Whole numbers are a mix of numbers including all natural numbers & 0. They are a part of real numbers that vì not include fractions, decimals, or negative numbers. Counting numbers are also considered as whole numbers. Let us learn everything about whole numbers in this article.

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 1 What are Whole Numbers? 2 Whole Numbers vs Natural Numbers 3 Whole Numbers on Number Line 4 Properties of Whole Numbers 5 FAQs on Whole Numbers

Natural numbers refer lớn a phối of counting numbers starting from 1 và on the other hand, natural numbers along with zero (0) khung a set, referred lớn as whole numbers. However, zero is an undefined identity that represents a null set or no result at all. In simple words, whole numbers are a set of numbers without fractions, decimals, or even negative integers. It is a collection of positive integers and zero. Or we can say that whole numbers are the phối of non-negative integers. The primary difference between natural and whole numbers is the presence of zero in the whole numbers set.

Whole Number Definition: Whole Numbers are the set of natural numbers along with the number 0. The set of whole numbers in mathematics is given as 0, 1, 2, 3, ... Which is denoted by the symbol W.

W = 0, 1, 2, 3, 4, …

All natural numbers are whole numbers.All counting numbers are whole numbers.All positive integers including zero are whole numbers.

### Whole Number Symbol

The symbol used to lớn represent whole numbers is the alphabet ‘W’ in uppercase, W = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …

### Smallest Whole Number

Whole numbers start from 0 (from the definition of whole numbers). Thus, 0 is the smallest whole number. The concept of zero was first defined by a Hindu astronomer & mathematician Brahmagupta in 628. In simple language, zero is a number that lies between the positive and negative numbers on a number line. Although zero carries no value, it is used as a placeholder. So, zero is neither a positive number nor a negative number.

## Whole Numbers vs Natural Numbers

From the above definitions, we can understand that every whole number other than 0 is a natural number. Also, every natural number is a whole number. So, the set of natural numbers is a part of the set of whole numbers or a subset of whole numbers. ### Difference Between Whole numbers and Natural numbers

Let us understand the difference between whole numbers & natural numbers through the table given below:

Whole NumberNatural Number
The phối of whole numbers is, W = 0,1,2,3,....The mix of natural numbers is, N = 1,2,3,....
The smallest whole number is 0.The smallest natural number is 1.
Every natural number is a whole number.Every whole number is a natural number, except 0.

## Whole Numbers on Number Line

The mix of natural numbers & the mix of whole numbers can be shown on the number line as given below. All the positive integers or the integers on the right-hand side of 0 represent the natural numbers, whereas all the positive integers along with zero, altogether represent the whole numbers. Both sets of numbers can be represented on the number line as follows: ## Properties of Whole Numbers

The basic operations on whole numbers: addition, subtraction, multiplication, and division, lead khổng lồ four main properties of whole numbers that are listed below:

Closure PropertyCommutative Property

Closure Property

The sum and product of two whole numbers is always a whole number. For example, 7 + 3 = 10 (whole number), 7 × 2 = 14 (whole number).

Associative Property

The sum or sản phẩm of any three whole numbers remains the same even if the grouping of numbers is changed. For example, when we địa chỉ the following numbers we get the same sum: 10 + (7 + 12) = (10 + 7) + 12 = (10 + 12) + 7 = 29. Similarly, when we multiply the following numbers we get the same sản phẩm no matter how the numbers are grouped: 3 × (2 × 4) = (3 × 2) × 4 = 24.

Commutative Property

The sum & the hàng hóa of two whole numbers remain the same even after interchanging the order of the numbers. This property states that a change in the order of addition does not change the value of the sum. Let 'a' and 'b' be two whole numbers. According lớn the commutative property a + b = b + a. For example, a = 10 and b = 19 ⇒ 10 + 19 = 29 = 19 + 10. This property also holds true for multiplication, but not for subtraction and division. For example: 7 × 9 = 63 and 9 × 7 = 63.

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. For example, 3 + 0 = 3 + 0 = 3.

Multiplicative Identity

When a whole number is multiplied by 1, its value remains unchanged, i.e., if x is a whole number then x × 1 = x = 1 × x. For example. 4 × 1 = 1 × 4 = 4.

Distributive Property

This property states that the multiplication of a whole number is distributed over the sum or difference of the whole numbers. It means that when two numbers, for example a & b are multiplied with the same number c and are then added, then the sum of a and b can be multiplied by c khổng lồ get the same answer. This property can be represented as: a × (b + c) = (a × b) + (a × c). Let a = 10, b = 20 & c = 7 ⇒ 10 × (20 + 7) = 270 và (10 × 20) + (10 × 7) = 200 + 70 = 270. The same property is true for subtraction as well. 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.

Multiplication by Zero

When a whole number is multiplied khổng lồ 0, the result is always 0, i.e., x × 0 = 0 × x = 0. For example, 4 × 0 = 0.

Division by Zero

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

Important Points

0 is a whole number but it is NOT a natural number.

☛ Related Topics

Check out a few more important articles related lớn whole numbers definition và examples.

Example 2: Is W closed under subtraction and division?

Solution:

Whole numbers include only the positive integers and zero. We know that on subtracting one positive integer by another, we may not get their difference as a positive integer, similarly, on dividing one positive number by another, we may not get the quotient as a counting number for example in the case of 13/5. Thus, for any two whole numbers, their difference và quotient obtained may not be whole numbers. Therefore, W is not closed under subtraction & division.

Example 3: For the whole number values of a, b, and c, that is, a = 3, b = 2, c = 1, prove a × (b + c) = (a × b) + (a × c).

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Solution:

Substituting the values of a, b, and c, we get: a × (b + c) = 3 × (2 + 1) = 3 × 3 = 9 and (a × b) + (a × c) = (3 × 2) + (3 × 1) = 6 + 3 = 9. Since, LHS = RHS (9 = 9), thus, a × (b + c) = (a × b) + (a × c), for the given whole number values. This is known as the distributive property of multiplication of whole numbers.