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The Closure Property


Addition and Multiplication

Math - Closure Property & Density Property

Closure Property & Density Property -

Properties of Real Numbers:

Real numbers have unique properties

which make them particularly useful in everyday life.

First , Real numbers are an ordered set of numbers. This means real numbers are sequential. The numerical value of every real number fits between the numerical values two other real numbers.

Everyone is familiar with this idea since all measurements (weight, the purchasing power of money, the speed of a car, etc.) depend upon the fact that some numbers have a higher value than other numbers. Ten is greater than five, and five is greater than four . . . and so on.

Second , we never run out of real numbers. The quantity of real numbers available is not fixed. There are an infinite number of values available. The availability of numbers expands without end. Real numbers are not simply a finite "row of separate points" on a number line. There is always another real number whose value falls between any two real numbers (this is called the "density" property).

Third , when real numbers are added or multiplied, the result is always another real number (this is called the "closure" property). [This is not the case with all arithmetic operations. For example, the square root of a -1 yields an imaginary number.]

With these three points in mind, the question is: How can we use real numbers in practical calculations? What rules apply?

  • How should numbers be added, subtracted, multiplied, and divided? What latitude do we have?

  • Does it matter what we do first? second? third? . . .

  • Can we add a series of numbers together in any order? Will the final answer be the same regardless of the order we choose?

  • Can we multiply a series of numbers together in any order? Will the final answer be the same regardless of the order we choose?

The following properties of real numbers answers these types of questions. The property characteristics which follow show how much latitude you have to change the mechanics of calculations which use real numbers without changing the results.

  • Associative Property

  • Commutative Property

  • Distributive Property

  • Identity Property

  • Inverse Property

The Seven Fundamental Properties of Real Numbers - click description

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Closure & Density Properties

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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:

Math - Closure Example - Closed Under Addition

"Blue Squares"

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

Math - Adding Real Numbers = Another Real Number

The number "21" is a real number.

Example 2: Multiplying two real numbers produces another real number

Math - Mult Real Numbers = 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.

Math - Number Line Comparison - Density Property