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# Whole Number|Definition & Meaning

## Definition

All natural numbers including zero are included in the category of whole numbers.

They are a subset of real numbers, which **exclude negative numbers**, decimals, and fractions. Whole numbers include **counting numerals** as well. Whole numbers are defined as natural numbers plus zero (0). We are aware that the term **“natural numbers”** refers to a group of counting numbers beginning with 1, 2, 3, 4, and so on.

**Whole numbers** are a **collection of numbers** that don’t contain any fractions, decimals, or even negative integers. It is made up of zero and positive integers. Alternately, we may define whole numbers as the collection of **non-negative integers**. The inclusion of zero in the set of whole numbers marks the main distinction between them and **natural numbers. Figure 1 **illustrates the real line of whole numbers.

**Whole numbers** are a kind of natural numbers that include the number 0. The letter W stands for the set of whole integers in **mathematics**, which are 0 through 9.

Due to the fact that whole numbers begin at 0, 0 is the **lowest whole number** (from the definition of whole numbers). On a **number line**, zero sits in the middle of the positive and negative numbers. Although it has no value, zero is **employed** as a stand-in. In light of this, zero is a number that cannot be either positive or negative.

We realize that a **characteristic number** is any entire number that isn’t zero. All regular numbers are entire numbers, as well. The set of **whole numbers** is therefore a subset of the set of natural numbers. **Natural numbers** are a subset of the whole number set, as shown in **Figure 2**.

There is no **biggest whole number**, which is an intriguing aspect of the entire set of numbers. B + 1 is a whole number if b is the highest whole number. However, **b + 1** is bigger than b. This method demonstrates that a larger **whole number** is always possible to find.

**Difference between Whole numbers and Natural number**

The complete set of all whole numbers is W = [0, 1, 2, 3,…]. The smallest total number is zero. A natural number is any whole number. The natural number range is given by N = 1, 2, 3, etc. 1. It is the lowest natural number. Except for 0, every whole number is a **natural number.**

**Properties of the Whole number**

Mathematical operations including combining, **subtracting, dividing, and multiplying** form the foundation of complete number **characteristics.** Two whole numbers can be joined, added, or subtracted to make the whole number. As a result, using the division method, we may also obtain a fraction. Here are some of the characteristics of **whole numbers.**

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

**Closure Property**

A whole number is always the result of two **whole numbers** when they are **added and multiplied** together.

**Associative Property**

The product or sum of any **three whole numbers** remains the same regardless of the numbers’ order.

**Commutative Property**

Even after changing the numbers’ order, the total and product of two whole numbers stay the same. According to this feature, the value of the total is **unaffected by changes** in the sequence of addition. A and B should be two entire numbers. A + B Equals B + A, according to the **commutative property.**

**Distributive Property**

According to this characteristic, a **whole number’s** multiplication is dispersed throughout the sum or difference of the whole numbers. It indicates that if two numbers, such as a and b, are multiplied by the same number c, and then added, the result may be obtained by **multiplying **the total of a and b by c. The formula for this attribute is: **a (b + c) = (axb) + (axc).**

## Further Properties

**Additive Identity**

If w is a whole number, then w + 0 equals w, and w + 0 equals w. This means that adding a whole number to 0 has no effect on its value.

**Multiplicative Identity**

If w is a whole number, then w $\times$ 1 equals w. Thus, the multiplicative identity of all whole numbers is unity (1).

**Multiplication of the Whole Number With Zero**

When a whole number is **multiplied** by zero the result will be **zero.**

**Division of the Whole Number With Zero**

When a whole number is **divided** by zero the result will be **undefined.**

A **whole number** cannot be considered **“large.”** There is either a number immediately preceding or immediately preceding every value. There is only a percentage or decimal for **two whole numbers;** there is no whole number. There is no natural number zero; It’s an **entire number.** There are five initial whole numbers: 0, 1, 2, 3, and 4. Zero is the lowest entire number. Unless they can be reduced to a natural number or **whole number, negative integers, fractions, and decimals** are not natural numbers or **whole numbers.**

**Some Examples of Whole Numbers**

**Example 1**

Identify whether {-2,-1,0,1,2} are whole numbers.

**Solution**

The set of the whole number starts from zero and whole numbers are always non-negative since the first two elements of the set are -1 and -2 and they are negative so {-2,-1,0,1,2} are not a set of whole numbers.

**Example 2**

Add three whole numbers 20, 30, and 40 using the associative property of Whole numbers.

**Solution**

By using the associative property, we have:

**First Method**

20 + (30 + 40)

= 20 + 70

= 90

**Second Method**

(20 + 30) + 40

= 50 + 40

= 90

**Third Method**

(20 + 40) + 30

= 60 + 30

= 90

**Example 3**

Solve 7(5+3) using the distributive property.

**Solution**

By the distributive property a (b+c)=a(b)+a(c) we have

** **By distributive property 7(5+3) can be written as

7(5) + 7(3)

= 35 + 21

= 56

**Example 4 **

Express the set {0,1,2,3,4,5} on the real line.

**Solution**

As {0,1,2,3,4,5} includes zero, and are positive so set is a whole number and figure 3 illustrates the real line of whole numbers.

*All images/mathematical drawings were created using GeoGebra.*