Table of Contents

This post is also available in: हिन्दी (Hindi)

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 $-56.0$.

DOWNLOAD FREE MATHS FLASHCARDS:

Beautifully designed and printable flashcards to help you remember all the important Maths concepts and formulas.

## 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.

Is your child struggling with Maths?

We can help!

## 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

## FAQs

### What is a 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.

### What is the 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.

### How do you find terminating and non terminating decimals?

Express the denominator of the fraction as the product of its prime factors. If the prime factors are only $2$ or $5$ and are expressed in the form $2^{m}5^{n}$, then the fraction is terminating, otherwise not.

### What are non terminating, non repeating decimals?

The decimal numbers where there are infinite decimal places and are not repeating or showing any fixed pattern are called non terminating, non repeating decimal numbers. For example $8.56425889209034…$.

### What are non terminating, repeating decimals?

The decimal numbers where there are infinite decimal places but are repeating or shows some fixed pattern is called non terminating but repeating decimals. For example $34.777777…$, $92.82828282…$.

## FAQs

### How do you know that the answer is a terminating decimal? ›

Just divide the numerator by the denominator . **If you end up with a remainder of 0** , then you have a terminating decimal. Otherwise, the remainders will begin to repeat after some point, and you have a repeating decimal.

**What are examples of terminating decimal? ›**

A terminating decimal is a decimal, that has an end digit. It is a decimal, which has a finite number of digits(or terms). Example: **0.15, 0.86**, etc.

**What is a terminating decimal decimal? ›**

A terminating decimal has **a finite number of digits after the decimal point**. For example, 3.782 and 0.25 are terminating decimals.

**Is 3.14 a terminating decimal? ›**

Pi is an irrational number, which means it cannot be represented as a simple fraction, and those numbers **cannot be represented as terminating or repeating decimals**. Therefore, the digits of pi go on forever in a seemingly random sequence.

**Is 0.33333 a terminating decimal? ›**

1/3 = 0.33333... is a recurring, **non-terminating decimal**. You can notice that the digits in the quotient keep repeating.

**Is 7 9 a terminating decimal? ›**

Answer and Explanation:

To convert seven-ninths to a decimal, simply divide 7 by 9. As is evident from the image above, 7/9 is a **non-terminating** decimal equal to 0.777777....

**What is a terminating number? ›**

A terminating number is **a decimal number which decimal ends**. It's a decimal with a finite number of digits. example 3.25, 7.545. terminating number also known as rational number.

**Is 0.5 a terminating number? ›**

For FREE access to this lesson, select your course from the categories below. Students learn that a terminating decimal is a decimal that ends. For example, 0.5 and 36.8924 are terminating decimals.

**Is 2 terminating a decimal? ›**

Hence Decimal Expansion of 2 is **Non-Terminating** and non-repeating.

**Is 0.7 a terminating decimal? ›**

**The decimal 0.7 is also a terminating decimal**, since it has a definite ending point and does not continue on forever. All terminating decimals are rational numbers.

### How do you change a terminating decimal? ›

Step 1: Determine the place value of the last digit in the decimal. Step 2: Write the decimal as a fraction with a denominator that matches the place value of the last digit in the decimal from the previous step. Step 3: Simplify the fraction into its simplest form.

**What are terminating and non-terminating decimal examples? ›**

A non-terminating decimal is a rational number that does not have a finite number of digits after the decimal point. Some examples of terminating decimals are **0.5, 0.7, and 0.9**. Some examples of non-terminating decimals are 0.6, 0.333, and 0.142857.

**Is 0.125 terminating or repeating? ›**

0.5 is a repeating decimal, so 0.5 is a rational number. 0.125 is a **terminating decimal**, so 0.125 is a rational number.

**Is 3 by 8 is a terminating decimal? ›**

And **if numbers after decimals are fixed, then they are said to be terminating, otherwise non-terminating**. Hence, the decimal representation of $\dfrac{3}{8}\to 0.375$ and it is terminating after the decimal.

**Is 7.1234 a irrational number? ›**

7.1234… **is irrational** because it is a nonterminating, nonrepeating decimal.

**Is 3.14159 rational or irrational? ›**

Students are usually introduced to the number pi as having an approximate value of 3.14 or 3.14159. Though it is an irrational number, some people use rational expressions, such as 22/7 or 333/106, to estimate pi. (These rational expressions are accurate only to a couple of decimal places.)

**Is 0.6 terminating or repeating? ›**

In addition, 0.6 is a **terminating decimal**. All terminating decimals are rational numbers.

**Is 3.33333 a terminating decimal? ›**

In fact any decimal number which ends after a limited number of places beyond decimal point, or in which digits repeat endlessly after decimal place, are rational number. Here in 3.33333............ , 3 gets repeated endlessly i.e. till infinity and hence is a rational number.

**Is 7 16 a terminating decimal? ›**

**7/16 has terminating decimal representation**.

**Is 7 by 75 terminating or non terminating? ›**

Hence, the given fraction is **non-terminating**.

### Is 2 3 a terminating decimal? ›

The quotient obtained on dividing 2 by 3 will be 0.666... So, the decimal form of 2/3 is a **non-terminating** and recurring decimal number 0.666...

**What is a terminating answer? ›**

Terminating decimals: Terminating decimals are **those numbers which come to an end after few repetitions after decimal point**. Example: 0.5, 2.456, 123.456, etc. are all examples of terminating decimals.

**Is 3 by 5 is terminating? ›**

Since, the remainder is zero. Therefore, **3/5 is terminating** and 0.6 is a terminating decimal.

**What does no terminating mean? ›**

non·ter·mi·nat·ing ˌnän-ˈtər-mə-ˌnā-tiŋ : **not terminating or ending**. especially : being a decimal for which there is no place to the right of the decimal point such that all places farther to the right contain the entry 0. ¹/₃ gives the nonterminating decimal .33333 …

**Does 3.232323 terminate or repeat? ›**

**It is going to repeat** when we look at the 3.2323. These 3 dots tell me that this pattern of . 23 is going to continue for ever and ever so. These 2 are gopeating.

**Is 0.375 a terminating decimal? ›**

a. = **0.375 is a terminating decimal**.

**Is 2.5 a terminating number? ›**

(i) 5/2 = 2.5, 2/8 = 0.25, 7 = 7.0, etc., are **rational numbers which are terminating decimals**.

**Which of the following is irrational √ 4 9 √ 12 √ 3 √ 7 √ 81? ›**

√81. Therefore, **√7** is irrational.

**Is 0.333 a terminating or repeating decimal? ›**

=13 is a **non-terminating but repeating number** that can be written in the form of pqwhere p and q belong to the set of integers and q is not equal to 0, making it a rational number.

**Is √ 2 terminating or non-terminating? ›**

As can be observed from the expansion, the decimal expansion of the number 2 is **non-terminating** non-recurring. Was this answer helpful?

### Is 0.333 a irrational number? ›

And also 0.3333 is non-terminating as the decimal is not ending or the remainder for 1/3 is not zero. So from 2) 0.333 is an irrational and it is non terminating.

**Is 3.14 rational or irrational? ›**

Yes, 3.14 is a rational number because it is terminating. But is not a rational number because the exact value of is 3.141592653589793238…which is non terminating non recurring.

**Is 0.3 Repeating a terminating decimal? ›**

If the repetend is anything other than 0, we say the decimal number is a repeating decimal. How about a question? Which of the following decimal numbers are repeating and which are terminating: 0.25, 0.3, 0.1212 … and 0.123123 … ? Answer: **the first two are terminating decimals**.

**Does a terminating decimal stop? ›**

**A terminating decimal is a decimal that ends**. It's a decimal with a finite number of digits.

**Is 2 by 7 a terminating decimal? ›**

2/7 in Decimal Form

2/7 in decimal notation has unlimited decimal places. That is, 2/7 as decimal is a **non-terminating**, repeating decimal.

**What is 1 8 as a terminating decimal? ›**

Answer: 1/8 as a decimal is written as **0.125**.

**What is an example of a non-terminating decimal? ›**

Decimals of this type cannot be represented as fractions, and as a result are irrational numbers. Pi is a non-terminating, non-repeating decimal. π = 3.141 592 653 589 793 238 462 643 383 279 ... e is a non-terminating, non-repeating decimal.

**What is repeating terminating and non-terminating decimal? ›**

A repeating decimal has an endless number of digits, yet all of the digits are known. For the decimal to be considered repeating, the digits after the decimal point cannot all be zero. Not all of the digits are known for non-terminating decimals that do not repeat.

**What is non-terminating repeating example? ›**

Non-terminating decimals are numbers that keep going after the decimal point. They go forever and do not end, and if they do, it happens after an extremely long interval. Example: **10/3 which equals 3.33333333…, 0.111111..., 0.233333**…, these are some examples.

**Is 0.4545 is a terminating decimal? ›**

**Terminating decimals, such as 0.4545, can repeat, but they must have an end**.

### Is 1.6 terminating or repeating? ›

Repeating decimals are the one, which has a set of terms in decimal to be repeated uniformly. We can write repeating 1.6 as 1.66666…, that is, the number after the decimal repeats. This can also be written as 1. ¯6 .

**Is 0.123123 a repeating decimal? ›**

For example, **0.123123123. . . is a repeating decimal**; the “123” will repeat endlessly. Any repeating decimal is equal to a rational number. For example, 0.123123. . . is equal to 123/999, or 41/333.

**What is 7/8 as a terminating decimal? ›**

Answer: 7/8 as a decimal is written as **0.875**.

**Is 5 12 a terminating decimal? ›**

5/12 Has the prime factors in the denominator as 2 and 3. Thus 5/12 is **not a terminating decimal**. The denominator is not in the form of 2𝑝 then 31/375 is not a terminating decimal.

**Is 4 11 a terminating decimal? ›**

On the left, the division process terminates with a zero remainder. Hence, 39/80=0.4875 is called a terminating decimal. On the right, the remainders repeat in a pattern and the quotient also repeats in blocks of two. Hence, 4/11=0.3636… **is called a repeating decimal**.

**Is 1.10100100010000 rational or irrational? ›**

Answer: The number,1.101001000100001…, is non-terminating non-repeating (non-recurring), it is an **irrational number**.

**Is 3.1415926535 rational or irrational? ›**

Non-terminating and non-repeating decimals are **irrational**. Hence, 3.1415926535... and √3 are irrational.

**Is 0.4014001400014 a irrational number? ›**

(d) 0.4014001400014... is a non-terminating and non-recurring decimal and therefore **is an irrational number**.

**Is 0.329 a terminating decimal? ›**

Assertion : 0.329 is a terminating decimal. Reason : **A decimal in which a digit or a set of digits is repeated periodically, is called a repeating, or a recurring, decimal**.

**Is 0.75 a terminating decimal? ›**

We find that **on long division 34=0.75 which is a terminating decimal**.

### Is 0.6 a terminating decimal? ›

In addition, **0.6 is a terminating decimal**. All terminating decimals are rational numbers.

**Is 0.1212 a terminating decimal? ›**

The decimal is **non-terminating**.

Therefore, the fraction form of 0.1212 is 433.

**Is 0.333 terminating or repeating? ›**

=13 is a **non-terminating but repeating** number that can be written in the form of pqwhere p and q belong to the set of integers and q is not equal to 0, making it a rational number.

**Is 34.12345 a terminating decimal fraction? ›**

The correct option is A True

Assertion :The denominator of 34.12345 is of the form 2n×5m, where m,n are non-negative integers. Reason: **34.12345 is a terminating decimal fraction**.

**Is 0.999 a terminating decimal? ›**

0.999… **represents a sequence of terminating decimals** where each number in the sequence is a string of 9's after the decimal point. The first number in the sequence is 0.9, the second number is 0.99, the third number is 0.999, the fourth number is 0.9999, and so on.

**Is 0.16666 a terminating decimal? ›**

The fraction 16=0.16666… =0.1˙6. Only the digit 6 repeats indefinitely. This is an example of an eventually **recurring decimal**.

**Is 0.666666666 rational or irrational? ›**

∴0.6666666...... is a **rational number**.