# Multiplicative Identity of Decimal Numbers (Definition & Examples)

This post is also available in: हिन्दी (Hindi)

In mathematics, you use different types of numbers and operators in your calculations. The operators in mathematical expressions along with the numbers result in different types of results.  You might have noticed that many times an operation performed on a certain number gives the same result. Such types of numbers(or elements) are known as identities.

All operators are associated with their own identities. The identity associated with multiplication is known as multiplicative identity. Let’s understand what is a multiplicative identity and what are its properties.

## What is  Multiplicative Identity?

The multiplicative identity is related to the process of multiplication and as the word ‘identity’ signifies, multiplicative identity is a number that when multiplied by a number gives an output that is the same as that of the number itself.

In the case of multiplication, such a number is ‘$1$’ (one). You might have noticed that when $1$ is multiplied by any number, the result is always the same number.

For example, $19 \times 1 = 1 \times 19 = 19$. You see here that when $1$ is multiplied by $19$, the result is the number itself i.e., $19$. Let’s consider one more example. $-27 \times 1 = 1 \times \left(-27 \right) = -27$. Here also the result remains the same after multiplying by $1$. It means for $-27$ also, the identity element is $1$.

In general, we can say that for any number $a$, $1$ is called a multiplicative identity, as $a \times 1 = 1 \times a = a$.

Multiplicative identity exists for most of the set of numbers. Still, there are few sets of numbers, where multiplicative identity does not exist. Let’s explore which of the set of numbers has the multiplicative identity and for which the multiplicative identity does not exist.

Note: $-1$ is not a multiplicative identity.

### Multiplicative Identity 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 $N = \{1, 2, 3, 4, …\}$.

Since the first element of the set of natural numbers is $1$, we see that the multiplicative identity exists in the case of natural numbers. Thus, the set of natural numbers has a multiplicative identity.

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

### Multiplicative Identity 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 $W = \{0, 1, 2, 3, …\}$.

Here, you’ve noticed that $1$ exists in the set of whole numbers which is a multiplicative identity. Thus, the set of whole numbers has a multiplicative identity.

For a set of whole numbers, the properties – closure property, commutative property, associative property,  distributive property,  multiplicative identity property, and additive identity property holds.

### Multiplicative Identity of Integers

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’ve noticed that the number $1$ exists in the set of integers $Z$, thus, the set of integers has a multiplicative identity.

For a set of integers, the properties – closure property, commutative property, associative property,  distributive property,  multiplicative identity property, and additive identity property holds.

### Multiplicative Identity 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 $q \ne 0$.

Also, every whole number such as $0$, $1$, $2$, … can be expressed as $\frac {0}{1}$, $\frac {1}{1}$, $\frac {2}{1}, …$, therefore, $1$ is a rational number and is a multiplicative identity, and hence, the set of rational numbers have a multiplicative identity.

For a set of rational numbers, the properties – closure property, commutative property, associative property,  distributive property,  multiplicative identity property, and additive identity property holds.

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

### Multiplicative Identity of Irrational Numbers

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

$1$ the multiplicative identity is a terminating decimal number $\left(1.0\right)$, therefore, $1$ is not an irrational number and hence, a set of irrational numbers does not have a multiplicative identity.

For a set of irrational numbers, the properties – commutative property, associative property, and distributive property holds, but closure property, additive identity property, and multiplicative identity property does not hold.

## Difference Between Additive Identity and Multiplicative Identity

These are the differences between additive identity and multiplicative identity of numbers.

## Practice Problems

1. Which of the following is the multiplicative identity for numbers?
• $1$
• $-1$
• $0$
• None of these
2. Which of the following sets have a multiplicative identity?
• Natural Numbers
• Whole Numbers
• Integers
• Rational Numbers
• Irrational Numbers
• Real Numbers
3. Which of the following sets does not have a multiplicative inverse?
• Natural Numbers
• Whole Numbers
• Integers
• Rational Numbers
• Irrational Numbers
• Real Numbers
4. The multiplicative identity which when added to a number gives the same result
• True
• False
5. The multiplicative identity which when subtracted from a number gives the same result
• True
• False
6. The multiplicative identity which when multiplied by a number gives the same result
• True
• False

## FAQs

### What is multiplicative identity?

A multiplicative identity is a number that when multiplied by any number gives the same number. The number $1$ is called the multiplicative identity of numbers, since for any number $a$, $a \times 0 = 0 \times a = a$.

### Which number in math is called the multiplicative identity?

$1$ is called the multiplicative identity of numbers, since for any number $a$, $a \times 1 = 1 \times a = a$.

### Is $-1$ also a multiplicative identity of numbers?

No, $-1$ is not a multiplicative identity of numbers, since when a number is multiplied by $-1$, its sign changes and hence no longer remains the same number.

### Are multiplicative identity and additive identity the same?

No, multiplicative identity and additive identity are two different concepts. The multiplicative identity is for multiplication operations,  whereas additive identity is for the addition operation.

The multiplicative identity is $1$, whereas the additive identity is $0$.

The multiplicative identity when multiplied by any number gives the same result, whereas the additive identity when added to any number gives the same result.

### Do all sets of numbers have a multiplicative identity?

No, all sets of numbers do not have a multiplicative identity. The sets having multiplicative identity are natural numbers, whole numbers, integers, rational numbers, and real numbers. The set that does not have a multiplicative identity is irrational numbers.

## Conclusion

A multiplicative identity is a number that when multiplied with a number gives the same result. $1$ is the multiplicative identity of numbers used in math. Some sets have a multiplicative identity, whereas there are some sets that do not have a multiplicative identity.