Real Analysis: Completeness of the Real Numbers

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  • 0:03 The Real Number System
  • 0:43 The Five Classifications
  • 2:46 Lesson Summary
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Lesson Transcript
Instructor: Michelle Vannoy
What is the real number system, and what does the system contain? Why does there need to be a classification of real numbers? Complete this lesson to learn the answers.

The Real Number System

The real number system is a classification of the numbers that we use every day. There are five subsets inside the real number system that divide the numbers into groups by the attributes of each number. Why is this important? When completing problems with unknown variables, it helps to be able to list a number type as a possible solution instead of listing all possible numbers that are in the set.

For example, if a variable can be stated as any number included in the set of natural numbers, that's much easier than having to list all the numbers from one to infinity. It would be impossible to write all of the natural numbers from one to infinity on a page. Classifying numbers makes describing them a much easier task.

The Five Classifications

There are five different classifications that belong to the real number system. Each classification has a set of rules for determining the numbers that are included. But, it's just vocabulary. The broadest classifications are irrational and rational numbers. All numbers are either rational or irrational, but cannot be both rational and irrational.

  • Irrational numbers are any non-repeating, non-terminating decimals. These types of numbers cannot be written as a fraction, so therefore, they are irrational. Examples of irrational numbers are π or √2.

  • Rational numbers are any number that can be written as a fraction a/b where b does not equal 0. Examples of rational numbers would be 3/4, 5, 1500, or 0.343434…. This includes terminating and repeating decimals because they can be written as a fraction.

Rational numbers are then split into the other three categories. Think of a nesting doll. The largest doll is on the outside. When you open that doll, there is a smaller one inside. When that doll is opened there is another one inside that's a little smaller. Rational numbers are the large doll, and next inside layer is integers, then the whole numbers, and then the natural numbers.

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