Computation with Real Numbers - Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST - Easy Algebra Step-by-Step

Easy Algebra Step-by-Step: Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST! (2012)

Chapter 2. Computation with Real Numbers

This chapter presents the rules for computing with real numbers—often called signed numbers. Before proceeding with addition, subtraction, multiplication, and division of signed numbers, the discussion begins with comparing numbers and finding the absolute value of a number.

Comparing Numbers and Absolute Value

Comparing numbers uses the inequality symbols shown in Table 2.1.

Table 2.1 Inequality Symbols

Image

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, they are equal; otherwise, they are unequal.

Problem Which is greater –7 or –2?

Solution

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

Image

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

–2 is to the right of –7, so image

The concept of absolute value plays an important role in computations with signed numbers. The absolute value of a real number 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.

Image

Figure 2.1 The 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: |–8| (read as “the absolute value of negative eight”). Thus, image

Problem Find the indicated absolute value.

image

Solution

image

Image Step 1. Recalling that the absolute value of a real number is its distance from 0 on the number line, determine the absolute value.

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

image

Image Step 1. Recalling that the absolute value of a real number is its distance from 0 on the number line, determine the absolute value.

image because 0.4 is 0.4 units from 0 on the number line.

c. Image


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.


Image Step 1. Recalling that the absolute value of a real number is its distance from 0 on the number line, determine the absolute value.

Image because Image units from 0 on the number line.

Problem Which number has the greater absolute value?

image

Solution

image

Image Step 1. Determine the absolute values.

image

Step 2. Compare the absolute values.

60 has the greater absolute value because 60 > 35.

image

Image Step 1. Determine the absolute values.

image

Step 2. Compare the absolute values.

–60 has the greater absolute value because 60 > 35.

c. Image

Image Step 1. Determine the absolute values.

Image

Step 2. Compare the absolute values.

Image has the greater absolute value because Image.

d. Image

Image Step 1. Determine the absolute values.

Image

Step 2. Compare the absolute values.

Image has the greater absolute value because Image.


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


Addition and Subtraction of Signed Numbers

Real numbers 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.

Addition of Signed Numbers

Image 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.

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


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.


Problem Find the sum.

image

Solution

image

Image Step 1. Determine which addition rule applies.

image

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

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

image

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

image

Image Step 1. Determine which addition rule applies.

image

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

Step 2. Subtract 35 from 60 because image

image

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

image

Image Step 1. Determine which addition rule applies.

image

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

Step 2. Subtract 35 from 60 because image

image

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

image

d. Image

Image Step 1. Determine which addition rule applies.

Image

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

Step 2. Subtract Image from Image because Image.

Image

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

Image

e. Image

Image Step 1. Determine which addition rule applies.

Image

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

Step 2. Subtract Image from Image because Image.

Image

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

Image

image

Image Step 1. Determine which addition rule applies.

image

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

Step 2. Add the absolute values 9.75 and 8.12.

image

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

image

Image Step 1. Determine which addition rule applies.

image

0 is added to a number, so the sum is the number (Rule 3).

image

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

Subtraction of Signed Numbers

Image 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.

image

Solution

image

Image Step 1. Keep –35.

–35

Step 2. Add the opposite of 60.

image

Image Step 1. Keep 35.

35

Step 2. Add the opposite of 60.

image

image

Image Step 1. Keep 60.

60

Step 2. Add the opposite of 35.

image

Image Step 1. Keep –35.

–35

Step 2. Add the opposite of –60.

image

Image Step 1. Keep 0.

0

Step 2. Add the opposite of 60.

image

Image Step 1. Keep –60.

–60

Step 2. Add the opposite of 0.

image


Remember 0 is its own opposite.


Problem Find the difference.

image


A helpful mnemonic to remember how to subtract signed numbers is “Keep, change, change.” You keep the first number, you change minus to plus, and you change the second number to its opposite.


Solution

image

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

image

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

= –95

Step 3. Review the main results.

image


Cultivate the habit of reviewing your main results. Doing so will help you catch careless mistakes.


image

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

image

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

= –25

Step 3. Review the main results.

image

image

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

image

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

= 25

Step 3. Review the main results.

image

image

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

image

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

= 25

Step 3. Review the main results.

image

image

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

image

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

= 60

Step 3. Review the main results.

image

image

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

image

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

= –60

Step 3. Review the main results.

image


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


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 Given the statement Image

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

b. Express the statement image in words.

Solution

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

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


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


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 image in words.

Image Step 1. Translate the statement into words.

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

Multiplication and Division of Signed Numbers

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

Multiplication of Signed Numbers

Image 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.


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


Problem Find the product.

image

Solution

image

Image Step 1. Determine which multiplication rule applies.

image

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

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

image

Step 3. Keep the product positive.

image

b. (3)(40)

Image Step 1. Determine which multiplication rule applies.

(3)(40)

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

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

(3)(40) = 120

Step 3. Keep the product positive.

image

Image Step 1. Determine which multiplication rule applies.

(–3)(40)

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

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

(3)(40) = 120

Step 3. Make the product negative.

(–3)(40) = –120

d. (3)(–40)

Image Step 1. Determine which multiplication rule applies.

(3) (–40)

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

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

(3)(40) = 120

Step 3. Make the product negative.

(3)(–40) = –120

e. (358)(0)

Image Step 1. Determine which multiplication rule applies.

(358)(0)

0 is one of the factors, so use Rule 7.

Step 2. Find the product.

(358)(0) = 0

Rules 5, 6, and 7 tell you how to multiply two 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.

image

Solution

image

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

image

Image Step 1. Find the product ignoring the signs.

image

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

image

Image Step 1. Find the product ignoring the signs.

image

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

image

Division of Signed Numbers

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


Problem Find the quotient.

a. Image

b. Image

c. Image

d. Image

e. Image


In algebra, division is commonly indicated by the fraction bar.


Solution

a. Image

Image Step 1. Divide 120 by 3.

Image

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

Image

b. Image

Image Step 1. Divide 120 by 3.

Image

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

Image

c. Image

Image Step 1. Divide 120 by 3.

Image

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

Image

d. Image

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

Image

e. Image

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

Image

To be successful in algebra, 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.

Image Exercise 2

For 1–3, simplify.

image

image

image

For 4 and 5, state in words.

image

image

For 6–20, compute as indicated.

image

image

Image

Image

image

Image

12. Image

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14. Image

15. Image

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17. Image

18. Image

19. Image

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