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BASIC CONCEPTS

In this chapter, we review some of the concepts that students are typically expected to know before learning algebra. Although we don't have the space to fully develop these concepts, we point out some common mistakes and other important points that you should keep in mind. Even if you think that you know these topics, we recommend that you work through this chapter.

1.1 Addition, Subtraction, Multiplication, Division

Throughout this book, we use visual models to represent mathematical ideas. One important model is a number line, a line on which each point represents exactly one number. The numbers always increase from left to right. To show the scale, numbers are marked off at equal intervals. We draw an arrow at the end to indicate that the numbers extend infinitely in that direction.

Positive numbers, which we indicate with a + in front of the number, are numbers greater than 0. Negative numbers, which we indicate with a - in front of the number, are numbers less than 0. The word sign refers to the property of being positive or negative. The term signed numbers refers to numbers and their signs. Numbers that don't have a sign in front of them are understood to be positive.

On a horizontal number line (Figure 1.1), positive numbers lie to the right of 0, and negative numbers lie to the left of 0:

Did You Know?

The idea of positive numbers, negative numbers, and 0 may seem obvious to us now, but they actually developed around the world over thousands of years. By the 3rd century BCE, the Chinese were using counting rods of different colors to represent positive and negative numbers in their calculations. The 7th-century Indian mathematician Brahmagupta described rules in terms of "fortunes" (positive numbers) and "debts" (negative numbers). Ancient societies understood the concept of nothing ("we have no water"), but many cultures, such as the Egyptians, Romans, and Greeks, created complex mathematics without 0. The use of 0 didn't fully develop until the 5th century CE in India.

Figure 1.1 Number line

The absolute value of a number is its distance from 0 on a number line. Since the absolute value represents distance, it is always positive (unless we're talking about 0, which has an absolute value of 0). We use vertical bars to indicate absolute value. We read |+2| as "the absolute value of positive two." For example, |+15| is equal to 15, |-15| is equal to 15, and |0| is equal to 0. Two numbers that are the same distance from 0 on the number line but have different signs, such as +2 and -2, are opposites. Zero is an exception-the opposite of 0 is itself.

In math, we have four basic operations (mathematical processes performed on quantities to get a result): addition, subtraction, multiplication, and division. When we combine quantities with operations, we make an expression, such as 5 + 3 and |+15| - 4.

Watch Out!

We use the + and - symbols to represent both addition and subtraction and the sign of a number.

• When + and - represent the sign of a number (which only occurs before a number), we read + as "positive" and - as "negative." We never put a space between the symbol and the number, so "negative 5" would be written -5, never - 5.
• When + and - represent addition or subtraction (which only occurs between two numbers), we read + as "plus" and - as "minus," and we put 1 space before and after the symbol. For example, 4 + 5, which is read as "4 plus 5," means 5 is added to 4 to get a sum of 9.

The + and - symbols can represent both operations and signs in the same mathematical sentence. For example, +5 - -3 is read "positive 5 minus negative 3," not "plus 5 minus minus 3." Sometimes, we put parentheses around signed numbers to separate them from the addition or subtraction symbols, so we write +5 - -3 as (+5) - (-3). The parentheses are not pronounced.

You may recall working with number lines in elementary school. In this book, we also use squares to model signed numbers because they enable us to represent far more complicated ideas that we need to work with in algebra. To represent +1, we use a square whose area is +1. To represent -1, we use a square whose area is -1. (Don't worry about what a square with a negative area actually "means"-it's just a model!) A square with area +1 and a square with area -1 have a total area of 0. We call this pair a zero pair. We can group zero pairs into rectangles (think of them as "jumbo packs" of +1 or -1 squares) and use them to add signed numbers, as shown in Example 1.1:

Example 1.1 Evaluate -40 + 54.

Solution: When we evaluate an expression, we perform mathematical calculations to get a single number.

In this example, we use the = symbol (which is called an equal sign and read "equals" or "is equal to"). The equal sign means that the expression on its left has the same value as the expression on its right. A mathematical statement containing an equal sign is called an equation. To make your work easier to read, do one part of the calculation at a time and write each step on a different line, starting each line with the equal sign.

Watch Out!

One common mistake when writing several equations on one line is to ignore the meaning of the equal sign. For example, when evaluating 2 + 3 + 4, some students write: 2 + 3 = 5 + 4 = 9. This "run-on" equation implies that 2 + 3, 5 + 4, and 9 are all equal, which isn't what we meant! Instead, write the following:

How to Add Signed Numbers

1. Determine the number with the larger absolute value.
2. Form zero pairs with the number with the smaller absolute value.
3. The remainder is the final answer, called the sum.

Addition and subtraction undo each other. For example, 5 + 3 - 3 equals 5. More formally, we say that addition and subtraction are inverse operations. This means that when we apply inverse operations on a number, the result is the original number. We can think of subtraction in terms of addition.

How to Subtract Signed Numbers

• To subtract a positive number, add a negative number with the same absolute value, so 5 - 3 = 5 + (-3). The result, called the difference, is 2. (This models real-world behavior-adding debt lowers your net worth.)
• To subtract a negative number, add a positive number with the same absolute value, so 5 - (-3) is the same as 5 + 3. The difference is 8. (This also models real-world behavior-removing debt raises your net worth.)

Example 1.2 illustrates how these rules work.

Example 1.2 Evaluate (-30) - (-46).

Solution:

1. From -30, we remove 46 negative unit squares. Since we don't have any more negative unit squares, add 46 zero pairs (46 negative and 46 positive unit squares) and remove the 46 negative unit squares.

1. After removing the 46 negative unit squares, we have 30 negative and 46 positive unit squares.

1. To determine what we have left in step 2, we separate the +46 into 30 positive and 16 positive unit squares (since 46 - 30 equals 16).

1. The 30 negative and 30 positive unit squares make 30 zero pairs, which add up to 0, leaving 16 positive unit squares.

Technology Tip

Many calculators have different buttons for subtraction and negative numbers. Often, the subtraction button is located next to the buttons for addition, multiplication, and division. To change the sign of an entry, they have a button labeled +/- or (-), where the - symbol on the button is shorter than the - symbol. Some...