$ —————————————————-(2)

Also, $2$ numbers $4$ and $5$ are repeating, therefore, multiply both sides by $100$

$100 \times 100x = 100 \times 627.454545…$ 

$=> 10000x = 62745.454545…$ ———————————————(3)

Now subtract (2) from (3)

$9900x = 62118 => x = \frac {62118}{9900} = \frac {31059}{4950} = 6 \frac {1359}{4950}$   

Identifying Terminating & Non Terminating Decimals

You can classify any rational number in the form $\frac {p}{q}$, where $p$ and $q$ are integers and $q \ne 0$ as recurring or non recurring without actually dividing the numerator and denominator.

The first step here is to reduce the fraction in its lowest or simplest form.

The next step is to check the denominator part of the fraction, i.e., $q$. 

If $q$ can be factorized in the form $2^{m}5^{n}$, then the number is terminating otherwise not.

Note: These types of  fractions are terminating

• The prime factor of the denominator is only $2$, such as $2$, $4$, $8$, $16$, … Fractions $\frac {1}{2}$, $\frac {3}{4}$, $\frac {5}{8}$, $\frac {11}{16}$ are all terminating decimals.
• The prime factor of the denominator is only $5$, such as $5$, $25$, $125$, $625$, … Fractions $\frac {3}{5}$, $\frac {19}{25}$, $\frac {79}{125}$, $\frac {107}{625}$ are all terminating decimals.
• The prime factors of the denominator are $2$ and $5$, such as $10$, $20$, $250$, … Fractions $\frac {3}{10}$, $\frac {17}{20}$, $\frac {127}{250}$ are all terminating decimals.

If the prime factors of the denominator are any number other than $2$ and $5$, the number is a non terminating decimal. For example, $\frac {7}{30}$ is a non terminating decimal. Although $2$ and $5$ are the prime factors of $30$, along with these prime factors, $3$ is also a prime factor of $30$. (Prime factorization of $30$ is $2 \times 3 \times 5$.

Finding Number of Decimal Places in Recurring Decimals

You can find the number of places in a terminating decimal number without actually dividing the numerator by the denominator. 

To do so express the denominator, i.e., $q$ in $\frac {p}{q}$ as $2^{m}5^{n}$. 

• If $m \gt n$, then the number will terminate after $m$ decimal places
• If $n \gt m$, then the number will terminate after $n$ decimal places

Examples

Ex 1: Find the number of decimal places in the expansion of $\frac {7}{40}$.

$40 = 2 \times 2 \times 2 \times 5 = 2^{3}\times 5^{1}$.

Since prime factors of $40$ are $2$ and $5$ only, $\frac {7}{40}$ is a terminating decimal number.

Power of $2$ is $3$ and that of $5$ is $1$, therefore, $\frac {7}{40}$ will terminate after $3$ decimal places.

$\frac {7}{40} = 0.175$.

Ex 2: Find the number of decimal places in the expansion of $\frac {51}{250}$.

$250 = 2 \times 5 \times 5 \times 5 = 2^{1}\times 5^{3}$.

Since, prime factors of $250$ are $2$ and $5$ only, so $\frac {51}{250}$ is a terminating decimal number.

Power of $2$ is $1$ and that of $5$ is $3$, therefore, $\frac {51}{250}$ will terminate after $3$ decimal places.

$\frac {51}{250} = 0.204$.

Ex 3: Find the number of decimal places in the expansion of $\frac {7}{100}$.

$100 = 2 \times 2 \times 5 \times 5 = 2^{2}\times 5^{2}$.

Since prime factors of $100$ are $2$ and $5$ only, $\frac {7}{100}$ is a terminating decimal number.

Power of $2$ is $2$ and that of $5$ is $2$, therefore, $\frac {7}{100}$ will terminate after $2$ decimal places.

$\frac {7}{100} = 0.07$.

Conclusion

The decimal numbers are of two types terminating and non terminating. The non terminating decimal numbers are further divided into two categories – non terminating but recurring and non terminating and non recurring. The terminating decimals and the non terminating but recurring decimals are also called rational numbers whereas the non terminating and non recurring decimal numbers are irrational numbers.

Practice Problems

Which of the following are terminating decimal numbers? Also, find the number of decimal places in terminating decimals without actually dividing.

• $\frac {5}{100}$
• $\frac {7}{75}$
• $\frac {49}{600}$
• $\frac {19}{200}$
• $\frac {127}{1450}$
• $\frac {68}{350}$

Recommended Reading

• What Are Decimals – Definition With Examples
• Reducing Fractions – Lowest Form of A Fraction
• Decimal Number System – With Types & Properties

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