Easy Mathematics Step-by-Step (2012)

Chapter 2. Integers

In this chapter, you learn about performing operations with integers. Before proceeding with addition, subtraction, multiplication, and division of integers, the discussion begins with comparing and finding the absolute value of numbers.

Comparing Integers

Recall that the integers are the numbers

…, –3, –2, –1, 0, 1, 2, 3,…

Comparing numbers uses the inequality symbols shown in Table 2.1.

Table 2.1 Inequality Symbols

Graphing the numbers on a number line is helpful when you compare two numbers. The number that is farther to the right is the greater number. If the numbers coincide, then they are equal; otherwise, they are unequal.

Problem Which is greater –7 or –2?

Solution

Step 1. Graph –7 and –2 on a number line.

Step 2. Identify the number that is farther to the right as the greater number.

–2 is to the right of –7, so .

Absolute Value

The concept of absolute value plays an important role in computations with signed numbers. The absolute value of an integer is its distance from 0 on the number line. For example, as shown in Figure 2.1, the absolute value of –8 is 8 because –8 is 8 units from 0.

Figure 2.1 Absolute value of –8

Absolute value is a distance, so it is never negative.

You indicate the absolute value of a number by placing the number between a pair of vertical bars like this: (read as “the absolute value of negative eight”). Thus, .

Problem Find the indicated absolute value.

Solution

Step 1. Recalling that the absolute value of an integer is its distance from 0 on the number line, determine the absolute value.

because –30 is 30 units from 0 on the number line.

Step 1. Recalling that the absolute value of an integer is its distance from 0 on the number line, determine the absolute value.

because 4 is 4 units from 0 on the number line.

Step 1. Recalling that the absolute value of an integer is its distance from 0 on the number line, determine the absolute value.

because –2 is 2 units from 0 on the number line.

As you likely noticed, the absolute value of a number is the value of the number with no sign attached. This strategy works for a number whose value you know, but do not use it when you don’t know the value of the number.

Problem Which number has the greater absolute value?

a. –35, 60

b. 35, –60

c. –7, 2

d. –21, 17

Solution

a. 35, 60

Step 1. Determine the absolute values.

Step 2. Compare the absolute values.

60 has the greater absolute value because .

b. 35, –60

Step 1. Determine the absolute values.

Step 2. Compare the absolute values.

c. –7, 2

Step 1. Determine the absolute values.

Step 2. Compare the absolute values.

d. 21, 17

Step 1. Determine the absolute values.

Step 2. Compare the absolute values.

Don’t make the mistake of trying to compare the numbers without first finding the absolute values.

Integers are called signed numbers because these numbers may be positive, negative, or 0. From your knowledge of arithmetic, you already know how to do addition, subtraction, multiplication, and division with positive numbers and 0. To do these operations with all signed numbers, you simply use the absolute values of the numbers and follow these eight rules.

Adding Signed Numbers

Addition of Signed Numbers

Rule 1. To add two numbers that have the same sign, add their absolute values and give the sum their common sign.

Rule 2. To add two numbers that have opposite signs, subtract the lesser absolute value from the greater absolute value and give the sum the sign of the number with the greater absolute value; if the two numbers have the same absolute value, their sum is 0.

These rules might sound complicated, but practice will make them your own. One helpful hint is that when you need the absolute value of a number, just use the value of the number with no sign attached.

Rule 3. The sum of 0 and any number is the number.

The number line is a good tool for illustrating the addition of signed numbers, as shown below.

Problem Add the two numbers. Illustrate the addition on a number line.

a. 4 and 7

b. –5 and 7

c. –3 and –4

Solution

a. 4 and –7

Step 1. Begin at 0 and move 4 units in the positive direction to 4.

Step 2. From that point, move 7 units in the negative direction to –3.

Step 3. Express the result.

b. –5 and 7

Step 1. Start at 0 and go 5 units in the negative direction to –5.

Step 2. From that point, go 7 units in the positive direction to 2.

Step 3. Express the result.

c. –3 and –4

Step 1. Start at 0 and move 3 units in the negative direction to –3.

Step 2. From that point, move 4 units in the negative direction to –7.

Step 3. Express the result.

Problem Find the sum.

Draw number line illustrations for parts d and e.

Solution

Step 1. Determine which addition rule applies.

The signs are the same (both negative), so use Rule 1.

Step 2. Add the absolute values, 35 and 60.

Step 3. Give the sum a negative sign (the common sign).

Step 1. Determine which addition rule applies.

The signs are opposites (one positive and one negative), so use Rule 2.

Step 2. Subtract 35 from 60 because .

Step 3. Make the sum negative because –60 has the greater absolute value.

Step 1. Determine which addition rule applies.

The signs are opposites (one negative and one positive), so use Rule 2.

Step 2. Subtract 35 from 60 because .

Step 3. Keep the sum positive because 60 has the greater absolute value.

Step 1. Start at 0 and go 3 units in the positive direction to 3.

Step 2. From that point, go 3 units in the negative direction to 0.

Step 3. Express the result.

Step 1. Start at 0 and go 5 units in the negative direction to –5.

Step 2. From that point, go 2 units in the positive direction to –3.

Step 3. Express the result.

Subtracting Signed Numbers

You subtract signed numbers by changing the subtraction problem in a special way to an addition problem, so that you can apply the rules for addition of signed numbers. Here is the rule.

Subtraction of Signed Numbers

Rule 4. To subtract two numbers, keep the first number and add the opposite of the second number.

To apply this rule, think of the minus sign, –, as “add the opposite of.” In other words, “subtracting a number” and “adding the opposite of the number” give the same answer.

Problem Change the subtraction problem to an addition problem.

Solution

Step 1. Keep –35.

Step 2. Add the opposite of 60.

Step 1. Keep 35.

Step 2. Add the opposite of 60.

Step 1. Keep 60.

Step 2. Add the opposite of 35.

Step 1. Keep –35.

Step 2. Add the opposite of –60.

Step 1. Keep 0.

0

Step 2. Add the opposite of 60.

Step 1. Keep –60.

Step 2. Add the opposite of 0.

Remember that 0 is its own opposite.

Problem Compute as indicated.

Solution

Step 1. Keep –35 and add the opposite of 60.

Step 2. The signs are the same (both negative), so use Rule 1 for addition.

Step 3. Review the main results.

Reviewing your work is a smart habit to cultivate. You will catch many careless errors with this strategy.

Step 1. Keep 35 and add the opposite of 60.

Step 2. The signs are opposites (one positive and one negative), so use Rule 2 for addition.

Step 3. Review the main results.

Step 1. Keep 60 and add the opposite of 35.

Step 2. The signs are opposites (one positive and one negative), so use Rule 2 for addition.

Step 3. Review the main results.

Step 1. Keep –35 and add the opposite of –60.

Step 2. The signs are opposites (one positive and one negative), so use Rule 2 for addition.

Step 3. Review the main results.

Step 1. Keep 0 and add the opposite of 60.

Step 2. 0 is added to a number, so the sum is the number (Rule 3 for addition).

Step 3. Review the main results.

Step 1. Keep –60 and add the opposite of 0.

Step 2. 0 is added to a number, so the sum is the number (Rule 3 for addition).

Step 3. Review the main results.

Notice that subtraction is not commutative. That is, in general, for real numbers a and b, .

Use of the – Symbol

Before going on, it is important that you distinguish the various uses of the short horizontal – symbol. Thus far, this symbol has three uses: (1) as part of a number to show that the number is negative, (2) as an indicator to find the opposite of the number that follows, and (3) as the minus symbol indicating subtraction.

Problem

a. Describe the use of the – symbols at (1), (2), (3), and (4).

b. Express the statement in words.

Solution

a. Describe the use of the – symbols at (1), (2), (3), and (4).

Step 1. Interpret each – symbol.

The – symbol at (1) is an indicator to find the opposite of –35.

Don’t make the error of referring to negative numbers as “minus numbers.”

The – symbol at (2) is part of the number –35 that shows –35 is negative.

The minus symbol always has a number to its immediate left.

The – symbol at (3) is the minus symbol indicating subtraction.

The – symbol at (4) is part of the number –60 that shows –60 is negative.

There is never a number to the immediate left of a negative sign.

b. Express the statement in words.

Step 1. Translate the statement into words.

is read as “the opposite of negative thirty-five minus sixty is thirty-five plus negative sixty.”

Multiplying Signed Numbers

For multiplication of signed numbers, use the following three rules.

Multiplication of Signed Numbers

Rule 5. To multiply two numbers that have the same sign, multiply their absolute values and keep the product positive.

Rule 6. To multiply two numbers that have opposite signs, multiply their absolute values and make the product negative.

Rule 7. The product of 0 and any number is 0.

Unlike in addition, when you multiply two positive or two negative numbers, the product is positive no matter what. Similarly, unlike in addition, when you multiply two numbers that have opposite signs, the product is negative—it doesn’t matter which number has the greater absolute value.

Problem Find the product.

Solution

Step 1. Determine which multiplication rule applies.

The signs are the same (both negative), so use Rule 5.

Step 2. Multiply the absolute values, 3 and 40.

Step 3. Keep the product positive.

Step 1. Determine which multiplication rule applies.

The signs are the same (both positive), so use Rule 5.

Step 2. Multiply the absolute values, 3 and 40.

Step 3. Keep the product positive.

Step 1. Determine which multiplication rule applies.

The signs are opposites (one negative and one positive), so use Rule 6.

Step 2. Multiply the absolute values, 3 and 40.

Step 3. Make the product negative.

Step 1. Determine which multiplication rule applies.

The signs are opposites (one positive and one negative), so use Rule 6.

Step 2. Multiply the absolute values, 3 and 40.

Step 3. Make the product negative.

Step 1. Determine which multiplication rule applies.

Step 2. Find the product.

Rules 5 and 6 tell you how to multiply two nonzero numbers, but often you will want to find the product of more than two numbers. To do this, multiply in pairs. You can keep track of the sign as you go along, or you simply can use the following guideline:

When 0 is one of the factors, the product is always 0; otherwise, products that have an even number of negative factors are positive, whereas, those that have an odd number of negative factors are negative.

Notice that if there is no zero factor, then the sign of the product is determined by how many negative factors you have.

Problem Find the product.

Solution

Step 1. 0 is one of the factors, so the product is 0.

Step 1. Find the product ignoring the signs.

Step 2. You have five negative factors, so make the product negative.

Step 1. Find the product ignoring the signs.

Step 2. You have four negative factors, so leave the product positive.

Dividing Signed Numbers

Division of Signed Numbers

Rule 8. To divide two numbers, divide their absolute values (being careful to make sure you don’t divide by 0) and then follow the rules for multiplication of signed numbers.

If 0 is the dividend, then the quotient is 0. For instance, . But, if 0 is the divisor, then the quotient is undefined. Thus, and . has no answer because division by 0 is undefined!

Problem Find the quotient.

Division is commonly indicated by the fraction bar.

Solution

Step 1. Divide 120 by 3.

Step 2. The signs are the same (both negative), so keep the quotient positive.

Step 1. Divide 120 by 3.

Step 2. The signs are opposites (one negative and one positive), so make the quotient negative.

Step 1. Divide 120 by 3.

Step 2. The signs are opposites (one positive and one negative), so make the quotient negative.

Step 1. The divisor (denominator) is 0, so the quotient is undefined.

Step 1. The dividend (numerator) is 0, so the quotient is 0.

To be successful in arithmetical computation, you must memorize the rules for adding, subtracting, multiplying, and dividing signed numbers. Of course, when you do a computation, you don’t have to write out all the steps. For instance, you can mentally ignore the signs to obtain the absolute values, do the necessary computation or computations, and then make sure your result has the correct sign.

Exercise 2

For 1–3, simplify.

For 4 and 5, state in words.

For 6–17, compute as indicated.

For 18–20, draw number line illustrations for the indicated sums.