Whole Numbers

Whole numbers are one the most studied type of numbers. These are also the first set of numbers that we’ll learn in mathematics. It’s also for a good reason – whole numbers are often used to quantify real world values and used as examples to showcase different properties in arithmetic and algebra.

In this article, we’ll show you what makes whole numbers special and we’ll also highlight how to identify whether a given number is a whole number or not. We’ll also cover some of the important representations and properties related to whole numbers. 

What Are Whole Numbers?

Whole numbers are the set of numbers that includes zero and the counting numbers (or the natural numbers). This means that the set of whole numbers start from 0 and increase by 1 each time, so it’s the set of numbers that contain 0, 1, 2, 3, 4, …, etc.

In mathematics, the real number system can be broken down into two main groups: rational and irrational numbers. The rational number system can be broken down into smaller groups:

  • Natural numbers or counting numbers are the smallest subset and only contain the numbers we use when counting.
N = \{1,2 , 3, 4, …\}
  • Whole numbers are subsets of positive integers plus zero.
W = \{0, 1, 2, 3, 4,…\}
  • Integers contain positive and negative whole numbers as well as zero.
Z = \{…, -2, -1, 0, 1, 2,…\}
  • Rational numbers cover all three groups of numbers and are numbers that can be written as ratio of two integers: \frac{p}{q}.
R = \{-0.15, 2/3, 1.25, 1/2, …\}

The diagram shown above is a helpful guide when trying to distinguish one group from another. We can see that whole numbers contain all positive integers and zero – this is also why whole numbers are oftentimes confused as counting or natural numbers. 

\begin{aligned}\textbf{Whole Numbers} &: \{0, 1, 2,3, 4, 5,...\}\\\textbf{Counting Numbers}&: \{\,\,\,\,\,1,2 , 3, 4, 5, ...\} \end{aligned}

Here’s a faster way to remember the difference between whole numbers and counting numbers: 0 is a whole number but it is not a natural (or counting) number. Now that we’ve established the process of identifying whole numbers, let’s test our knowledge by trying out the problem below.

Problem 1

For each set of numbers, which of the following are not whole numbers?

a. \{-4, 6, 8, 10\}

Since whole numbers contain either positive integers of 0, the negative integer, -4 is not a whole number.

b. \{10, 2/3, -12.5, 0.5, 12, 500 \}

Whole numbers must either be part of the counting number or zero, so any number not meeting those conditions are not whole numbers. Hence, 2/3 , -12.5, and 0.5 are not whole numbers.

c. \{-12, 12, -24, 24, 0.6, 0, 18 \}

Applying the same thought process, negative integers and decimals are not whole numbers. This means that \{-12, -24, 0.6\} are not whole numbers.

How To Classify Whole Numbers as Odd or Even?

Aside from zero, whole numbers can be classified as odd or even by inspecting their last digits. A whole number is said to be even when it be divided by 2 and it returns no remainder. 

\begin{aligned}2 \times 1 = 2 &\Rightarrow 2 \div 2 = 1\\2 \times 2 = 4 &\Rightarrow 4 \div 2 = 2\\2 \times 3 = 6 &\Rightarrow 6 \div 2 = 3\\2 \times 4 = 8 &\Rightarrow 8 \div 2 = 4\\2 \times 5 = 10 &\Rightarrow 10 \div 2 = 5\end{aligned}

This means that the one-digit whole numbers, \{2, 4, 6, 8\}, are even numbers. Meanwhile, the remaining one-digit whole numbers, \{1, 3, 5, 7, 9\}, are odd numbers.  Extending this concept to larger whole numbers, we can classify whole numbers based on their last digits:

  • A whole number is even when it has a last digit of: \{0, 2, 4, 6, 8\}.
  • A whole number is odd when it has a last digit of: \{1, 3, 5, 7, 9\}.

Here are some examples odd and even whole numbers:

Even NumberOdd Number
\begin{aligned}\{212, 434, 506, 728, 980 \}\end{aligned} \begin{aligned}{111, 423, 505, 757, 879}\end{aligned}

As can be seen from these examples, even numbers will have even last digits while odd numbers will have always have an odd number as their last digit. When divided by 2, the even numbers will return a whole number for a quotient. When odd numbers are divided by 2, the remainder will always be equal to 1.

Even NumberOdd Number
\begin{aligned}212 \div 2&=106\\434\div 2&= 217\\506\div 2&= 253\\728\div 2&= 364\\980 \div 2&= 490\end{aligned} \begin{aligned}111\div 2&=55 \,R1\\434\div 2&= 211\,R1\\505\div 2&= 252\,R1\\757\div 2&= 378\,R1\\879\div 2&= 439\,R1\end{aligned}

Now, let’s use our knowledge to classify whole numbers as odd or even.

Problem 2

Check the numbers shown below and classify the numbers that are odd and numbers that are even.

\begin{aligned}{145, \,239,\,540,\,612,\,593,\,817,\,918,\,976,\,861}\end{aligned}

When classifying whole numbers as odd or even, inspect their last digits and determine whether the last digit is either even or odd. 

Even NumberOdd Number
\begin{aligned}\{540,\,612,\,918,\,976\}\end{aligned} \begin{aligned}{145, \,239,\,593,\,817,\,861}\end{aligned}

What Are Some Important Properties of Whole Numbers?

Whole numbers also exhibit interesting properties when arithmetic operations are involved. Here are some helpful properties to remember when working with whole numbers:

  1. Closure Property: When two or more whole numbers are added or multiplied, the result will still be a whole number.
\begin{aligned}11 + 2 &= 13 \,(\text{Whole Number})\\101 + 45 &= 146 \,(\text{Whole Number})\end{aligned}
  1. Commutative Property: When reverse the order of two whole numbers that are multiplied or added, the result will still be the same.
\begin{aligned}21 + 12 &= 12 + 21\\37 \times 4 &= 4 \times 37\end{aligned}
  1. Additive Identity: When we add zero to a whole number, the result will still the same whole number.
\begin{aligned}28 + 0 &=28\\102 +0&= 102\end{aligned}
  1. Multiplicative Identity: When we multiply a whole number by 1, the product will still be the whole number.
\begin{aligned}48 \times 1 &=48\\120 \times 1 &= 120\end{aligned}
  1. Associative Property: When we add or multiply numbers in a group, the sum will still be the same when we rearrange the numbers within the parentheses.
\begin{aligned}12 + (34 + 45) &= 34 + (12 + 45)\\8 \times (10 \times 12) &= 10 \times (8 \times 12)\end{aligned}
  1. Zero Property: Any whole number multiplied to zero will result to a product of zero.
\begin{aligned}23\times 0&= 0\\ 4238 \times 0 &= 0\end{aligned}

There are still other properties involving whole numbers, but these are six of the most frequently applied properties. Why don’t we check out understanding by identifying the properties applied in the problem shown below?

Problem 3

Fill in the blanks to make the following mathematical equations true. Identify the properties that were used for each of the equations as well.

a. 45 \times 36 = \_\_\_\_\_\_\_\_ \times 45

Let’s begin with the first equation: 45 \times 36 = \_\_\_\_\_\_\_\_ \times 45. Recall that through the commutative property, we can reverse the order of the whole numbers when multiplying them. The result will still be the same, so we have 45 \times 36 = \textbf{36} \times 45.

b. 128 + 0 = \_\_\_\_\_\_\_\_

Now, when a whole number and zero are added, the sum will still be the whole number. We call this the identity property. Hence, we have 128 + 0 = \textbf{128} .

c. 12 \times (3 \times 18) = \_\_\_\_\_\_\_\_\times (12 \times 18)

When a group of whole numbers are multiplied, we can still rearrange the terms inside the parenthesis. Through the associative property, the resulting product will still be the same. The missing factor on the right-hand side is 3, so we have 12 \times (3 \times 18) = \textbf{3}\times (12 \times 18) .

d. 235 \times  \_\_\_\_\_\_\_\_ = 0

According to the zero property, when we multiply a whole number by zero, the product will always be equal to zero. Hence, 235 \times  \textbf{0} = 0 .

e. \_\_\_\_\_\_\_\_  + (12 + 45) = 45 + (12 + 36)

Similar to the third item, associative property also applies to addition of whole numbers grouped as a set. Applying the same process and seeing that 36 is missing from the left-hand side, we have \textbf{36}  + (12 + 45) = 45 + (12 + 36).

Another helpful property involving whole numbers is the distributive property

\begin{aligned}4 \times (12 + 8) &= 4\times 12 + 4\times 8\\3 \times(8 - 5) &= 3\times 8 - 3\times 5 \end{aligned}

As can be seen, we can also distribute the factor outside the sum and multiply each factor first before adding them. This comes in handy when factors can be manipulated to make the process simpler such as finding the product of 40 and 51.

Problem 3

Use the distributive property to simplify 40 \times 51.

\begin{aligned}40\times 51&= 40 \times(50 + 1)\\&= 40\times 50 + 40\times1\\&= 2000 + 40\\&= 2040\end{aligned}

This shows how important it is to understand these properties and know how to work with whole numbers. This will make more advanced topics easier to understand and learn as well.