The **whole numbers** are the part of the number system which includes all the positive integers from 0 to infinity. These numbers exist in the number line. Hence, they are all real numbers. We can say, all the whole numbers are real numbers, but not all the real numbers are whole numbers. Thus, we can define whole numbers as the set of natural numbers and 0. Integers are the set of whole numbers and negative of natural numbers. Hence, integers include both positive and negative numbers including 0. Real numbers are the set of all these types of numbers, i.e., natural numbers, whole numbers, integers and fractions.

The complete set of natural numbers along with ‘0’ are called whole numbers. The examples are: 0, 11, 25, 36, 999, 1200, etc.

Learn more about numbers here.

**Table of contents:**

- Definition
- Symbol

- Properties
- Closure
- Commutative
- Additive
- Multiplicative
- Associative
- Distributive

- Whole Numbers and Natural numbers
- Solved Examples
- Practice Problems
- Video Lesson
- FAQs

## Whole Numbers Definition

The **whole numbers** are the numbers without fractions and it is a collection of positive integers and zero. It is represented by the symbol “W” and the set of numbers are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,……………}. Zero as a whole represents nothing or a null value.

- Whole Numbers: W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10……}
- Natural Numbers: N = {1, 2, 3, 4, 5, 6, 7, 8, 9,…}
- Integers: Z = {….-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,…}
- Counting Numbers: {1, 2, 3, 4, 5, 6, 7,….}

These numbers are positive integers including zero and do not include fractional or decimal parts (3/4, 2.2 and 5.3 are not whole numbers). Also, arithmetic operations such as addition, subtraction, multiplication and division are possible on whole numbers.

### Symbol

The symbol to represent whole numbers is the alphabet ‘W’ in capital letters.

W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,…}

Thus, the **whole numbers list** includes 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ….

**Facts:**

- All the natural numbers are whole numbers
- All counting numbers are whole numbers
- All positive integers including zero are whole numbers
- All whole numbers are real numbers

If you still have doubt, What is a whole number in maths? A more comprehensive understanding of the whole numbers can be obtained from the following chart:

- Whole Numbers and Natural Numbers
- Natural Numbers
- Difference Between Natural and Whole numbers
- Important Questions For Class 6 Maths

## Whole Numbers Properties

The properties of whole numbers are based on arithmetic operations such as addition, subtraction, division and multiplication. Two whole numbers if added or multiplied will give a whole number itself. Subtraction of two whole numbers may not result in whole numbers, i.e. it can be an integer too. Also, the division of two whole numbers results in getting a fraction in some cases. Now, let us see some more properties of whole numbers and their proofs with the help of examples here.

### Closure Property

They can be closed under addition and multiplication, i.e., if x and y are two whole numbers then x. y or x + y is also a whole number.

**Example:**

5 and 8 are whole numbers.

5 + 8 = 13; a whole number

5 × 8 = 40; a whole number

Therefore, the whole numbers are closed under addition and multiplication.

### Commutative Property of Addition and Multiplication

The sum and product of two whole numbers will be the same whatever the order they are added or multiplied in, i.e., if x and y are two whole numbers, then x + y = y + x and x . y = y . x

**Example:**

Consider two whole numbers 3 and 7.

3 + 7 = 10

7 + 3 = 10

Thus, 3 + 7 = 7 + 3 .

Also,

3 × 7 = 21

7 × 3 = 21

Thus, 3 × 7 = 7 × 3

Therefore, the whole numbers are commutative under addition and multiplication.

### Additive identity

When a whole number is added to 0, its value remains unchanged, i.e., if x is a whole number then x + 0 = 0 + x = x

**Example:**

Consider two whole numbers 0 and 11.

0 + 11 = 11

11 + 0 = 11

Here, 0 + 11 = 11 + 0 = 11

Therefore, 0 is called the additive identity of whole numbers.

### Multiplicative identity

When a whole number is multiplied by 1, its value remains unchanged, i.e., if x is a whole number thenx.1 = x = 1.x

**Example:**

Consider two whole numbers 1 and 15.

1 × 15 = 15

15 × 1 = 15

Here, 1 × 15 = 15 = 15 × 1

Therefore, 1 is the multiplicative identity of whole numbers.

### Associative Property

When whole numbers are being added or multiplied as a set, they can be grouped in any order, and the result will be the same, i.e. if x, y and z are whole numbers then x + (y + z) = (x + y) + z and x. (y.z)=(x.y).z

**Example:**

Consider three whole numbers 2, 3, and 4.

2 + (3 + 4) = 2 + 7 = 9

(2 + 3) + 4 = 5 + 4 = 9

Thus, 2 + (3 + 4) = (2 + 3) + 4

2 × (3 × 4) = 2 × 12 = 24

(2 × 3) × 4 = 6 × 4 = 24

Here, 2 × (3 × 4) = (2 × 3) × 4

Therefore, the whole numbers are associative under addition and multiplication.

### Distributive Property

If x, y and z are three whole numbers, the distributive property of multiplication over addition is x. (y + z) = (x.y) + (x.z), similarly, the distributive property of multiplication over subtraction is x. (y – z) = (x.y) – (x.z)

**Example:**

Let us consider three whole numbers 9, 11 and 6.

9 × (11 + 6) = 9 × 17 = 153

(9 × 11) + (9 × 6) = 99 + 54 = 153

Here, 9 × (11 + 6) = (9 × 11) + (9 × 6)

Also,

9 × (11 – 6) = 9 × 5 = 45

(9 × 11) – (9 × 6) = 99 – 54 = 45

So, 9 × (11 – 6) = (9 × 11) – (9 × 6)

Hence, verified the distributive property of whole numbers.

### Multiplication by zero

When a whole number is multiplied to 0, the result is always 0, i.e., x.0 = 0.x = 0

**Example:**

0 × 12 = 0

12 × 0 = 0

Here, 0 × 12 = 12 × 0 = 0

Thus, for any whole number multiplied by 0, the result is always 0.

### Division by zero

The division of a whole number by o is not defined, i.e., if x is a whole number then x/0 is not defined.

**Also, check:** Whole number calculator

## Difference Between Whole Numbers and Natural Numbers

| |
---|---|

Whole Numbers | Natural Numbers |

Whole Numbers: {0, 1, 2, 3, 4, 5, 6,…..} | Natural Numbers: {1, 2, 3, 4, 5, 6,……} |

Counting starts from 0 | Counting starts from 1 |

All whole numbers are not natural numbers | All Natural numbers are whole numbers |

The below figure will help us to understand the difference between the whole number and natural numbers :

### Can Whole Numbers be negative?

The whole number can’t be negative!

As per definition: {0, 1, 2, 3, 4, 5, 6, 7,……till positive infinity} are whole numbers. There is no place for negative numbers.

### Is 0 a whole number?

Whole numbers are the set of all the natural numbers including zero. So yes, 0 (zero) is not only a whole number but the first whole number.

## Solved Examples

**Example 1:**Are 100, 227, 198, 4321 whole numbers?

**Solution:**Yes. 100, 227, 198, and 4321 are all whole numbers.

**Example 2**:Solve 10× (5 + 10) using the distributive property.

**Solution:** Distributive property of multiplication over the addition of whole numbers is:

x × (y + z) = (x × y) + (x × z)

10× (5 + 10) = (10× 5) + (10× 10)

= 50 + 100

= 150

Therefore, 10× (5 + 10) = 150

However, we can show several examples of whole numbers using the properties of the whole numbers.

### Practice Problems

- Write whole numbers between 12 and 25.
- What is the additive inverse of the whole number 98?
- How many whole numbers are there between -1 and 14?

To learn more concepts like natural numbers, and real numbers in a more engaging way, register at BYJU’S. Also, watch interesting videos on various maths topics by downloading BYJU’S– The Learning App from Google Play Store or the app store.

## Video lesson

## Frequently Asked Questions on Whole Numbers

### What are whole numbers?

The whole numbers are defined as positive integers including zero. The whole number does not contain any decimal or fractional part. It means that it represents the entire thing without pieces. The set of whole numbers is mathematically represented as:

W = (0, 1, 2, 3, 4, 5,……}

### Can whole numbers be negative?

No, the whole numbers cannot be negative. The whole numbers start from 0, 1, 2, 3, … and so on. All the natural numbers are considered as whole numbers, but all the whole numbers are not natural numbers. Thus, the negative numbers are not considered as whole numbers.

### What are the properties of whole numbers?

The properties of whole numbers are:

Whole numbers are closed under addition and multiplication

The addition and multiplication of whole numbers is commutative

The addition and multiplication of whole numbers is associative

It obeys the distributive property of multiplication over addition

The additive identity of whole numbers is 0

The multiplicative identity of whole numbers is 1

### Is 10 a whole number?

10 is a whole as well as a natural number. It is written as Ten in words. Although -10 also represents a whole and not a fraction.

### Which numbers are not whole numbers?

The numbers which do not exist between 0 and infinity are not whole numbers. Negative integers, fractions or rational numbers are not whole numbers. Examples are -1, -5, ½, 9/4, pi, etc. are not whole numbers.

### Are all whole numbers real numbers?

Real numbers are those numbers that include rational numbers, integers, whole numbers and natural numbers. All whole numbers are real numbers but not all real numbers are whole.

### Are all natural numbers, whole numbers?

Natural numbers are those which start from 1 and end at infinity, whereas whole numbers start from 0 and end at infinity. All the natural numbers are whole numbers but not all whole numbers are natural.

### Are natural numbers and counting numbers the same?

Natural numbers are the numbers starting from 1 and extend up to infinity. Counting numbers are used to count the objects or people or anything which is countable. Hence, we always start counting from 1.

## FAQs

### What are the properties of whole numbers with examples? ›

Whole numbers are **closed under addition and multiplication**. It means the addition and multiplication of two or more whole numbers results in a whole number only. 20 + 30 = 50, a whole number. 20 x 30 = 600, a whole number.

**What are the properties of whole numbers? ›**

**The four properties of whole numbers are as follows:**

- Closure Property.
- Associative Property.
- Commutative Property.
- Distributive Property.

**What are symbols used to write whole numbers? ›**

Answer and Explanation: The symbol for whole numbers is **a capital W**. The set of whole numbers would be represented as: W: 0, 1, 2, 3, 4, 5...

**What are 4 examples of properties? ›**

Familiar examples of physical properties include **density, color, hardness, melting and boiling points, and electrical conductivity**.

**What are the properties with examples? ›**

Property can be tangible items, such as **houses, cars, or appliances**, or it can refer to intangible items that carry the promise of future worth, such as stock and bond certificates. Intellectual property refers to ideas such as logo designs and patents.

**What is a whole property? ›**

Property as a whole means **all property assembled as one investment or to further one development plan**. The property as a whole may include more than one municipal tax block or lot.

**What are the 5 properties? ›**

**Commutative Property, Associative Property, Distributive Property, Identity Property of Multiplication, And Identity Property of Addition**.

**What is an example of a number property? ›**

When we talk about number properties being commutative within addition, this means that no matter which way around you add the numbers, the answer will always be the same. For example, **4+9 is exactly the same as 9+4**. Both will give you the answer of 13. In multiplication, the same rule applies.

**What is a number symbol example? ›**

A numeral is a symbol or name that stands for a number. Examples: **3, 49 and twelve** are all numerals.

**What is whole number give an example? ›**

Whole numbers include natural numbers that begin from 1 onwards. Let us look at some examples of whole numbers. The set of whole numbers is denoted by the alphabet 'W'. W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,.…}

### What is a symbol for numbers? ›

The symbol **#** is known variously in English-speaking regions as the number sign, hash, or pound sign.

**What are the 3 types of property? ›**

In economics and political economy, there are three broad forms of property: **private property, public property, and collective property** (also called cooperative property).

**What is called property? ›**

Property definition describes it as **anything over which a business or a person has lawful rights**. They may have some enforceable privileges over the items. Most of the properties contain present or future monetary value and that is why they are considered assets.

**What is property and its types? ›**

Property is basically of two categories : **Corporeal Property and Incorporeal Property**. Corporeal Property is visible and tangible, whereas incorporeal Property is not. Moreover, corporeal Property is the right of ownership in material things, whereas incorporeal Property is an incorporeal right in rem.

**What are 3 examples of real property? ›**

Examples of real property are **buildings, canals, crops, fences, land, landscaping, machinery, minerals, ponds, railroad tracks, and roads**.

**What are the 4 types of math properties? ›**

**Let A , B , C are three integers.**

- . Commutative property :
- 2 . Associative property :
- 3 . Distributive property :
- 4 . Identity property :

**What are the 9 properties of numbers? ›**

**TutorMe Blog**

- Additive identity.
- Multiplicative identity.
- Commutative property of addition.
- Commutative property of multiplication.
- Associative property of addition.
- Associative property of multiplication.
- Distributive property of multiplication.

**Which is the best definition of property? ›**

: **something owned or possessed**. specifically : a piece of real estate. : the exclusive right to possess, enjoy, and dispose of a thing : ownership. : something to which a person or business has a legal title.

**What is the difference between property and properties? ›**

Definition of properties

**the plural of property**.

**What are the 5 traditional property types? ›**

There are five main types of real estate investment properties according to the National Council of Real Estate Investment Fiduciaries: **multifamily, industrial, office, retail, and hotel** (and sub-types in each group).

### What are the 8 properties of math? ›

**Properties of Mathematics**

- Properties of Mathematics.
- Identity Property of Addition.
- Identity Property of Multiplication.
- Commutative Property of Addition.
- Commutative Property of Multiplication.
- Associative Property of Addition.
- Associative Property of Multiplication.
- ***Distributive Property.

**What are basic properties? ›**

There are four basic properties of numbers: **commutative, associative, distributive, and identity**. You should be familiar with each of these. It is especially important to understand these properties once you reach advanced math such as algebra and calculus.

**What are six properties? ›**

**The Six Properties of Life**

- Living things are made up of cells.
- Living things respond to stimuli.
- Living things turn food into energy.
- Living things reproduce.
- Living things grow and develop.
- Living things maintain homeostasis.

**What are the 7 properties? ›**

**Terms in this set (7)**

- cellular organization. all living things are composed of cells, some have unicellular organisms, some have multicellular.
- homeostasis. ...
- metabolism. ...
- responsiveness. ...
- reproduction. ...
- heredity. ...
- growth.

**What are the 3 properties in math? ›**

**Commutative, Associative, and Distributive Properties**.

**What are the list properties? ›**

**The CSS list properties allow you to:**

- Set different list item markers for ordered lists.
- Set different list item markers for unordered lists.
- Set an image as the list item marker.
- Add background colors to lists and list items.

**What is a symbol example? ›**

Symbols take the form of words, sounds, gestures, ideas, or visual images and are used to convey other ideas and beliefs. For example, **a red octagon is a common symbol for "STOP"**; on maps, blue lines often represent rivers; and a red rose often symbolizes love and compassion.

**What does symbol mean examples? ›**

A symbol (pronounced SIM-bull) is **any image or thing that stands for something else**. It could be as simple as a letter, which is a symbol for a given sound (or set of sounds). Similarly, every word is a symbol for the idea it represents. Flags are symbols for nations.

**What is a whole number simple definition? ›**

Whole Numbers **The numbers that include natural numbers and zero**. Not a fraction or decimal. {0, 2, 3, 4, 5 6, 7, 8, 9, 10, 11 …}

**What is a whole number called? ›**

The whole numbers are also called the **positive integers** (or the nonnegative integers, if zero is included).

### What are the properties of whole numbers Class 6? ›

**The property of whole number includes:**

- Closure for addition and multiplication.
- Commutative property for addition and multiplication.
- Associative property for addition and multiplication.
- Distributive property of multiplication over addition.
- Identity for addition and multiplication.

**How do you write a number symbol? ›**

The number sign ( # ) is made on an American keyboard by **holding the SHIFT key and pressing the number 3 on the top number line**. The number sign can also be inserted into a document by choosing INSERT SYMBOL and selecting the # symbol.

**What is word symbol number? ›**

The numero sign or numero symbol, №, (also represented as Nº, No, No. or no.), is **a typographic abbreviation of the word number(s) indicating ordinal numeration, especially in names and titles**.

**What is the 8 symbol called? ›**

**Infinity** in mathematics is considered as a number and represented as ∞. It is an endlessness and limitless concept which describes something without any bound or larger than any number. Sometimes it is also known as lemniscate, has a shape of number 8 on its side.

**What are the 4 basic properties of numbers? ›**

**What are Number Properties?**

**Definition, Types, Chart, Examples**

- Commutative Property.
- Associative Property.
- Identity Property.
- Distributive Property.

**What are the 5 properties in math? ›**

- Properties of. Math Foldable.
- Commutative Property. “Ordering”
- Associative Property. “Grouping”
- Identity Property. “Staying the Same”
- ▶Zero is the Additive Identity.
- ▶One is the Multiplicative.
- Inverse Property. “Undoing”
- “Distributing Through”

**What are integer properties? ›**

The three properties of integers are: **Closure Property**. **Commutativity Property**. **Associative Property**.

**How many properties are there in math? ›**

There are **four basic properties**: commutative, associative, distributive, and identity.

**What is properties of irrational? ›**

An irrational number is a type of real number which **cannot be represented as a simple fraction**. It cannot be expressed in the form of a ratio. If N is irrational, then N is not equal to p/q where p and q are integers and q is not equal to 0. Example: √2, √3, √5, √11, √21, π(Pi) are all irrational.