# Chapter 1: The Real Number System

**Advanced**Created by: CK-12

## Introduction

Here you'll review how to perform the basic operations of addition, subtraction, multiplication and division with real numbers. All of these operations will then be applied to the order of operations. You will also learn how to express a fraction as a decimal and a decimal number as a fraction. Finally, you will represent real numbers on a number line.

## Chapter Outline

- 1.1. Addition of Integers
- 1.2. Addition of Fractions
- 1.3. Addition of Decimals
- 1.4. Subtraction of Integers
- 1.5. Subtraction of Fractions
- 1.6. Subtraction of Decimals
- 1.7. Multiplication of Real Numbers
- 1.8. Division of Real Numbers
- 1.9. Properties of Real Number Addition
- 1.10. Properties of Real Number Multiplication
- 1.11. PEMDAS with Positive Real Numbers
- 1.12. PEMDAS with Negative Real Numbers
- 1.13. Decimal Notation
- 1.14. Real Number Line Graphs
- 1.15. Real Number Comparisons

### Chapter Summary

## Summary

You reviewed how to add real numbers using two rules:

- Real numbers with unlike signs must be subtracted and the answer will have a sign the same as that of the higher number.
- Real numbers with the same sign must be added and the answer will have a sign the same as that of the numbers being added.

Then you reviewed how to subtract real numbers by adding the opposite. You learned that a number has both direction and magnitude. The direction of a positive number is to the right and that of a negative number is to the left. This direction is determined by the numbers location with respect to zero on the number line. The magnitude of a number is simply its size with no regard to its sign.

Next you reviewed multiplication and division of real numbers using the following rules:

- The product/quotient of two positive numbers is always positive.
- The product/quotient of two negative numbers is always positive.
- The product/quotient of a positive number and a negative number is always negative.

You reviewed the standard order of operations represented by the letters PEMDAS.

- P – Parenthesis – Do all the calculations within parenthesis.
- E – Exponents – Do all calculations that involve exponents.
- M/D – Multiplication/Division – Do all multiplication/division, in the order it occurs, working from left to right.
- A/S – Addition/Subtraction – Do all addition/subtraction, in the order it occurs, working from left to right.
- S – Subtraction – Do all subtraction, in the order it occurs, working from left to right.

Next you learned how to distinguish between a terminating decimal number and a periodic decimal number. You were also shown that a given fraction can be changed to a decimal number by long division.

Last you revisited the real number system . The real number system is made up of rational and irrational numbers. The rational numbers include the natural numbers, whole numbers, integers and rational numbers. The natural numbers are the counting numbers and consist of all positive, whole numbers.