Whole Numbers : Properties & Difference

Whole numbers are defined as natural numbers plus zero (0). The term "natural numbers" refers to a group of counting numbers beginning with 1, 2, 3, 4, and so forth. 

• A group of numbers known as whole numbers are those without any fractions, decimals, or even negative integers. 
• It is composed of positive integers and zeros. 
• A different definition for whole numbers is the grouping of non-negative integers. 
• The major distinction between whole numbers and natural numbers is that zero is included in the set of whole numbers.

Key terms: Whole numbers, Natural numbers, Closure, Associative, Commutative, Distributive, Zero

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Definition of Whole Number

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A whole number is any positive number that doesn't have a fractional or decimal component. All whole numbers, including 0, 1, 2, 3, 4, 5, 6, and 7, are thus all whole numbers, according to this. The closest that some numbers come to being whole numbers are -3, 2.7, and 3 1/2.

• A whole number must be positive and cannot be a negative number (also known as a minus number).
• This indicates that it must be greater than zero. As an illustration, the whole integers 0, 1, 2, and 3 are, whereas -1, -2, and -3 are not.
• A complete integer cannot include any fractional elements. 
• A whole integer cannot contain any decimal elements. As a result, while 3, 7, and 11 are whole numbers, numbers like 3.4, 7.9, and 11.234 are not.
• Although all integers are whole numbers, not all whole numbers are considered to be integers.

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Set of Whole Numbers

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Along with the number 0, the set of whole numbers also contains the set of natural numbers. In mathematics, the set of whole numbers is given as [0, 1, 2, 3,...], and is represented by the letter W.

W= [0, 1, 2, 3,...]

Consider the following details to better understand whole numbers:

• Whole numbers make up all natural numbers.
• There are only whole numbers when counting.
• All positive integers and Zero are whole numbers.
• Real numbers are all entire numbers.
• A rational number is any entire number.

Read More: Real Numbers Formula

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Smallest Whole Number

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Whole numbers do not include negative values and range from 0 to infinity. Since there is no positive number less than 0, zero is the smallest whole number. 

• When employed as a counting number, zero, which is an integer with the symbol 0 denotes the absence of all items. 
• It is the only integer that is neither negative nor positive, making it the only real number as well. 
• A number that is not zero is referred to as nonzero. 
• A function's zero is also referred to as its root. 
• Since it is neither negative nor positive, zero is a whole number and an integer. It is referred to as a neutral integer.

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Properties of Whole Number 

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The foundation of the properties of whole numbers is laid by arithmetic operations including addition, subtraction, division, and multiplication. 

• A whole number will result from the addition or multiplication of two whole numbers.
• The outcome of subtracting two whole numbers could either be an integer or a value that is not a whole number. 
• Sometimes dividing two whole numbers results in a fraction. 
• Let's look at some more whole number quantities and their justifications now using these examples. 

Closure Property

A set is closed for a certain mathematical operation if it possesses the closure attribute. 

• A set is considered closed concerning that operation if it can always be finished using pieces from the set. 
• As a result, a set either possesses closure concerning a particular operation or it does not.

Example: 8 + 10 = 18 is a natural number.

Commutative Property

One can change or swap the numbers used without affecting the solution because of the commutative characteristic. For addition and multiplication, the principle is true, but not for division or subtraction.

Example: 5 + 4 = 9 & 4 + 5 = 9

Hence, 2 + 4 = 4 + 2

Associative Property

Rearranging the parenthesis in an expression will not change the outcome since some binary operations have the associative property. Associativity is a legitimate rule of replacement for expressions in logical proofs in propositional logic.

Example: 2 + (4 + 1) = 2 + (5) = 7 

(2 + 4) + 1 + (6) + 1 = 7 

Hence, 2 + (4 + 1) = (2 + 4) + 1

Distributive Property

The distributive principle of binary operations generalises the distributive law, which claims that equality is always true in elementary algebra.

Example: 3 x (4 + 5) = (3 x 4) + (3 x 5) = 12 + 15 = 27

2 x (3 + 4) = 2x (7) = 14 

Hence, 2 x (3 + 4) = (2 x 3) + (2 x 4)

Therefore, Multiplication is distributive over addition 

Multiplicative Identity

According to the concept of multiplicative identity, if a number is multiplied by one, the outcome will be the original number. A number's multiplicative identity is 1 or 1.

Example: -8 x 1 = 1 x (-8) = – 8

Multiplication by Zero

A Whole number multiplied by 0 always gives a result of 0, i.e., a × 0 = 0 × a = 0. 

Example: 4 × 0 = 0.

Division by Zero- It is not defined how to divide a whole number by zero, hence if an is a whole number, a/0 is also not defined.

Read More: Differentiation and Integration

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Natural Numbers and Whole Numbers

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One 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.

Natural Number- 1, 2, 3, 4, 5, 6, 7, …….. 

Whole Number- 0, 2, 3, 4, 5, 6, 7, ……..

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Difference Between Natural Numbers and the Whole Numbers

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The difference between Natural & Whole Numbers are mentioned below:

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Solved Examples

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Example 1- Using the distributive property, find the product. (a) 237 × 103

Solution- 237 × 103

237 × (100 + 3)

Property: a × (b + c) = a × b + a × c

Therefore, 237 × (100 + 3)

=237 × 100 + 237 × 3

23700 + 711

=24411

Example 2: Out of the following numbers, pick out the whole numbers (-1, 0, 3, 1/2, 5).

Solution: In mathematics, the set of whole numbers is [0, 1, 2, 3, 4, …..](inclusive). The whole numbers among the supplied numbers are therefore (0, 3, and 5).

Read More:

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Things to Remember 

• Natural numbers (counting numbers), as well as zero, are considered whole numbers.
• The commutative property holds for addition and multiplication with whole numbers.
• Whole numbers follow the associative property for addition.
• The distributive property applies to multiplication over addition in whole numbers.
• The multiplicative identity is 1, and multiplying any whole number by zero results in zero.
• Division by zero is not defined for whole numbers.
• Natural numbers are a subset of whole numbers because they all consist of whole numbers (1, 2, 3, etc.).
• Natural numbers begin at 1, whereas whole numbers also include zero, which is the main distinction between them.

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