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Whole Numbers – Definition, Examples, List, Symbols

Math evolves into more complex structures that would confuse anyone who has no basic foundation in the subject. In kindergarten and through the grade levels, kids explore these foundations and use them as a stepping stone before they counter more challenging math problems. One of these basic structures is whole numbers. 

What Are Whole Numbers?

What is a whole number? Whole numbers are a set of positive integers which can be described as the primary number sequence, 1,2,3… and their negative counterparts -1, -2, -3, … A simple whole numbers definition is that they are numbers that can also be called non-negative integers or counting numbers. They do not include fractions or decimals. 

Whole numbers are essential in mathematics as they are used to count, label, and order items or values. In addition, they are the foundation for arithmetic operations such as addition, subtraction, division, and multiplication.

Whole Number Symbol 

The symbol used to represent whole numbers is “W” or “ℤ⁺” (pronounced as “Z plus”). “ℤ” represents the set of all integers, including positive and negative whole numbers, while “ℤ⁺” represents only the positive numbers. 

Whole Numbers on a Number Line 

The number line visually represents the actual number system, and whole numbers are plotted on this line. On a number line, each whole number is represented by a point placed at an equal distance from the other points. The number line starts at zero and extends in both directions to infinity.

To plot a whole number on a number line, we can find its position by counting the number of units to the right of zero for positive whole numbers or to the left of zero for negative whole numbers. For example, to plot the whole number 4, we start at zero and count four units to the right. The point that corresponds to 4 is then marked on the number line. Similarly, to plot the whole number -3, we start at zero, count three units to the left, and mark the point corresponding to -3 on the number line.

The number line helps us understand the relationship between whole numbers and their position on the natural number line. It also aids in performing operations with whole numbers, such as addition and subtraction, and comparing and ordering them.

Here is what a number line looks like 

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Properties of Whole Numbers 

Here are some of the properties of whole numbers:

Closure property

The closure property states that the sum and product of any two whole numbers is also a whole number. For example, we have two whole numbers, 2 and 5. If we add them together, we get 7, which is also a whole number. If we multiply them, we get 10, which is also a whole number. This pattern holds for any two numbers you add or multiply. So, if you take any two whole numbers, a and b, and add them together (a + b) or multiply them (a x b), the result will always be a whole number.

Commutative property

The commutative property states that the order of the numbers does not affect the sum and product of any two whole numbers. That is, a + b = b + a and a x b = b x a, where a and b are whole numbers. Let’s say we have two whole numbers, 2 and 3. According to the commutative property, we can add them in any order and still get the same answer. So, 2 + 3 equals 3 + 2, equaling 5.

Associative property

The principle of associative property states that the sum and product of three or more whole numbers remain the same irrespective of how you group them. That is (a + b) + c = a + (b + c) and (a x b) x c = a x (b x c), where a, b, and c are whole numbers.

For example, we have three whole numbers, 2, 3, and 4. According to the associative property, we can group them in any way we want when performing addition. So, (2 + 3) + 4 is the same as 2 + (3 + 4). Looking at the left side, we get 5 + 4 = 9. And if we look at the right side, we get 2 + 7 = 9. Both results are the same.

Distributive property

According to distributive property, the product of a whole number and the sum or difference of two whole numbers is the same as the sum or difference of the product of that whole number and each of the whole numbers. That is, a x (b + c) = a x b + a x c and a x (b – c) = a x b – a x c, where a, b, and c are whole numbers.

If we have 2 and want to multiply it by the sum of 3 and 4, we can use the distributive property to simplify the calculation. We can distribute the 2 to the 3 and 4 inside the parentheses, then add the results. So, 2 x (3 + 4) = 2 x 3 + 2 x 4 = 6 + 8 = 14.

There are other properties of whole numbers which you can add to your lessons, and they are:

• Identity property
• Division algorithm

Difference between Whole Numbers and Natural Numbers

Natural numbers are a subset of whole numbers. They start at 1 and continue counting upwards infinitely. They represent counting numbers in real-life scenarios, such as counting apples or students in a classroom. Natural numbers are only positive integers and do not include 0 or negative ones.

On the other hand, whole numbers include 0 along with positive integers. They start at 0 and continue counting upwards infinitely. Whole numbers represent a broader set of integers, including natural numbers and 0. They are used in mathematical calculations that involve measurements, quantities, and quantities that cannot be negative.

It is important to note that all natural numbers are whole numbers, but not all whole numbers are natural numbers. Natural numbers are a proper subset of whole numbers. Therefore, any number that is a natural number is automatically a whole number, but a whole number may or may not be a natural number. For example, 0 is a whole number but not a natural number, and -5 is an integer but not a whole number or a natural number.

Solved Examples on Whole Numbers

Here are some of the solved examples of whole numbers in their different operations and properties: 

Example 1

Find the sum of 25 and 48.

Solution: To find the sum of two whole numbers, we add them together.

25 + 48 = 73

Therefore, the sum of 25 and 48 is 73.

Example 2 

Multiply 8 by 13.

Solution: To multiply two whole numbers, we multiply the first number by the second number.

8 x 13 = 104

Therefore, the product of 8 and 13 is 104.

Example 3 

Find the value of x in the equation: 7x – 5 = 23.

Solution: To solve for x in an equation involving whole numbers, first move x to one side of the equation.

Then add 5 to both sides of the equation to get: 7x  = 23 + 5

7x = 28

Then, divide both sides by 7 to get: x = 4

Therefore, the value of x in equation 7x – 5 = 23 is 4.

Frequently Asked Questions

Give some examples and non-examples of whole numbers.

Examples of whole numbers include 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and so on.

Non-examples of whole numbers include fractions (such as 1/2, 3/4, and 5/6), decimals (such as 0.5, 0.75, and 1.25), and negative numbers (such as -1, -2, and -3). The examples of whole numbers provide the answer for the question, “what is a whole number in math?”

What is whole number used for?

Whole numbers are used for counting objects, doing basic arithmetic operations, measuring quantities such as length or area, and analyzing data in fields like mathematics, science, and daily life. Examples of using whole numbers include:

• Counting the number of students in a class.
• Measuring the length of a bookshelf in whole-number units.
• Representing the number of people who responded to a survey question.

Can whole numbers be negative?

No, whole numbers cannot be negative. The basic idea behind whole numbers is that they are a set of positive integers, and since you cannot add negative numbers to a set of positive numbers, negative numbers cannot be whole numbers. Negative numbers, however, belong to a set of numbers called integers. If someone asks you “what’s a whole number?”, you can simply use the above definition. 

Which number do whole numbers start from?

Whole numbers start from the number 0 and continue from 1,2,3,4,5 to infinity. 

What are some examples of whole numbers?

Some examples of whole numbers include 1,2,3,4,5,6,7,8,9 to infinity. As far as you can count the significant sequence of numbers, that is as far as whole numbers go. Any positive integer or 0 can be considered a whole number. 

Which whole number is not a natural number?

0 is not a natural number because natural numbers start from 1 and continue to infinity, while whole numbers start from 0 and continue to infinity. 

Are all whole numbers integers?

Yes, all whole numbers are integers. A whole number definition could be “numbers which are integers”.