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In This Chapter

Using the number line

Getting the numbers in order

Operating on signed numbers: adding, subtracting, multiplying, and dividing

Numbers have many characteristics: They can be big, little, even, odd, whole, fraction, positive, negative, and sometimes cold and indifferent. (I’m kidding about that last one.) Chapter 1 describes numbers’ different names and categories. But this chapter concentrates mainly on the positive and negative characteristics of numbers and how a number’s sign reacts to different manipulations. This chapter tells you how to add, subtract, multiply, and divide signed numbers, no matter whether all the numbers are all the same sign or a combination of positive and negative.

Assigning Numbers Their Place

Positive numbers are greater than 0. They’re on the opposite side of 0 from the negative numbers. If you were to arrange a tug-of-war between positive and negative numbers, the positive numbers would line up on the right side of 0. Negative numbers get smaller and smaller, the farther they are from 0. This situation can get confusing because you may think that –400 is bigger than –12. But just think of –400°F and –12°F. Neither is anything pleasant to think about, but –400°F is definitely less pleasant – colder, lower, smaller.

Remember: When comparing negative numbers, the number closer to 0 is the bigger or greater number. You may think that identifying that 16 is bigger than 10 is an easy concept. But what about –1.6 and –1.04? Which of these numbers is bigger?

Remember: The easiest way to compare numbers and to tell which is bigger or has a greater value is to find each number’s position on the number line. The number line goes from negatives on the left to positives on the right (see Figure 2-1). Whichever number is farther to the right has the greater value, meaning it’s bigger.

© John Wiley & Sons, Inc.

Figure 2-1: A number line.

Examples

Q. Using the number line in Figure 2-1, determine which is larger, –16 or –10.

A. The number –10 is to the right of –16, so it’s the bigger of the two numbers.

Q. Which is larger, –1.6 or –1.04?

A. The number –1.04 is to the right of –1.6, so it’s larger. A nice way to compare decimals is to write them with the same number of decimal places. So rewrite –1.6 as –1.60; it’s easier to compare to –1.04 in this format.

Comparing Positives and Negatives with Symbols

Although my mom always told me not to compare myself to other people, comparing numbers to other numbers is often useful. And, when you compare numbers, the greater-than sign (>) and less-than sign (<) come in handy, which is why I use them in Table 2-1, where I put some positive- and negative-signed numbers in perspective.

Table 2-1 Comparing Positive and Negative Numbers

Two other signs related to the greater-than and less-than signs are the greater-than-or-equal-to sign (≥) and the less-than-or-equal-to sign (≤).

So, putting the numbers 6, –2, –18, 3, 16, and –11 in order from smallest to biggest gives you: –18, –11, –2, 3, 6, and 16, which are shown as dots on a number line in Figure 2-2.

Figure 2-2: Positive and negative numbers on a number line.

Zeroing in on zero

But what about 0? I keep comparing numbers to see how far they are from 0. Is 0 positive or negative? The answer is that it’s neither. Zero has the unique distinction of being neither positive nor negative. Zero separates the positive numbers from the negative ones – what a job!

Practice Questions

1. Which is larger, –2 or –8?

2. Which has the greater value, –13 or 2?

3. Which is bigger, –0.003 or –0.03?

4. Which is larger, or

Practice Answers

1. -2. The following number line shows that the number –2 is to the right of –8. So –2 is bigger than –8. This is written –2 > –8.

2. 2. The number 2 is to the right of –13. So 2 has a greater value than –13. This is written 2 > –13.

3. – 0.003. The following number line shows that the number –0.003 is to the right of –0.03, which means –0.003 is bigger than –0.03. You can also rewrite –0.03 as –0.030 for easier comparison. The original statement is written –0.003 > –0.03.

1. . The number , and is to the left of on the following number line. So is larger than .

Going In for Operations

Operations in algebra are nothing like operations in hospitals. Well, you get to dissect things in both, but dissecting numbers is a whole lot easier (and a lot less messy) than dissecting things in a hospital.

Algebra is just a way of generalizing arithmetic, so the operations and rules used in arithmetic work the same for algebra. Some new operations do crop up in algebra, though, just to make things more interesting than adding, subtracting, multiplying, and dividing. I introduce three of those new operations after explaining the difference between a binary operation and a nonbinary operation.

Breaking into binary operations

Bi means two. A bicycle has two wheels. A bigamist has two spouses. A binary operation involves two numbers. Addition, subtraction, multiplication, and division are all binary operations because you need two numbers to perform them. You can add 3 + 4, but you can’t add 3 + if there’s nothing after the plus sign. You need another number.

Introducing nonbinary operations

A nonbinary operation needs just one number to accomplish what it does. A nonbinary operation performs a task and spits out the answer. Square roots are nonbinary operations. You find by performing this operation on just one number (see Chapter 6 for more on square roots). In the following sections, I show you three nonbinary operations.

Getting it absolutely right with absolute value

One of the most frequently used nonbinary operations is the one that finds the absolute value of a number – its value without a sign. The absolute value tells you how far a number is from 0. It doesn’t pay any attention to whether the number is less than or greater than 0; it just determines how far it is from 0.

Remember: The symbol for absolute value is two vertical bars: . The absolute value of a, where a represents any real number – positive, negative, or zero – is

✓ , where a ≥ 0.

✓ , where a < 0 (negative), and –a is positive.

Here are some examples of the absolute-value operation:

✓

✓

✓

✓

Basically, the absolute-value operation gives you an undirected distance – the distance from 0 without regard to direction.

Getting the facts straight with factorial

The factorial operation looks like someone took you by surprise. You indicate that you want to perform the operation by putting an exclamation point after a number. If you want 6 factorial, you write 6!. Okay, I’ve given you the symbol, but you need to know what to do with it.

Remember: To find the value of n!, you multiply that number by every positive integer smaller than n.

Here are some examples of the factorial operation:

✓ 3! = 3 · 2 · 1 = 6

✓ 6! = 6 · 5 · 4 · 3 · 2 · 1 = 720

✓ 7! = 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5,040

The value of 0! is 1. This result doesn’t really fit the rule for computing the factorial, but the mathematicians who first described the factorial operation designated that 0! is equal to 1 so that it worked with their formulas involving permutations, combinations, and probability.

Getting the most for your math with the greatest integer

You may have never used the greatest integer function before, but you’ve certainly been its victim. Utility and phone companies and sales tax schedules use this function to get rid of fractional values. Do the fractions get dropped off? Why, of course not. The amount is rounded up to the next greatest integer.

Remember: The greatest integer function takes any real number that isn’t an integer and changes it to the greatest integer it exceeds. If the number is already an integer, then it stays the same.

Here are some examples of the greatest integer function at work:

✓

✓

✓

You may have done a double-take for the result of using the function on –3.87. Just picture the number line. The number –3.87 is to the right of –4, so the greatest integer not exceeding –3.87 is –4. In fact, a good way to compute the greatest integer is to picture the value’s position on the number line and slide back to the closest integer to the left – if the value isn’t already an integer.

Practice Questions

1.

2.

3.

4.

Practice Answers

1. 8. 8 > 0.

2. 6. –6 < 0 and 6 is the opposite of –6.

3. 3. 3 is the largest integer smaller than 3.25.

4. – 4. –4 is smaller than –3.25, is to the left of –3.25, and is the largest integer that’s smaller than –3.25.

Adding Signed Numbers

If you’re on an elevator in a building with four floors above the ground floor and five floors below ground level, you can have a grand time riding the elevator all day, pushing buttons, and actually “operating” with signed numbers. If you want to go up five floors from the third sub-basement, you end up on the second floor above ground level.

You’re probably too young to remember this, but people actually used to get paid to ride elevators and push buttons all day. I wonder if these people had to understand algebra first…

Adding like to like: Same-signed numbers

When your first-grade teacher taught you that 1 + 1 = 2, she probably didn’t tell you that this was just one part of the whole big addition story. She didn’t mention that adding one positive number to another positive number is really a special case. If she had told you this big-story stuff – that you can add positive and negative numbers together or add any combination of positive and negative numbers together – you might have packed up your little school bag and sack lunch and left the room right then and there.

Adding positive numbers to positive numbers is just a small part of the whole addition story, but it was enough to get you started at that time. This section gives you the big story – all the information you need to add numbers of any sign. The first thing to consider in adding signed numbers is to start with the easiest situation – when the numbers have the same sign. Look at what happens:

✓ You have three CDs and your friend gives you four new CDs:

(+3) + (+4) = +7

You now have seven CDs.

✓ You owed Jon $8 and had to borrow $2 more from him:

(–8) + (–2) = –10

Now you’re $10 in debt.

Tip: There’s a nice S rule for addition of positives to positives and negatives to negatives. See if you can say it quickly three times in a row: When the signs are the same, you find the sum, and the sign of the sum is the same as the signs. This rule holds when a and b represent any two real numbers:

I wish I had something as alliterative for all the rules, but this is math, not poetry!

Say you’re adding –3 and –2. The signs are the same; so you find the sum of 3 and 2, which is 5. The sign of this sum is the same as the signs of –3 and –2, so the sum is also a negative.

Here are some examples of finding the sums of same-signed numbers:

✓ (+8) + (+11) = +19: The signs are all positive.

✓ (–14) + (–100) = –114: The sign of the sum is the same as the signs.

✓ (+4) + (+7) + (+2) = +13: Because all the numbers are positive, add them and make the sum positive, too.

✓ (–5) + (–2) + (–3) + (–1) = –11: This time all the numbers are negative, so add them and give the sum a minus sign.

Adding same-signed numbers is a snap! (A little more alliteration for you.)

Adding different signs

Can a relationship between a Leo and a Gemini ever add up to anything? I don’t know the answer to that question, but I do know that numbers with different signs add up very nicely. You just have to know how to do the computation, and, in this section, I tell you.

Tip: When the signs of two numbers are different, forget the signs for a while and find the difference between the numbers. This is the difference between their absolute values (see the “Getting it absolutely right with absolute value” section, earlier in this chapter). The number farther from 0 determines the sign of the answer.

if the positive a is farther from 0.

if the negative b is farther from 0.

Look what happens when you add numbers with different signs:

✓ You had $20 in your wallet and spent $12 for your movie ticket:

(+20) + (–12) = +8

✓ After settling up, you have $8 left.

✓ I have $20, but it costs $32 to fill my car’s gas tank:

(+20) + (–32) = –12

I’ll have to borrow $12 to fill the tank.

Here’s how to solve the two situations above using the rules for adding signed numbers.

✓ (+20) + (–12) = +8: The difference between 20 and 12 is 8. Because 20 is farther from 0 than 12, and 20 is positive, the answer is +8.

✓ (+20) + (–32) = –12: The difference between 20 and 32 is 12. Because 32 is farther from 0 than 20 and is a negative number, the answer is –12.

Here are some more examples of finding the sums of numbers with different signs:

✓ (+6) + (–7) = –1: The difference between 6 and 7 is 1. Seven is farther from 0 than 6 is, and 7 is negative, so the answer is –1.

✓ (–6) + (+7) = +1: This time the 7 is positive. It’s still farther from 0 than 6 is. The answer this time is +1.

✓ (–4) + (+3) + (+7) + (–5) = +1: If you take these in order from left to right (although you can add in any order you like), you add the first two together to get –1. Add –1 to the next number to get +6. Then add +6 to the last number to get +1.

Examples

Q. (–6) + (–4) = –(6 + 4) =

A. The signs are the same, so you find the sum and apply the common sign. The answer is –10.

Q. (+8) + (–15) = –(15 – 8) =

A. The signs are different, so you find the difference and use the sign of the number with the larger absolute value. The answer is –7.

Practice Questions

1. 4 + (–3) =

2. 5 + (–11) =

3. (–18) + (–5) =

4. 47 + (–33) =

5. (–3) + 5 + (–2) =

6. (–4) + (–6) + (–10) =

7. 5 + (–18) + (10) =

8. (–4) + 4 + (–5) + 5 + (–6) =

Practice Answers

1. 1. 4 is the greater absolute value.

2. – 6. –11 has the greater absolute value.

3. – 23. Both of the numbers have negative signs; when the signs are the same, find the sum of their absolute values.

4. 14. 47 has the greater absolute value.

5. 0.

6. – 20.

7. – 3.

Or you may prefer to add the two numbers with the same sign first, like this:

You can do this because order and grouping (association) don’t matter in addition.

8. – 6.

Making a Difference with Signed Numbers

Subtracting signed numbers is really easy to do: You don’t! Instead of inventing a new set of rules for subtracting signed numbers, mathematicians determined that it’s easier to change the subtraction problems to addition problems and use the rules I explain in the previous section. Think of it as an original form of recycling.

Consider the method for subtracting signed numbers for a moment. Just change the subtraction problem into an addition problem? It doesn’t make much sense, does it? Everybody knows that you can’t just change an arithmetic operation and expect to get the same or right answer. You found out a long time ago that 10 – 4 isn’t the same as 10 + 4. You can’t just change the operation and expect it to come out correctly.

So, to make this work, you really change two things. (It almost seems to fly in the face of two wrongs don’t make a right, doesn’t it?)

Tip: When subtracting signed numbers, change the minus sign to a plus sign and change the number that the minus sign was in front of to its opposite. Then just add the numbers using the rules for adding signed numbers.

✓ (+a) – (+b) = (+a) + (–b)

✓ (+a) – (–b) = (+a) + (+b)

✓ (–a) – (+b) = (–a) + (–b)

✓ (–a) – (–b) = (–a) + (+b)

The following examples put the process of subtracting signed numbers into real-life terms:

✓ The submarine was 60 feet below the surface when the skipper shouted, “Dive!” It went down another 40 feet:

– 60 – (+40) = –60 + (–40) = –100

Change from subtraction to addition. Change the 40 to its opposite, –40. Then use the addition rule. The submarine is now 100 feet below the surface.

✓ Some kids are pretending that they’re on a reality-TV program and clinging to some footholds on a climbing wall. A team challenges the position of the opposing team’s player. “You were supposed to go down 3 feet, then up 8 feet, then down 4 feet. You shouldn’t be 1 foot higher than you started!” The referee decides to check by having the player go backward – do the opposite moves. Making the player do the opposite, or subtracting the moves:

– (–3) – (+8) – (–4) = +(+3) + (–8) + (+4) = –5 + (+4) = –1

The player ended up 1 foot lower than where he started, so he had moved correctly in the first place.

Here are some examples of subtracting signed numbers:

✓ –16 – 4 = –16 + (–4) = –20: The subtraction becomes addition, and the +4 becomes negative. Then, because you’re adding two signed numbers with the same sign, you find the sum and attach their common negative sign.

✓ –3 – (–5) = –3 + (+5) = 2: The subtraction becomes addition, and the –5 becomes positive. When adding numbers with opposite signs, you find their difference. The 2 is positive because the +5 is farther from 0.

✓ 9 – (–7) = 9 + (+7) = 16: The subtraction becomes addition, and the –7 becomes positive. When adding numbers with the same sign, you find their sum. The two numbers are now both positive, so the answer is positive.

Remember: To subtract two signed numbers:

Examples

Q. (–8) – (–5) =

A. Change the problem to (–8) + (+5) =. The answer is –3.

Q. 6 – (+11) =

A. Change the problem to 6 + (–11) =. The answer is –5.

Practice Questions

1. 5 – (–2) =

2. –6 – (–8) =

3. 4 – 87 =

4. 0 – (–15) =

5. 2.4 – (–6.8) =

6. –15 – (–11) =

Practice Answers

1. 7.

2. 2.

3. – 83.

4. 15.

5. 9.2.

6. – 4.

Multiplying Signed Numbers

When you multiply two or more numbers, you just multiply them without worrying about the sign of the answer until the end. Then to assign the sign, just count the number of negative signs in the problem. If the number of negative signs is an even number, the answer is positive. If the number of negative signs is odd, the answer is negative.

Remember: The product of two signed numbers:

The product of more than two signed numbers:

(+)(+)(+)(–)(–)(–)(–) has a positive answer because there are an even number of negative factors.

(+)(+)(+)(–)(–)(–) has a negative answer because there are an odd number of negative factors.

Examples

Q. (–2)(–3) =

A. There are two negative signs in the problem. The answer is +6.

Q. (–2)(+3)(–1)(+1)(–4) =

A. There are three negative signs in the problem. The answer is –24.

Practice Questions

1. (–6)(3) =

2. (14)(–1) =

3. (–6)(–3) =

4. (6)(–3)(4)(–2) =

5. (–1)(–1)(–1)(–1)(–1)(2) =

6. (–10)(2)(3)(1)(–1) =

Practice Answers

1. – 18. The multiplication problem has one negative, and 1 is an odd number.

2. – 14. The multiplication problem has one negative, and 1 is an odd number.

3. 18. The multiplication problem has two negatives, and 2 is an even number.

4. 144. The multiplication problem has two negatives.

5. – 2. The multiplication problem has five negatives.

6. 60. The multiplication problem has two negatives.

Dividing Signed Numbers

The rules for dividing signed numbers are exactly the same as those for multiplying signed numbers – as far as the sign goes (see “Multiplying Signed Numbers” earlier in this chapter.) The rules do differ though because you have to divide, of course.

Remember: When you divide signed numbers, just count the number of negative signs in the problem – in the numerator, in the denominator, and perhaps in front of the problem. If you have an even number of negative signs, the answer is positive. If you have an odd number of negative signs, the answer is negative.

Examples

Q.

A. There are two negative signs in the problem, which is even, so the answer is positive. The answer is +4.

Q.

A. There are three negative signs in the problem, which is odd, so the answer is negative. The answer is –9.

Practice Questions

1.

2.

3.

4.

5.

6.

Practice Answers

1. 2. The division problem has two negatives.

2. – 8. The division problem has one negative.

3. – 6. Three negatives result in a negative.

4. 30. The division problem has two negatives.

5. – 4. The division problem has five negatives.

6. – 1. The division problem has one negative.

Working with Nothing: Zero and Signed Numbers

What role does 0 play in the signed-number show? What does 0 do to the signs of the answers? Well, when you’re doing addition or subtraction, what 0 does depends on where it is in the problem. When you multiply or divide, 0 tends to just wipe out the numbers and leave you with nothing.

Here are some general guidelines about 0:

✓ Adding zero: 0 + a is just a. Zero doesn’t change the value of a. (This is also true for a + 0.)

✓ Subtracting zero: 0 – a = –a. Use the rule for subtracting signed numbers: Change the operation from subtraction to addition and change the sign of the second number, giving you 0 + (–a). But changing the order, a – 0 = a. It doesn’t change the value of a to subtract 0 from it.

✓ Multiplying by 0: a × 0 = 0. Twice nothing is nothing; three times nothing is nothing; multiply nothing and you get nothing: Likewise, 0 × a = 0.

✓ Dividing 0 by a number: 0 ÷ a = 0. Take you and your friends: If none of you has anything, dividing that nothing into shares just means that each share has nothing.

Remember: You can’t use 0 as a divisor. Numbers can’t be divided by 0; not even 0 can be divided by 0. The answers just don’t exist.

So, working with 0 isn’t too tricky. You follow normal addition and subtraction rules, and just keep in mind that multiplying and dividing with 0 (0 being divided) leaves you with nothing – literally.

Practice Questions

1. 4 + 0 =

2. 0 – 4 =

3. 4 × 0 =

4.

Practice Answers

1. 4. Adding 0 to a number doesn’t change the number.

2. – 4. Change the problem to 0 + (–4) and add.

3. 0. Multiplying by 0 always gives you 0 as a result.

4. 0. Dividing 0 by a nonzero number always gives you 0.