• Home
• /
• Blog
• /
• Additive Inverse of Decimal Numbers – Definition & Property (With Examples)

# Additive Inverse of Decimal Numbers – Definition & Property (With Examples)

August 10, 2022

This post is also available in: हिन्दी (Hindi)

What comes to your mind when you think of the word ‘inverse’? Inverse means something opposite. So, what could be the meaning of the word ‘additive inverse’?

The term additive inverse is related to the term additive identity. The additive inverse of a number is a number that when added to the number gives an additive identity.

For example, additive inverse of $5$ is $-5$, since $5 + \left(-5 \right) = 0$.

The additive inverse of a number is its opposite number. If a number is added to its additive inverse, the sum of both the numbers becomes $0$ (zero).

The simple rule to get an additive inverse of a number is to change the sign of a number, i.e.,

• change the positive number to a negative number
• change the negative number to a positive number

For example, the additive inverse of $8$ is $-8$ and the additive inverse of $-6$ is $6$.

Observe that $8 + \left(-8 \right) = 0$ and $-6 + 6 = 0$. For more on addition and subtraction check Addition and Subtraction of Integers

Note: $0$ is an additive identity of a number.

The additive inverse property of numbers states that when a number is added to its additive inverse, the sum obtained is an additive identity, i.e., zero ($0$).

Mathematically it is represented as: For any number $a$, if $a + b = 0$, then $b$ is the additive inverse of $a$.

For example, $9 + \left(-9\right) = 0$, therefore, $-9$ is additive inverse of $9$.

Let’s find out which sets of numbers have an additive inverse.

Maths can be really interesting for kids

### Additive Inverse of Natural Numbers

A set of natural numbers is a set of numbers that starts with $1$ and moves on to $2$, $3$, $4$, and so on. That is $N = \{1, 2, 3, 4, …\}$.

You have seen above that for any positive number, its additive inverse is a negative number. Since the set of natural numbers does not have any negative numbers, therefore, the numbers in the set of natural numbers do not have an additive inverse.

For a set of natural numbers, the properties – closure property, commutative property, associative propertydistributive property, and multiplicative identity property hold, but additive identity property, additive inverse property, and multiplicative inverse property do not hold.

### Additive Inverse of Whole Numbers

A set of whole numbers is a set of numbers that starts with $0$ and moves on to $1$, $2$, $3$, and so on. That is $W = \{0, 1, 2, 3, …\}$.

You have seen above that for any positive number, its additive inverse is a negative number. Since the set of whole numbers does not have any negative numbers, therefore, the numbers in the set of whole numbers do not have an additive inverse.

For a set of whole numbers, the properties – closure property, commutative property, associative propertydistributive propertyadditive identity property, and multiplicative identity property hold but, additive inverse property, and multiplicative inverse property do not hold.

A set of integers is a set of numbers that contains all the natural numbers $1$, $2$, $3$, … and their corresponding negative values $-1$, $-2$, $-3$, …, along with $0$, ie., $Z = \{…, -3, -2, -1, 0, 1, 2, 3, … \}$.

You have seen above that for any positive number, its additive inverse is a negative number.  And in a set of integers, for every positive integer, there is a corresponding negative number and for every negative number and for every negative number, its additive inverse is a positive number, and there is a corresponding positive number, therefore, the set of integers has an additive inverse for every number.

For a set of whole numbers, the properties – closure property, commutative property, associative propertydistributive property, additive identity property, multiplicative identity property, and additive inverse property hold but, the multiplicative inverse property does not hold.

### Additive Inverse of Rational Numbers

Rational numbers are the numbers that can be expressed in the form $\frac {p}{q}$, where both $p$ and $q$ are integers and $q \ne 0$.

As the set of rational numbers contains both negative and positive numbers and as seen above that for any positive number, its additive inverse is a negative number and for every negative number, its additive inverse is a positive number, therefore the set of rational numbers has an additive inverse for every number.

For a set of rational numbers, the properties – closure property, commutative property, associative propertydistributive propertyadditive identity propertymultiplicative identity propertyadditive inverse property, and multiplicative inverse property hold.

Is your child struggling with Maths?
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
• 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
• Cape Verde 238
• Caribbean Netherlands 599
• Cayman Islands 1-345
• Central African Republic 236
• 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
• Egypt 20
• 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
• 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
• 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
• 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
• 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
• Less Than 6 Years
• 6 To 10 Years
• 11 To 16 Years
• Greater Than 16 Years

### Additive Inverse of Irrational Numbers

Irrational numbers is the numbers that when expressed in decimal form have non-terminating and non-recurring decimal places.

The additive inverse of any irrational number will be a number with an opposite sign, therefore the set of an irrational number will have an additive inverse for every irrational number.

For a set of irrational numbers, the properties – commutative property, associative propertydistributive property, additive inverse property, and multiplicative inverse property holds, whereas, closure property, additive identity property, and multiplicative identity property do not hold.

### Additive Inverse of Real Number

A set of real numbers is a set consisting of all the sets – natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Therefore, the set of real numbers will have an additive inverse for every real number.

For a set of real numbers, all the properties – closure property, commutative property, associative propertydistributive property, additive identity property, multiplicative identity property, additive inverse property, and multiplicative inverse property holds.

## Difference Between Additive Inverse and Multiplicative Inverse

Following are the differences between additive inverse and multiplicative inverse

## Conclusion

An additive inverse of a number is a number that when added with the number results in an additive identity. Some sets have additive inverse for the numbers whereas some sets do not have additive inverse for the numbers.

## Practice Problems

1. Which of the following sets have additive inverse for the numbers?

Natural numbers, Whole numbers, Integers, Rational numbers, Irrational numbers, Real numbers

1. Which of the following sets do not have additive inverse for the numbers?

Natural numbers, Whole numbers, Integers, Rational numbers, Irrational numbers, Real numbers

1. Find the additive inverse of
• $19$
• $-25$
• $56.8$
• $-0.004$
• $\frac {3}{4}$
• $-\frac {45}{21}$
• $2 \frac {1}{5}$
• $\sqrt{2}$
• $9 + \sqrt{3}$
• $-12 + \sqrt{5}$

## FAQs

### What is the additive inverse?

The additive inverse of a number is its opposite number. If a number is added to its additive inverse, the sum of both the numbers becomes $0$ (zero). $0$ is also called the additive identity of a number.

### How do you find the additive inverse of a number?

To find the additive inverse of a number, you have to just change the sign of a number. The additive inverse of a positive number is a negative number and that of a negative number is a positive number.

For example, additive inverse of $47$ is $-47$ and additive inverse of $-91$ is $91$.

Also, the additive inverse of $\frac {2}{11}$ is $-\frac {2}{11}$ and additive inverse of $-\frac {7}{15}$ is $\frac {7}{15}$.

### What is the additive inverse formula?

For any number $a$, the additive inverse is $-1 \times a$.

For example, additive inverse of $4$ is $-1 \times 4 = -4$ and additive inverse of $-12$ is $-1 \times \left(-12 \right) = 12$.

### What is the additive inverse of zero?

Since zero ($0$) does not have any sign, neither positive nor negative, therefore, the additive identity of $0$ is $0$.

Note:  You do not have to add anything to $0$ to get $0$ and nothing means $0$.

### What is the additive inverse property?

The additive inverse property states that the sum of a number and its additive inverse is always $0$. For example, $7 + \left( -7 \right) = 0$.

No, the additive inverse is not the same as the additive identity. The additive identity for every number is $0$, whereas the additive inverse is a number with an opposite sign.
The additive identity for all the numbers $5$, $-8$ and $\frac {2}{3}$ is $0$, whereas the additive inverse of $5$ is $-5$, $-8$ is $8$ and that of $\frac {2}{3}$ is $-\frac {2}{3}$.
The additive inverse is what we add to a number to make the sum zero ($0$), whereas, the multiplicative inverse is the reciprocal of the given number, which when multiplied together, gives the product as $1$.