638 Chapter 8 An Introduction to Algebra
SECTION 8.1
The Language of Algebra
Objectives
1 Use variables to state
properties of addition,
multiplication, and division.
2 Identify terms and coefficients
of terms.
3 Translate word phrases to
algebraic expressions.
4 Evaluate algebraic expressions.The first seven chapters of this textbook have been an in-depth study of arithmetic.
It’s now time to begin the move toward algebra. Algebra is the language of
mathematics. It can be used to solve many types of problems. In this chapter, you will
learn more about thinking and writing in the language of algebra using its most
important component—a variable.The Language of Mathematics The word algebracomes from the title of the
book Ihm Al-jabr wa’l muqabalah, written by an Arabian mathematician
around A.D. 800.Use variables to state properties of addition, multiplication,
and division.
One of the major differences between arithmetic and algebra is the use of
variables. Recall that a variableis a letter (or symbol) that stands for a number. In
this course, we have used variables on several occasions. For example, in Chapter 1,
we let lstand for the length and wstand for the width in the formula for the area
of a rectangle:A= lw.In Chapter 6, we let xrepresent the unknown number in
percent problems.The Language of Mathematics The word variableis based on the root word
vary,which means change or changing. For example, the length and width of
rectangles vary, and the unknown numbers in percent problems vary.Many symbols used in arithmetic are also used in algebra. For example, a plus
symbol is used to indicate addition, a minus symbol – is used to indicate subtraction,
and an symbol means is equal to.
Since the letter xis often used in algebra and could be confused with the
multiplication symbol , we usually write multiplication using a raised dotor
parentheses.When multiplying a variable by a number, or a variable by another
variable, we can omit the symbol for multiplication. For example,
2 bmeans 2bxymeans xy 8 abcmeans 8abc
In the notation 2b, the number 2 is an example of a constantbecause it does not
change value.
Many of the patterns that we have seen while working with whole numbers,
integers, fractions, and decimals can be generalized and stated in symbols using
variables. Here are some familiar properties of addition written in a very compact
form, where the variables aand brepresent any numbers.- The Commutative Property of Addition
abba- The Associative Property of Addition
(ab) ca(bc)
Changing the grouping when adding
does not affect the answer.Changing the order when adding
does not affect the answer.1