What are Real Numbers? - Definition & Properties

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  • 0:02 What Are Real Numbers?
  • 0:28 Rules Describing Real Numbers
  • 2:55 Types
  • 4:12 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe
There are many definitions of real numbers, but they all lead to the same conclusion. Real numbers are numbers that have a measurable value. Learn more about real numbers with some examples and a quiz to test your knowledge.

What are Real Numbers?

Numbers can be grouped in many different ways. There are rational and irrational numbers, positive and negative numbers, integers, natural numbers and real or imaginary numbers. It can be difficult to keep them all straight. For this lesson, we will define real numbers and give some examples. Real numbers can be defined in many different ways; here are a few of the different types of ways to describe the set of real numbers.

3 Rules Describing Real Numbers

One way you can define real numbers is through the rules that govern them. There are three main rules:

First, real numbers are measurable.

This means that the set of real numbers are those numbers that can be mapped on a number line. The number line has three parts: a negative side, a positive side and the zero in between. Each side of the number line goes on infinitely; there is no end to the positive or negative numbers that make up the set of real numbers.

Second, real numbers have a concrete value.

You know what 15 marbles looks like, or half a cake. You can even know how many pieces are in the square root of 5 pizzas (if you have a calculator). The negative side works for this as well, especially when you are working with money. If your checking account balance is $-2.27, you probably should start to worry.

Finally, real numbers can be manipulated.

For example, all real numbers can be rewritten as a decimal. This just means that even if a number looks odd, or has a strange symbol or Greek letter associated with it, it's still a real number. Let's look at a couple of examples:

  • 1/4 = 0.25
  • The square root of 7 = 2.645751311
  • pi = 3.14159…

Another way in which real numbers can be manipulated is by mathematical operations. All of the set of real numbers can be added, subtracted, multiplied or divided with each other, and the result will be another real number, which can also be written as a decimal. For example: 5 + 2 = 7, 3/4 * 1/8 = 0.09375, or the square root of 17 - pi = 0.98

In addition to our three rules, another way to understand the definition of real numbers is to remember that they are not imaginary. This might seem obvious, but the need to define a number as real was not necessary until imaginary numbers were discovered. Imaginary numbers are used in mathematics to describe numbers that do not actually exist, but can be used in high level calculations.

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