What is a whole number? | Definition, symbols, and example

Math requires the ability to define whole numbers and distinguish them from natural numbers. Whole numbers such as 0, 1, and 2 are the foundation for understanding more complex number identifiers such as real, rational, and irrational numbers. Rounding to the closest digit can help you perform calculations or mental math faster. Keep reading to learn what is a whole number, its definition, and its symbol.

What is a whole number?

Whole numbers include natural numbers and zero (0). Natural numbers are a collection of counting numbers beginning with 1, 2, 3, 4, and so on. Whole numbers don’t contain fractions, decimals, or negative integers. It is a set of positive integers and zeroes. 

Alternatively, whole numbers are the set of non-negative integers. The presence of zero in the whole numbers set is the primary distinction between natural and whole numbers.

Definition of whole numbers

The group of natural numbers that includes 0 is known as whole numbers. 

Here are some facts to help you understand them better-

• Natural numbers are all whole numbers.
• Every counting number is a whole number.
• Whole numbers are all positive integers, in addition to zero.
• Whole numbers are all real numbers.

Symbol

The alphabet ‘W’ in uppercase represents whole numbers, with W = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11, 12,13, and so on.

Smallest whole number

Whole numbers begin with 0; hence, 0 is the smallest whole number (from the definition of whole numbers). On a number line, zero is the number that lies between the positive and negative numbers. Even though zero has no value, it is used as a placeholder. Hence, zero is neither a positive nor a negative number.

Natural numbers and whole numbers

According to the definitions above, each number other than 0 is a natural number. Furthermore, each natural number is a whole number. As a result, the set of natural numbers is a subset of the whole numbers.

Difference between whole numbers and natural numbers

Let us examine the differences-

Whole number	Natural number
The set of whole numbers is, W = {0,1,2,3,4,5…}.	The set of natural numbers is, N = {1,2,3,4,5…}.
The smallest whole number is 0.	1 is the smallest natural number.
Each natural number is a whole number.	Except for 0, each whole number is a natural number.

Whole numbers on the number line

Whole numbers are easily visible on a number line because the number line generally represents integers from – to +, and the whole numbers are positive integers, including 0.

A number line shows that the integers on the right side of 0, including 0, are all whole numbers, while the integers on the right side of 0, excluding 0, are all natural numbers.

Properties of whole numbers

The four basic operations on whole numbers (addition, subtraction, multiplication, and division) result in four major properties of whole numbers. They are as follows-

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

Closure Property

The product and sum of two whole numbers will always be whole numbers. For example, 6 + 4 = 10 , 8 x 2 = 16.

Associative Property

Even if the order of the numbers is changed, the sum or product of any three whole numbers remains constant. For example, adding the following numbers yields the same result- 10 + (7 + 12) = (10 + 7) + 12 = (10 + 12) + 7 = 29.

Similarly, no matter how the numbers are grouped, multiplying them yields the same result. For example, 3 × (2 × 4) = (3 × 2) × 4 = 24.

Commutative Property

The sum of two whole digits is the same according to the commutative property of addition. In other words, the order of addition is irrelevant.

Let ‘x’ and ‘y’ be two whole numbers.

According to the addition commutative property-

x + y = y + x

5 + 8 = 13

8 + 5 = 13

As a result, 5 + 8 = 8 + 5

Additive Identity

When a whole number is added to zero, its value remains constant; that is, if ‘a’ is a whole number, a + 0 = 0 + a = a. For instance, 4 + 0 = 0 + 4 = 4.

Multiplicative Identity

When multiplied by one, its value remains constant; if ‘a’ is a whole number, then a × 1 = a = 1 × a. As an example, 5 × 1 = 1 × 5 = 5.

Distributive Property

The distributive property states that when a number is multiplied and distributed in any order, the value of the result remains the same.

(x × y + z) = (x × y) + (x × z)

The distributive property states-

(x × y + z) = (x × y) + (x × z)

3 × (6 + 8) = (3 × 6) + (3 × 8)

3 × (14) = 18 + 24

42 = 42

As a result, proved.

Multiplication by zero

The result is always zero when you multiply a whole digit by 0. That is, a × 0 = 0 × a = 0. For example, 4 × 0 = 0.

Division by zero

If divided by 0 is not defined; if ‘a’ is a whole number, then a/0 is not defined.

Key takeaways

• You can define whole numbers as the group of natural numbers that includes 0.
• There are four major properties of whole numbers- Closure, Associative, Commutative, and Distributive.
• All positive or integers on the right-hand side of 0 represent the natural numbers. All the positive integers, in addition to zero, represent the whole numbers.

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FAQs

Q1. Who is the father of math?

Answer- Archimedes, the great Greek mathematician and philosopher is the father of Mathematics.

Q2. Which is the greatest whole number?

Answer- Because whole numbers range from 0 to infinity, the greatest whole number is infinity.

Q3. Is 0 a real number?

Answer- Real numbers can be positive or negative, and they include the number zero. They are called real numbers because they do not belong to the imaginary number system.