3
ADDING AND SUBTRACTING
SIGNED NUMBERS
WE MUST GIVE A 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 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 -- 8 + (−2) -- to separate
the operation sign + from the algebraic sign − . It would
be very bad form to write 8 + −2.
Problem 1. Sandra had $100 in the bank, and she made a "deposit" of −$25. What is her balance now?
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Do the problem yourself first!
$100 + (−$25) = $75.
To "deposit" −$25 mean to subtract!
That is the way we speak in algebra.
Naming terms
Upon understanding that rule, 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 could remove the parentheses as follows:
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 minus sign as part of the name of the term.
Problem 2. Name each term.
| 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. | |
"Adding" terms
In algebra, we speak of "adding," even though there are minus signs. With that understanding, we can now state a rule 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
from the larger, and
keep the sign of the larger.
| 2 + (−3) = −1. | −2 + 3 = 1. |
Algebra, after all, is an imitation of 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 this 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 3. You borrow $5 from Sandra and then borrow $10 from Sarah. Express that algebraically.
−5 − 10 = −15.
Note: 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 4. 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 5. 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 | ||
Zero
Here is a fundamental rule for 0:
a + 0 = 0 + a = a
Adding 0 to any term does not change it.
Problem 6.
| 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."
We can quote the rule of Lesson 2:
−(−5) = +5
-- and that is how to deal with it
| 2 − (−5) | = | 2 + 5 |
| = | 7. | |
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.)
If we name the terms of a − (−b), they are a and −(−b).
But −(−b) is equal to +b. Hence the terms are a and +b,
and the rule follows.
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 7. 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 8. 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 9. Evaluate −x when x = −4.
−x = −(−4) = 4.
Problem 10. 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. | |
That method is usually more skillful.
Problem 11. Add each series.
a) 2 − 3 + 4 − 5 = 6 − 8 = −2
b) 8 − 10 − 4 + 12 − 5 = 20 − 19 = 1
c) −3 + 5 − 6 − 4 + 8 = −13 + 13 = 0
Canceling
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 12. Add each series. Cancel if possible.
a) 2 − 6 + 4 − 2 + 3 + 5 − 4 = 2
b) 12 − 3 − 7 + 10 − 5 − 12 = −5
c) 7 − 17 + 2 − 4 + 15 + 2 = 5
d) −10 + 6 − 3 + 4 + 2 − 5 + 3 = −3
Problem 13. 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 14. 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 15. Make this
x + 5
take this form:
x − a.
| Solution. | x + 5 = x − (−5). |
Next Lesson: Multiplying and dividing signed numbers
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