Closure Property & Density Property -
Closure Property -
The term "closure" is used in many diverse fields. Generally, these definitions are not compatible with the precise definition used in mathematics.
Outside the field of mathematics, closure can mean many different things. For example, it can mean something is enclosed (such as a chair is enclosed in a room), or a crime has been solved (we have "closure"). Visual Closure means that you mentally fill in gaps in the incomplete images you see. These kinds of concepts should not be carried over to mathematics.
When the term "closure" is used in mathematics, it applies to sets and mathematical operations. The sets can include ordinary numbers, vectors, to matrices, algebra, and other elements. The operations can include any operation (addition, multiplication, square root, etc.).
However . . . here, our concern is only with the closure property as it applies to real numbers .
The Closure Property states that when you perform an operation (such as addition, multiplication, etc.) on any two numbers in a set, the result of the computation is another number in the same set.
As an example, consider the set of all blue squares, highlighted on a yellow background, below:
The set of blue squares is closed under addition because when any blue squares are added, the result is more blue squares. You have not moved outside the set of all blue squares.
However, the set of blue squares is NOT closed under multiplication. If you multiply some blue squares by ½ the result sometimes includes ½ a blue square. The ½ blue square does not belong to the set of all blue squares.
Real numbers are closed with respect to addition and multiplication. Real numbers are not closed with respect to division (a real number cannot be divided by 0).
Example 1: Adding two real numbers produces another real number
The number "21" is a real number.
Example 2: Multiplying two real numbers produces another real number
The number "312" is a real number.
Why is this important to know?
It is important because equations which only involve addition and multiplication have a solution which is also a real number - you know that in advance.
For example, you know for certain that you can add the costs of all the items in a shopping basket and get a "real number" answer. This is due to the property of closure.
If this were not the case, you could never be sure that addition would quite work.
Density Property -
The Density Property states that there is always a another real number between any two real numbers : there is a limitless supply of real numbers.
This idea is illustrated by the number lines shown below . No matter what two numbers are chosen, there are always more numbers in between the two.
Closure Property & Density Property