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# Whole Numbers

Whole numbers may be a simple concept, but mastering them is vital for success in math! Whole numbers are usually the first type of number system that students will work with. Simply put, whole numbers are a set of numbers that include the natural numbers and 0. They're sometimes called "counting numbers" because they do not include decimals or fractions. Let's learn more about whole numbers below.

## What are whole numbers?

The set of whole numbers includes 0 as well as the natural numbers 1, 2, 3,.. up to infinity. All natural numbers are also whole numbers, and all whole numbers are real numbers. But unlike real numbers, every whole number is positive.

The same can be said for whole numbers' relationship to integers: because integers can be positive or negative, all whole numbers are integers, but not all integers are whole numbers.

Whole numbers also do not include any numbers with a fractional or decimal part. It's easy to remember this if you think of whole numbers literally-they're whole, not a part.

## The set of whole numbers

In mathematics, the set of whole numbers is shown as $\left\{0,1,2,3,4,...\right\}$ . It's represented by the symbol W. So:

$W=\left\{0,1,2,3,4,...\right\}$

The set of whole numbers has a property called well ordered, which simply means that we can list them from smallest to largest. The number 0 is the smallest whole number because that's where whole numbers begin. But there is no highest whole number-the number line tends towards infinity! For each whole number you can think of, there is always a number bigger than it.

## Other properties of whole numbers

The set of whole numbers is said to be closed and commutative. By "closed," we mean that when two whole numbers are added or multiplied, the result is also a closed number. For example, $5+10=15$ , where 5 and 10 are both whole numbers, as is 15. Likewise, $7×3=21$ , with 3, 7, and 21 all in the set of whole numbers. By "commutative", we mean that you can swap the order of numbers on either side of a plus or multiplication sign; for example, $5+6$ is the same as $6+5$ because they are both 11. The order does not change the result.

## Whole numbers practice questions

a. Which whole numbers are between the numbers 6 and 13?

Since the numbers are well ordered, we can start at the first number, 6, and list them one at a time until we get to 13.

$\left\{7,8,9,10,11,12,13\right\}$

b. Which of the following four numbers are whole numbers? $\left\{4,\frac{1}{3},29,8,1\right\}$

We know that whole numbers can't have nonzero fractional or decimal parts, so $\frac{1}{3}$ can be eliminated, as can 8.1; this leaves us with:

$\left\{4,29\right\}$

c. Can whole numbers be negative?

The smallest whole number is 0, so there are no negative whole numbers.

d. What symbol represents the set of whole numbers?

We represent the set of whole numbers with W.

e. What is the smallest whole number?

Zero

## The set of whole numbers in the real world

We use counting numbers every day and in all sorts of different situations, from counting out ingredients in an apple pie (3 apples cut into 8 pieces each) to figuring out how many stamps we need to send out birthday party invitations. These sorts of counting problems show up all over the place and will become very important when we learn about the fundamental counting principle later down the line.

## Learn whole numbers with Varsity Tutors

Whole numbers are frequently one of the first math lessons taught to students, and success in math depends on understanding its components, including whole numbers. Because math concepts build upon one another as they increase in complexity, it's important that students have a strong grasp of this basic concept to support them going forward.

If your student is struggling with their whole numbers, working with an experienced tutor can make a big difference. With personalized support and customized lessons, tutors can help your student lay a solid foundation before moving on to more advanced math concepts. At Varsity Tutors, our experienced Educational Directors will find your student a tutor who has the right experience and the right personality. Get in touch with us today!

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