1. Operations with Real Numbers
1.1 Algebraic Expressions
Learning Objectives
By the end of this section it is expected that you will be able to:
- Use variables and algebraic symbols
- Identify expressions and equations
- Simplify expressions with exponents
- Simplify expressions using the order of operations
- Evaluate algebraic expressions
Use Variables and Algebraic Symbols
In algebra, letters of the alphabet are used to represent variables.
Letters often used for variables are .
Variables and Constants
A variable is a letter that represents a number or quantity whose value may change.
A constant is a number whose value always stays the same.
To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. There are four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.
| Operation | Notation | Say: | The result is… |
|---|---|---|---|
| Addition | the sum of and | ||
| Subtraction | the difference of and | ||
| Multiplication | The product of and | ||
| Division | divided by | The quotient of and |
In algebra, the cross symbol, ×, is not used to show multiplication because that symbol may cause confusion. Does mean (three times ) or (three times )? To make it clear, use • or parentheses for multiplication.When two quantities have the same value, we say they are equal and connect them with an equal sign.
Equality Symbol
The symbol is called the equal sign.
An inequality is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities. Remember that on the number line the numbers get larger as they go from left to right. So if we know that is greater than , it means that is to the right of on the number line. We use the symbols < and > for inequalities.
Inequality
< is read is less than
is to the left of on the number line
> is read is greater than
is to the right of on the number line
The expressions < > can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,
When we write an inequality symbol with a line under it, such as , it means or . We read this is less than or equal to . Also, if we put a slash through an equal sign, it means not equal.
We summarize the symbols of equality and inequality in the table below.
| Algebraic Notation | Say |
|---|---|
| is equal to | |
| is not equal to | |
| < | is less than |
| > | is greater than |
| is less than or equal to | |
| is greater than or equal to |
Symbols < and >
The symbols < and > each have a smaller side and a larger side.
smaller side < larger side
larger side > smaller side
The smaller side of the symbol faces the smaller number and the larger faces the larger number.
Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They