• Home
  • /
  • Blog
  • /
  • Types of Decimal Numbers(With Examples)

Types of Decimal Numbers(With Examples)

types of decimal numbers

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.

girl-with-teacher-happy
Maths can be really interesting for kids

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, 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, 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, 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 $\frac {7129}{500} = 14 \frac{129}{500}$.

Is your child struggling with Maths?
frustrated-kid
We can help!
Country
  • Afghanistan 93
  • Albania 355
  • Algeria 213
  • American Samoa 1-684
  • Andorra 376
  • Angola 244
  • Anguilla 1-264
  • Antarctica 672
  • Antigua & Barbuda 1-268
  • Argentina 54
  • Armenia 374
  • Aruba 297
  • Australia 61
  • Austria 43
  • Azerbaijan 994
  • Bahamas 1-242
  • Bahrain 973
  • Bangladesh 880
  • Barbados 1-246
  • Belarus 375
  • Belgium 32
  • Belize 501
  • Benin 229
  • Bermuda 1-441
  • Bhutan 975
  • Bolivia 591
  • Bosnia 387
  • Botswana 267
  • Bouvet Island 47
  • Brazil 55
  • British Indian Ocean Territory 246
  • British Virgin Islands 1-284
  • Brunei 673
  • Bulgaria 359
  • Burkina Faso 226
  • Burundi 257
  • Cambodia 855
  • Cameroon 237
  • Canada 1
  • Cape Verde 238
  • Caribbean Netherlands 599
  • Cayman Islands 1-345
  • Central African Republic 236
  • Chad 235
  • Chile 56
  • China 86
  • Christmas Island 61
  • Cocos (Keeling) Islands 61
  • Colombia 57
  • Comoros 269
  • Congo - Brazzaville 242
  • Congo - Kinshasa 243
  • Cook Islands 682
  • Costa Rica 506
  • Croatia 385
  • Cuba 53
  • Cyprus 357
  • Czech Republic 420
  • Denmark 45
  • Djibouti 253
  • Dominica 1-767
  • Ecuador 593
  • Egypt 20
  • El Salvador 503
  • Equatorial Guinea 240
  • Eritrea 291
  • Estonia 372
  • Ethiopia 251
  • Falkland Islands 500
  • Faroe Islands 298
  • Fiji 679
  • Finland 358
  • France 33
  • French Guiana 594
  • French Polynesia 689
  • French Southern Territories 262
  • Gabon 241
  • Gambia 220
  • Georgia 995
  • Germany 49
  • Ghana 233
  • Gibraltar 350
  • Greece 30
  • Greenland 299
  • Grenada 1-473
  • Guadeloupe 590
  • Guam 1-671
  • Guatemala 502
  • Guernsey 44
  • Guinea 224
  • Guinea-Bissau 245
  • Guyana 592
  • Haiti 509
  • Heard & McDonald Islands 672
  • Honduras 504
  • Hong Kong 852
  • Hungary 36
  • Iceland 354
  • India 91
  • Indonesia 62
  • Iran 98
  • Iraq 964
  • Ireland 353
  • Isle of Man 44
  • Israel 972
  • Italy 39
  • Jamaica 1-876
  • Japan 81
  • Jersey 44
  • Jordan 962
  • Kazakhstan 7
  • Kenya 254
  • Kiribati 686
  • Kuwait 965
  • Kyrgyzstan 996
  • Laos 856
  • Latvia 371
  • Lebanon 961
  • Lesotho 266
  • Liberia 231
  • Libya 218
  • Liechtenstein 423
  • Lithuania 370
  • Luxembourg 352
  • Macau 853
  • Macedonia 389
  • Madagascar 261
  • Malawi 265
  • Malaysia 60
  • Maldives 960
  • Mali 223
  • Malta 356
  • Marshall Islands 692
  • Martinique 596
  • Mauritania 222
  • Mauritius 230
  • Mayotte 262
  • Mexico 52
  • Micronesia 691
  • Moldova 373
  • Monaco 377
  • Mongolia 976
  • Montenegro 382
  • Montserrat 1-664
  • Morocco 212
  • Mozambique 258
  • Myanmar 95
  • Namibia 264
  • Nauru 674
  • Nepal 977
  • Netherlands 31
  • New Caledonia 687
  • New Zealand 64
  • Nicaragua 505
  • Niger 227
  • Nigeria 234
  • Niue 683
  • Norfolk Island 672
  • North Korea 850
  • Northern Mariana Islands 1-670
  • Norway 47
  • Oman 968
  • Pakistan 92
  • Palau 680
  • Palestine 970
  • Panama 507
  • Papua New Guinea 675
  • Paraguay 595
  • Peru 51
  • Philippines 63
  • Pitcairn Islands 870
  • Poland 48
  • Portugal 351
  • Puerto Rico 1
  • Qatar 974
  • Romania 40
  • Russia 7
  • Rwanda 250
  • Réunion 262
  • Samoa 685
  • San Marino 378
  • Saudi Arabia 966
  • Senegal 221
  • Serbia 381 p
  • Seychelles 248
  • Sierra Leone 232
  • Singapore 65
  • Slovakia 421
  • Slovenia 386
  • Solomon Islands 677
  • Somalia 252
  • South Africa 27
  • South Georgia & South Sandwich Islands 500
  • South Korea 82
  • South Sudan 211
  • Spain 34
  • Sri Lanka 94
  • Sudan 249
  • Suriname 597
  • Svalbard & Jan Mayen 47
  • Swaziland 268
  • Sweden 46
  • Switzerland 41
  • Syria 963
  • Sao Tome and Principe 239
  • Taiwan 886
  • Tajikistan 992
  • Tanzania 255
  • Thailand 66
  • Timor-Leste 670
  • Togo 228
  • Tokelau 690
  • Tonga 676
  • Trinidad & Tobago 1-868
  • Tunisia 216
  • Turkey 90
  • Turkmenistan 993
  • Turks & Caicos Islands 1-649
  • Tuvalu 688
  • U.S. Outlying Islands
  • U.S. Virgin Islands 1-340
  • UK 44
  • US 1
  • Uganda 256
  • Ukraine 380
  • United Arab Emirates 971
  • Uruguay 598
  • Uzbekistan 998
  • Vanuatu 678
  • Vatican City 39-06
  • Venezuela 58
  • Vietnam 84
  • Wallis & Futuna 681
  • Western Sahara 212
  • Yemen 967
  • Zambia 260
  • Zimbabwe 263
Age Of Your Child
  • Less Than 6 Years
  • 6 To 10 Years
  • 11 To 16 Years
  • Greater Than 16 Years

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. For example, $2.3333…$, $11.121212…$, $5.6765675….$, etc. are 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.

Difference Between Infinity & Not Defined

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. 

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 $6$ digits. 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 decimal such as $0.3333…$ then multiply by $10$, if 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, $0.4333333… \left( 0.4\overline{3} \right) = \frac {13}{30}$

6 Amazing Facts About Numbers

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 $\pi$.

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

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 $\frac {p}{q}$.

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.
    • $0.1111… = 0.\overline{1}$.
    • $0.121212… = 0.\overline{12}$.
    • $0.53333… = 0.5\overline{3}$.
    • $0.724444… = 0.72\overline{4}$.
    • $0.6323232… = 0.6\overline{32}$.

Recommended Reading

FAQs

What are decimal numbers and examples?

A decimal number consists of two parts separated by a decimal point ($.$) – a whole part and a fractional part. The whole part is written to the left of the decimal point, while the fractional (or decimal) part is written to the right of the decimal point.

For example, $54.96$ is a decimal number where ‘$54$’ is the whole part and ‘$96$’ is the fractional (decimal) part.

How many decimal numbers are there?

There are two main categories of decimal numbers – terminating decimal numbers and non-terminating decimal numbers. The non-terminating decimal numbers are again divided into two types – non-terminating but recurring and non-terminating and recurring.

What is a terminating decimal number?

A terminating decimal number is a decimal number that has a countable (or finite) number of decimal places.

For example, $3.8$, $7.52$, and $98.8976$ are all terminating decimal numbers.
$3.8$ terminating after $1$ decimal places, $7.52$ terminating after $2$ decimal places, and $98.8976$ terminating after $4$ decimal places.

What is an example of a terminating decimal?

An example of a terminating decimal is $98.654$ terminating after $3$ decimal places.

What is a non-terminating decimal number?

A non-terminating decimal number is a decimal number that has an uncountable (or infinite) number of decimal places, i.e., the decimal place never terminates.

For example, $6.564379879583…$ is a non-terminating decimal number.

What is an example of a non-terminating decimal?

An example of a terminating decimal is $12.75648654…$. The most commonly used non-terminating decimal number is $\pi$. Its value is $3.1415926535 8979323846 2643383279 5028841971…$.

What is a recurring decimal and example?

A recurring decimal is a type of non-terminating decimal number in which the decimal places do not terminates but it repeats in a fixed pattern. 

For example, $5.22222…$ is a non-terminating but recurring decimal number with a $1$ digit recurring (or repeating). It can also be written as $5.\overline{2}$.

Another example of a non-terminating but recurring decimal number is $73.23232323…$, where $2$ decimal places are recurring.  It can also be written as $73.\overline{23}$.

What is a non-recurring decimal and example?

A non-recurring decimal is a type of non-terminating decimal number in which the decimal places do not terminates and do not recur(or repeat) also. 

For example, $5.938504834949203823…$ is a non-terminating and recurring decimal number.

The most widely used non-terminating and non-recurring decimal number in mathematics is $\pi = 3.1415926535 8979323846 2643383279 5028841971…$.

{"email":"Email address invalid","url":"Website address invalid","required":"Required field missing"}
>