3
ADDING AND SUBTRACTING
SIGNED NUMBERS
WE MUST GIVE AN ALGEBRAIC MEANING to "adding" a negative number:
8 + (−2).
Now, when we add a positive number, we get more. Therefore, when we "add" a negative number, we must get less. It means to subtract.
8 + (−2) = 8 − 2 = 6.
Algebraically, here is the rule:
a + (−b) = a − b
Note that we use parentheses, a + (−b), to separate the operation sign + from the algebraic sign − . It would be bad form to write a + −b.
We are about to learn how to add signed numbers. But first we must learn how to name the terms.
Naming terms
By applying this rule—
a + (−b) = a − b
—we can now name the terms of any sum.
Here is a sum of four terms:
1 + (−2) + 3 + (−4).
The terms are 1, −2, 3, and −4.
But according to the rule, we can remove the parentheses:
1 + (−2) + 3 + (−4) = 1 − 2 + 3 − 4.
We say that the sum on the right has the same four terms:
1, −2, 3, and −4.
In other words, we include the subtraction sign as part of the name of the term.
1 − 2 + 3 − 4
1 and 3 are the positive terms. −2 and −4 are the negative terms.
When a term appears with no sign in front—namely the first term, 1—we must understand it to be positive. 1 = +1.
Again, with positive numbers we typically do not write the algebraic sign + . (Lesson 2.)
Problem 1. Name each term.
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| a) | 3 + (−4) + 5 + (−6). 3, −4, 5, −6. | |||
| b) | 3 − 4 + 5 − 6. 3, −4, 5, −6. | |||
| c) | −2 − 5. −2, −5. | d) | −a − b + c − d. −a, −b, c, −d. | |
In algebra we speak of "adding," even though there are minus signs. With that understanding, we can now state the rules for "adding" terms.
1) If the terms have the same sign, add their absolute values, and keep that same sign.
| 2 + 3 = 5. | −2 + (−3) = −5. | −2 − 3 = −5. |
2) If the terms have opposite signs, subtract the smaller in absolute value from the larger, and keep the sign of the larger.
| 2 + (−3) = −1. | −2 + 3 = 1. |
Algebra, after all, imitates arithmetic, and it is easy to justify these rules by considering money coming in or going out. For example, if you borrow $10 and then pay back $4, we express that algebraically as
−10 + 4 = −6.
You now owe $6.
Or, if you lose $6 and then win $8,
−6 + 8 = 2.
You're now ahead $2.
Problem 2. You borrow $5 from Sandra, and then borrow another $10. Express that algebraically.
−5 − 10 = −15.
Note: Again, in algebra we say that we "add" terms, even when there are subtraction signs. And we call the terms themselves—and the answer—a "sum." In other words, we always speak of a sum of terms.
Problem 3. Add according to the rules for adding terms.
| a) | 6 + 2 = 8. | b) −6 + (−2) = −8. | |
| c) | −6 − 2 = −8. | d) −4 − 1 = −5. | |
| e) | −6 + 2 = −4. | f) 6 + (−2) = 4. | |
| g) | 2 + (−6) = −4. | h) −2 + 6 = 4. | |
Problem 4. Add these terms.
| a) | 8 + (−3) = 5 | b) | −8 + 3 = −5 | c) | −8 + (−3) = −11 | ||
| d) | −8 − 3 = −11 | e) | 2 + (− 5) = −3 | f) | −2 + (− 5) = −7 | ||
| g) | −2 − 5 = −7 | h) | 8 + (− 11) = −3 | i) | −7 + (− 6) = −13 | ||
| j) | 9 + (− 2) = 7 | k) | −9 − 2 = −11 | l) | −9 + (− 2) = −11 | ||
| m) | 6 + (− 10) = −4 | n) | −6 − 10 = −16 | o) | −6 + 10 = 4 | ||
| p) | −9 + 9 = 0 | q) | −9 − 9 = −18 | r) | 9 + 9 = 18 | ||
Here is a fundamental rule for 0:
a + 0 = 0 + a = a
Adding 0 to any term does not change it.
Problem 5.
| a) | 0 + 6 = 6 | b) | 0 + (−6) = −6 | |
| c) | 0 − 6 = −6 | d) | −6 + 0 = −6 | |
Subtracting a negative number
What sense can we make of
2 − (−5) ?
"2 subtract negative 5."
Let us name the terms. The first term is 2. The second term is −(−5)—for we include the minus sign as part of the name of the term. But
−(−5) = +5.
Lesson 2. Therefore,
| 2 − (−5) | = | 2 + 5 |
| = | 7. | |
Here is the rule:
a − (−b) = a + b
Any problem that looks like this—
a − (−b)
—rewrite so that it looks like this:
a + b.
That is the only form that the student should have to rewrite.
(Please don't cross out. Rewrite. If you cross out, you can't read the original problem.)
Note again that we use parentheses: a − (−b), to
separate the operation sign − from the algebraic sign − .
| Examples. | 10 − (−3) | = | 10 + 3 = 13. |
| −10 − (−3) | = | −10 + 3 = −7. | |
The first number a does not change. Look at the rule. Change only −(−3) to + 3.
Problem 6. Rewrite without parentheses and calculate.
| a) | 7 − (− 4) = 7 + 4 = 11 | b) | 1 − (− 9) = 1 + 9 = 10 | |
| c) | 8 − (− 5) = 8 + 5 = 13 | d) | −8 − (− 5) = −8 + 5 = −3 | |
| e) | −5 − (− 7) = −5 + 7 = 2 | f) | 2 − (− 10) = 2 + 10 = 12 | |
| g) | −9 − (− 8) = −9 + 8 = −1 | h) | −20 − (− 1) = −20 + 1 = −19 | |
| i) | 4 − (−4) = 4 + 4 = 8 | j) | −4 − (−4) = −4 + 4 = 0 | |
Problem 7. Review.
| a) | 8 + (− 2) = 6 | b) | 8 − (− 2) = 10 | |
| c) | −8 + (− 2) = −10 | d) | −8 − 2 = −10 | |
| e) | 12 − 20 = −8 | f) | −12 − 20 = −32 | |
| g) | −12 + (− 20) = −32 | h) | −12 − (− 20) = 8 | |
| i) | 6 + (− 10) = −4 | j) | −5 − 9 = −14 | |
| k) | −30 − (− 6) = −24 | l) | 4 − 28 = −24 | |
| m) | 0 − 9 = −9 | n) | 0 + 9 = 9 | |
| o) | 9 + (− 9) = 0 | p) | −1 − 9 = −10 | |
Problem 8. Evaluate −x when x = −4.
−x = −(−4) = 4.
Problem 9. Evaluate x − y when
a) x = 5, y = −2. 5 − (−2) = 5 + 2 = 7
b) x = −5, y = −2. −5 − (−2) = −5 + 2 = −3
Adding a series of terms
Consider the following series of terms:
1 − 3 + 5 − 6 + 9 − 2
We could, of course, add these in the order in which they appear:
"1 − 3 = −2. −2 + 5 = 3. 3 − 6 = −3." And so on.
Or, we could add the positive and negative terms separately:
| 1 − 3 + 5 − 6 + 9 − 2 | = | 15 − 11 |
| = | 4. | |
Again, the order of the terms does not matter. And that method is usually more skillful.
Problem 10. Add each series.
a) 2 − 3 + 4 − 5 = 2 + 4 − 3 − 5 = 6 − 8 = −2.
b) 8 − 10 − 4 + 12 − 5 = 8 + 12 − 10 − 4 − 5 = 20 − 19 = 1.
c) −3 + 5 − 6 − 4 + 8 = −13 + 13 = 0.
When numbers add up to 0, we may "cancel" them.
Example 1. 5 − 2 + 3 − 5
5 + (−5) = 0. Therefore, we may cancel -- that is, ignore -- them. We are left with −2 + 3 = 1.
Example 2. 8 − 10 + 5 − 3 + 2
8 − 10 = −2, which we may cancel with +2. We are left with
5 − 3 = 2.
Or, 8 + 2 = 10, which we could cancel with −10. The order of terms never matters
Problem 11. Add each series. Cancel if possible.
a) 2 − 6 + 4 − 2 + 3 + 5 − 4 = (2 − 2) + (4 − 4) − 6 + (3 + 5) = 2.
b) 12 − 3 − 7 + 10 − 5 − 12 = (12 − 12) − 3 − 7 + 10 − 5 = −5.
c) 7 − 17 + 2 − 4 + 15 + 2 = 5
d) −10 + 6 − 3 + 4 + 2 − 5 + 3 = −3
Problem 12. Rewrite without parentheses:
| a + (−b) | = | a − b |
| a − (−b) | = | a + b |
Example 3. Rewrite without parentheses, then calculate:
2 + (− 3) − (− 4) + 5 + (− 6).
Solution. We will remove the parentheses according to the previous problem.
| 2 + (− 3) − (− 4) + 5 + (− 6) | |
| = | 2 − 3 + 4 + 5 − 6. |
Now, 2 + 4 will cancel with −6. We are left with
−3 + 5 = 2.
Problem 13. Rewrite without parentheses, then calculate.
a) −1 − (− 2) + (− 3) − 4 + 5 = −1 + 2 − 3 − 4 + 5 = −1
b) 8 − (− 2) + (−3) − (− 4) − 7 = 4
c) −10 − (− 8) + (− 3) − 1 + (− 8) = −14
Problem 14. Make this
x + 5
look like this:
x − a.
| Solution. | x + 5 = x − (−5). |
We will see that the rules of algebra go both ways. Since
a − (−b) = a + b,
then
a + b = a − (−b).
Next Lesson: Multiplying and dividing signed numbers
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