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The Closure Property of Real Numbers

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will look at real numbers, closure properties, and the closure properties of real numbers. We will also see an example of why it is useful to know what operations real numbers are closed under.

Closure Properties of Real Numbers

Get excited because we are about to learn about a really fun property of real numbers - the closure property of real numbers. This property is fun to explore. It gives us a chance to become more familiar with real numbers. Before we get to the actual closure property of real numbers, let's familiarize ourselves with the set of real numbers and the closure property itself.

Sets of Numbers and The Closure Property

I'm certain that you are familiar with numbers. After all, you use them everyday in one way or another. However, did you know that numbers actually have classifications?

We classify different types of numbers using different properties of those numbers, and we call them sets. We can break all numbers in to the sets of natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, and imaginary numbers. These are all defined in the following image.


closreal1


In this lesson, we are going to be working with real numbers. From the image, we see that real numbers consist of all of the sets of numbers that we normally work with. That is, they include the natural numbers {1,2,3, ...}, the whole numbers {0,1,2,3, ...}, the integers {...,-3,-2,-1,0,1,2,3,....}, the rational numbers {p/q, where p and q are integers}, and the irrational numbers {all non-repeating and non-terminal decimals}. In fact, the real numbers consist of all of the sets of numbers except imaginary numbers {a + bi, where a and b are real numbers and i = sqrt(-1)}. This is why they are called real numbers - they aren't imaginary!

Now that we are familiar with real numbers, let's explore some certain properties of these numbers. I'm talking about closure properties. A set of numbers is said to be closed under a certain operation if when that operation is performed on two numbers from the set, we get another number from that set as an answer.

Closure Properties of Real Numbers

Real numbers are closed under two operations - addition and multiplication. We could also say that real numbers are closed under subtraction and division, but this is actually covered by addition and multiplication because we can turn any subtraction or division problem into an addition or multiplication problem, respectively, due to the nature of real numbers.

That being said, you may wonder about the number 0 when it comes to division because we can't divide by 0. Well, here's an interesting fact! Since x / 0 is considered to be undefined, the real numbers are closed under division and it just so happens that division by zero was defined this way so that the real numbers could be closed under division.

Let's take a look at the addition and multiplication closure properties of real numbers. Because real numbers are closed under addition, if we add two real numbers together, we will always get a real number as our answer. This is shown in the image.


closreal3


As we said, any subtraction problem of real numbers can be turned into an addition problem, and since real numbers are closed under addition, we can also be assured they are closed under subtraction. Changing subtraction to addition is done as follows.

x - y = x + (-y)

Real numbers are also closed under multiplication, so if we multiply any two real numbers together, the answer will be a real number, as shown in the image.


closreal5


Again, we mentioned that any division problem of real numbers can be turned into a multiplication problem of real numbers, so real numbers are also closed under division (excluding division by zero since it is undefined). Changing division to multiplication is done as follows.

x / y = x * (1/y)

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