Irrational Numbers – Definition, Properties & Examples

This post is also available in: हिन्दी (Hindi)

There are real numbers that can be expressed in the form of ratios. These numbers are called rational numbers. But there are real numbers that cannot be expressed in the form of ratios. Such numbers are called irrational numbers. The irrational numbers are the numbers that cannot be expressed in the form of $\frac {p}{q}$, where $p$ and $q$ are integers.

Let’s understand irrational numbers and their properties.

What are Irrational Numbers?

Irrational numbers are real numbers that are not rational numbers. These numbers cannot be expressed as a ratio, i.e., in the form of $\frac {p}{q}$, where $p$ and $q$ are integers and $q \ne 0$. 

Also, the decimal expansion of irrational numbers is non-terminating and non-recurring decimals.

Note: Decimal expansion of rational numbers is either terminating or non-terminating but a recurring decimal number.

Examples of Irrational Numbers

The numbers like $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt[3]{4}$, $\sqrt[3]{6}$, $\sqrt[4]{7}$, $\sqrt[5]{8}$ are all irrational numbers. All these numbers have non terminating and non recurring decimal expansions.

Some of the widely used irrational numbers are

  • $\pi = 3⋅14159265…$. Since the value of $\pi$ is closer to the fraction $\frac {22}{7}$, we take the value of $\pi$ as $\frac {22}{7}$ or $3.14$
  • Euler’s number $e = 2⋅718281⋅⋅⋅⋅$
  • Golden ratio, $\phi = 1.61803398874989….$

Locating Irrational Numbers On a Number Line

The irrational numbers along with the rational numbers are real numbers. Hence, a unique point is considered to represent them on the number line. Some irrational numbers in the form of $\sqrt{n}$, where $n$ is a positive integer can be represented on a number line by using the following steps.

Step 1: Split the number inside the square root such that the sum adds up to the number

Step 2: The distance between these two natural numbers should be equal on the number line starting from the origin. One line should be perpendicular to the other

Step 3: Use the Pythagoras Theorem

Step 4: Represent the area as the desired measurement

Let us consider the example of $\sqrt{2}$ to understand this better.

Step 1: Draw a number line with the center as zero, left of zero as $-1$, and right of zero as $1$

Step 2: Keeping the same length as between $0$ and $1$, draw a line perpendicular to point $1$, such that the new line has a length of $1$ unit.

Step 3: Draw a line from $0$ to the end of the perpendicular line constructing a right-angled triangle ABC. With AB as height, BC as the base, and AC as the hypotenuse

irrational numbers

Step 4: Find the length of the hypotenuse i.e. AC by applying the Pythagoras theorem

$AC^{2} = AB^{2} + BC^{2} => AC^{2} = 1^{2} + 1^{2} => AC^2 = 2 => AC = \sqrt {2}$

irrational numbers

Step 5: Keeping AC as the radius with C as the center, cut an arc on the same number line naming it D. CD will also become the radius of the arc with length $\sqrt{2}$.

Step 6: Therefore, point D represents $\sqrt{2}$ on the number line.

irrational numbers
Amazing Facts About Numbers

Properties of Irrational Numbers

You can perform any of the following four basic operations on irrational numbers. 

  • Addition
  • Subtraction
  • Multiplication 
  • Division

Each of these operations shows one or more of the following properties:

  • Commutative Property
  • Associative Property

Let’s understand these properties of irrational numbers in detail.

Commutative Property of Irrational Numbers

The commutative property deals with the ordering of numbers in an operation. It states that the result remains the same even if the order of numbers in the operation is changed or swapped.

The commutative property is exhibited by the operations of addition and multiplication in the set of irrational numbers. The operations subtraction and division do not show exhibit commutative property in the set of irrational numbers.

Commutative Property of Addition of Irrational Numbers

It states that for any two irrational numbers their sum remains the same even if the positions of the numbers are interchanged or swapped.

Mathematically, it is represented as if $a, b \in \overline{Q}, \text {then  } a + b = b + a$.

For example, $\sqrt{7}$ and $\sqrt{11}$ are two irrational numbers. $\sqrt{7} + \sqrt{11} = \sqrt{11} + \sqrt{7}$ 

Commutative Property of Multiplication of Irrational Numbers

It states that for any two irrational numbers their product remains the same even if the positions of the numbers are interchanged or swapped.

Mathematically, it is represented as if $a, b \in \overline{Q}, \text {then  } a \times b = b \times a$.

For example, $\sqrt{3}$ and $\sqrt{5}$ are two irrational numbers. $\sqrt{3} \times \sqrt{5} = \sqrt{15}$ and also $\sqrt{5} \times \sqrt{3} = \sqrt{15}$.

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

Associative Property of Irrational Numbers

The associative property deals with the grouping of irrational numbers in an operation. It states that the result remains the same even if the grouping of numbers is changed while performing the operation.

The associative property is exhibited by the operations addition and multiplication in the set of irrational numbers. The operations subtraction and division do not show exhibit associative property in the set of irrational numbers.

Associative Property of Addition of Irrational Numbers

It states that the sum of any three irrational numbers remains the same even if the grouping of the numbers is changed. 

Mathematically, it is represented as if $a, b, c \in \overline{Q}, \text {then  } \left (a + b \right) + c = a + \left(b + c \right)$. 

For example, for three irrational numbers $\sqrt{2}$, $\sqrt{5}$ and $\sqrt{7}$, $\left( \sqrt{2} + \sqrt{5} \right) + \sqrt{7}$ and $\sqrt{2} + \left(\sqrt{5} + \sqrt{7} \right)$ are equal.

Associative Property of Multiplication of Irrational Numbers

It states that the product of any three irrational numbers remains the same even if the grouping of the numbers is changed. 

Mathematically, it is represented as if $a, b, c \in \overline{Q}, \text {then  } \left (a \times b \right) \times c = a \times \left(b \times c \right)$. 

For example, for three irrational numbers $\sqrt{3}$, $\sqrt{5}$ and $\sqrt{11}$, $\left( \sqrt{3} \times \sqrt{5} \right) \times \sqrt{11} = \sqrt{15} \times \sqrt{11} = \sqrt{165}$ and $\sqrt{3} \times \left(\sqrt{5} \times \sqrt{11} \right) = \sqrt{3} \times \sqrt{55} = \sqrt{165}$.

Difference Between Rational Numbers & Irrational Numbers

Following is the differences between rational and irrational numbers.

ParameterRational NumbersIrrational Numbers
MeaningRational numbers are the numbers that can be represented as a ratio of two numbers. They exist in the form of $\frac {p}{q}$ where $p$ and $q$ are integers and $q \ne 0$
$7$ can be represented $\frac {7}{1}$
Irrational numbers are numbers that cannot be represented as a ratio of two numbers. The number $\pi = 3.14159265358979…$ cannot be represented in $\frac {p}{q}$ form
PropertiesThe rational numbers exhibit 
a) closure property of addition, subtraction, multiplication, and division
b) commutative property of addition and multiplication
c) associative property of addition and multiplication
d) distributive property of multiplication over addition and subtraction
e) existence of additive identity
f) existence of multiplicative identity
g) existence of the additive inverse
h) existence of the multiplicative inverse 
The irrational numbers exhibit
a) closure property of addition and subtraction (closure property of  multiplication and division do not exist)
b) commutative property of addition and multiplication
c) associative property of addition and multiplication
d) additive identity of irrational numbers does not exist ($0$ is a rational number)
e) multiplicative identity of irrational numbers does not exist ($1$ is a rational number)
f) existence of the additive inverse
g) existence of the multiplicative inverse
FormBoth the numerator and the denominators of the rational numbers are whole numbers, where the denominator of a rational number is not equal to zeroThey cannot be represented in fractional form
Decimal formThey are numbers having decimal places that are finite or infinite but recurring in nature. eg: 2.9, 7, 8.66…, 9.1212…They are non-terminating or non-recurring in nature. e.g: 3.141592…, 1.61803398…

Conclusion

Irrational numbers are numbers that are not rational numbers, i.e., these numbers cannot be expressed in $\frac {p}{q}$ form. When expanded in a decimal form they result in non terminating and non recurring decimal numbers.

Practice Problems

  1. Which of the following are irrational numbers
    • $\sqrt{4}$
    • $\sqrt{8}$
    • $\sqrt{6}$
    • $\sqrt[3]{3}$
    • $\sqrt[3]{9}$
    • $\sqrt[3]{18}$
    • $\sqrt[4]{2}$
    • $\sqrt[4]{4}$
    • $\sqrt[4]{6}$
  2. State True or False
    • The sum of two irrational numbers is always an irrational number
    • The difference between two irrational numbers is always an irrational number
    • The product of two irrational numbers is always an irrational number
    • The quotient of two irrational numbers is always an irrational number

Recommended Reading

FAQs

What are irrational numbers in math?

Irrational numbers are real numbers that are not rational numbers. These numbers cannot be expressed as a ratio, i.e., in the form of $\frac {p}{q}$, where $p$ and $q$ are integers and $q \ne 0$. 

Also, the decimal expansion of irrational numbers is non-terminating and non-recurring decimals.

Are rational numbers and irrational numbers the same?

No, rational numbers and irrational numbers are two different categories of real numbers. The numbers which are rational numbers are not irrational numbers and vice-versa.

How can you identify an irrational number?

The numbers that cannot be written in the form of fractions are irrational numbers and when expressed in decimal form have non terminating and non recurring expansion.

Why $\pi$ is an irrational number?

The decimal expansion of $pi$ is a non terminating and non recurring decimal number. The usual form of $\pi$ as $\frac {22}{7}$ is just an approximation to make calculation easy.

Are irrational numbers non terminating and non recurring only?

Yes, irrational numbers are non terminating and non recurring only. If the decimal expansion of a number is terminating decimal then it’s a rational number and even if the decimal expansion is non terminating but recurring, then also the number is a rational number.

How many irrational numbers lie between $\sqrt{2}$ and $\sqrt{3}$?

There lie an infinite number of irrational numbers between $\sqrt{2}$ and $\sqrt{3}$. In fact, there are always an infinite number of irrational numbers between any two irrational numbers.

How many irrational numbers lie between any two rational numbers?

Between any two rational numbers, you can find an infinite number of irrational numbers.

2 thoughts on “Irrational Numbers – Definition, Properties & Examples”

  1. I think there is an error. I do not believe that “irrational numbers exhibit the closure property” under addition or subtraction. sqrt{2} – sqrt{2} = 0 is a counter-example because 0 is rational.

    Reply

Leave a Comment