Learner Description: This page is intended for students in an Algebra 1 class.

What Numbers Make Up the Real Numbers?

Diagram showing the relationship of the natural numbers, whole numbers, integers, rational numbers, and irrational numbers to the set of real numbers. Link to a Khan Academy video that explains the sets that make up the set of real numbers. Link to a MathIsFun.com page that explains what irrational numbers are. Link to a page that defines the natural numbers, and that provides interesting facts about the first 1,000 natural numbers. A link to a definition of whole numbers from iCoachMath.com. Link to a definition of integer from FactMonster.com. Link to a definition and examples of rational numbers from MathIsFun.com.

Click the different parts of the image above to learn more about the sets of numbers. All links open in a new window.

What is this thing called the set of real numbers? Mathematicians have all sorts of complicated ways to define it. For example, PlanetMath.org has a definition of real numbers that might curl your hair. Why do mathematicians make things so complicated? They do it because without precise definitions, they may end up with errors and invalid results.

If you care less about a formal definition, and care more about simply understanding the set of real numbers, then you can take a different approach. You can look at some of the sets that are contained within the set of real numbers. The diagram on the right illustrates some of these sets and how they are related to each other. Notice that the sets on the left side of the diagram are contained inside one another, like a set of Russian dolls. Click each part of the diagram to learn more about each of the sets that make up the set of real numbers. You can also click the title of the diagram to see a video from Khan Academy that explains each of these sets. As you understand these subsets better, you will also gain a better understanding of the set of real numbers.

Natural Numbers

When you click this set, you will see a page that explains the set of natural numbers, which are also sometimes called counting numbers. The site also has links to interesting facts about each number from 1 through 1,000.

Whole Numbers

Whole numbers are the set of natural numbers plus the number 0. The link will take you to an explanation on the iCoachMath.com site, which also has links to example problems using whole numbers.


Integers including the positive whole numbers, the negatives of the whole numbers, and 0. This link will take you to FactMonster.com, where you will find a definition along with some helpful tips for adding, subtracting, multiplying, and dividing integers.

Rational Numbers

Though the site name may make you cringe, MathIsFun.com has a clear and simple definition of rational numbers, along with a few examples.

Irrational Numbers

MathIsFun.com also has a great explanation of irrational numbers. It explains how they are different from rational numbers, and how to tell if a number is rational or irrational. Note that while a number like 3 can belong to the natural numbers, whole numbers, integers, and rational numbers, there are no numbers that can be both rational and irrational.