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High School Algebra I: Help and Review25 chapters | 292 lessons

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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.

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.

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 = 3.14159â€¦

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.

The set of real numbers can be divided into many different groups. Each of these groups has their own special set of characteristics, but they are all still real numbers.

**Whole numbers**are positive numbers that are not fractions or decimals. They include zero.**Positive numbers**include all numbers that are greater than zero. They can be fractions, decimals or whole numbers.**Negative numbers**include all numbers that are less than zero. They can also be fractions or decimals.**Integers**are whole numbers and their opposite negative numbers (without fractions or decimals).**Natural numbers**are the counting numbers (1,2,3,â€¦). They are whole numbers that exclude zero.**Rational numbers**are fractions that when written as a decimal either have an end point (0.5) or repeat (1.3333333. . .).**Irrational numbers**are numbers that when written as a decimal have no end point (2.5463489762547. . .). Pi is a well-known irrational number. Although it is often abbreviated as 3.14, the numbers after the decimal go on forever.

Let's review. Even though there are many definitions that point to the different properties of real numbers, they all describe the same thing. A set of **real numbers** obeys these three rules:

- The numbers are measurable
- The numbers have concrete value
- The numbers can be manipulated

And, of course, the numbers are real, not imaginary. Types of real numbers include **whole** numbers, **rational** numbers and **irrational** numbers.

When you are finished, you should be able to:

- Determine if a number is a real number using the three rules of real numbers
- Name the types of real numbers

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High School Algebra I: Help and Review25 chapters | 292 lessons

- What are the Different Types of Numbers? 6:56
- Graphing Rational Numbers on a Number Line 5:02
- Notation for Rational Numbers, Fractions & Decimals 6:16
- The Order of Real Numbers: Inequalities 4:36
- Finding the Absolute Value of a Real Number 3:11
- What Are Composite Numbers? - Definition & Examples 4:38
- What Are Odd & Even Numbers? - Definition & Examples 6:38
- What are Real Numbers? - Definition & Properties 4:50
- Go to High School Algebra - Real Numbers: Help and Review

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