What are Whole numbers: Definition, Concepts, Properties, Even, Odd and Examples

Whole Numbers Overview

All positive integers from 0 to infinity are included in the group of numbers known as whole numbers. The number line contains these figures. Therefore, they are all real numbers. Although not all real numbers are whole numbers, we can say that all whole numbers are real numbers. As a result, the set of natural numbers plus zero can be used to define whole numbers. The category of whole numbers and the negative of natural numbers is known as integers. 

What are whole numbers?

All natural numbers and 0 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 numbers as well. Positive and negative numbers, including zero, are included in integers. Natural numbers, whole numbers, integers, and fractions all fall under the category of real numbers.

Facts about Whole Numbers

To better understand whole numbers, consider the following information:

1. Whole numbers make up all natural numbers.
2. There are only whole numbers when counting.
3. Zero and all positive integers are whole numbers.
4. All whole numbers are real numbers.

Whole Number Symbol

The letter "W" in uppercase is the symbol used to represent whole numbers. W = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,...

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Smallest and Largest Whole Number

Since the definition of a whole number states that a whole number begins at 0, 0 is the smallest whole number. On a number line, zero sits in the middle of the positive and negative numbers. Although it has no value, zero is used as a stand-in. Therefore, zero is a number that can neither be positive nor negative. There is no "largest" whole number.

Natural Numbers and Whole Numbers

We can infer from the definitions given above that all whole numbers other than 0 are natural numbers. All natural numbers are whole numbers, too. As a result, the set of natural numbers is a subset of the set of whole numbers.

Difference Between Whole numbers and Natural Numbers

Let's use the table below to clarify the distinction between whole numbers and natural numbers:

Whole Numbers on Number Line

The number line can display both the set of whole numbers and the set of natural numbers. The natural numbers are represented by all positive integers, or those on the right-hand side of zero, while all positive integers, including zero, collectively represent the whole numbers. 

Can whole numbers be negative?

The whole number can't be negative!

According to the definition, whole numbers are 0, 1, 2, 3, 4, 5, 6, 7,... positive infinity. Negative numbers have no place.

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Is 0 a whole number?

All natural numbers, including zero, are included in the category of whole numbers. So, yes, zero (zero) is the first whole number in addition to being a whole number.

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Properties of Whole Numbers

The four fundamental operations on whole numbers—addition, subtraction, multiplication, and division—lead to the following four main characteristics of whole numbers:

• Closure Property
• Associative Property
• Commutative Property
• Distributive Property

1. Closure Property

A whole number is always the result of two whole numbers when added together. For instance, 7 + 3 = 10 (a whole number), and 7 + 2 = 14.

2. Associative Property

Even if the order of the numbers is altered, the sum or product of any three whole numbers stays the same. For example, when we add the following numbers, we get the same sum: 10 + (7 + 12) = (10 + 7) + 12 = (10 + 12) + 7 = 29. Similar to that, no matter how the following numbers are grouped, multiplying them produces the same result: 3 × (2 × 4) = (3 × 2) × 4 = 24.

3. Commutative Property

Even after changing the numbers' order, the sum and product of two whole numbers remain the same. According to this property, the value of the sum is unaffected by changes in the order of addition. Let a and b each be a whole number. According to the commutative property, A + B = B + A. For example, if a = 10 and b = 19, then 10 + 19 = 19 + 10 = 29. This property also applies to multiplication, but not to division or subtraction. For example: 7 × 9 = 63, and 9 × 7 = 63.

a. Additive Identity

If a is a whole number, then a + 0 = 0 + a = a, meaning that when a whole number is added to 0, its value does not change. For example, 3 + 0 = 3 + 0 = 3.

b. Multiplicative Identity

A whole number's value does not change when it is multiplied by one, so if a is a whole number, then a  1 = a = 1  a. For example. 4 × 1 = 1 × 4 = 4.

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4. Distributive Property

According to this characteristic, a whole number's multiplication is distributed over the sum or difference of the whole numbers. It means that if two numbers, such as a and b, are multiplied by the same number, c, and then added, the result can be obtained by multiplying the sum of a and b by c. This property can be represented as: a × (b + c) = (a × b) + (a × c). Let a = 10, b = 20 and c = 7 ⇒10 × (20 + 7) = 270 and (10 × 20) + (10 × 7) = 200 + 70 = 270. The same characteristic also applies to subtraction. For example, we have a × (b − c) = (a × b) − (a × c). Let a = 10, b = 20 and c = 7 ⇒10 × (20 − 7) = 130 and (10 × 20) − (10 × 7) = 200 − 70 = 130.

a. Multiplication by Zero

A whole number multiplied by 0 always yields a result of 0, i.e., a × 0 = 0 × a = 0. For example, 4 × 0 = 0.

b. Division by Zero

It is not defined how to divide a whole number by zero, so if a is a whole number, a/0 is also not defined.

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Important Points

1. 0 is not a natural number; it is a whole number.
2. There are five initial whole numbers: 0, 1, 2, 3, and 4.
3. Zero is the smallest whole number.
4. Unless they can be reduced to a natural number or whole number, negative numbers, fractions, and decimals are not natural numbers or whole numbers.