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A L G E B R A

3

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)  =  ab

Problem 1.   Sandra had \$100 in the bank, and she made a "deposit" of −\$25.  What is her balance now?

Do the problem yourself first!

\$100 + (−\$25) = \$75.
To deposit −\$25 mean to subtract.

Naming terms

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.

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

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 could express this algebraically as

−10 + 4 = −6

You now owe \$6.

Or, if you lose \$6 and then win \$8,

−6 + 8 = 2

Problem 3.   You borrow \$5 from Sandra and then borrow \$10 from Sarah.   Express this algebraically.

−5 − 10 = −15

 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

 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) ?

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.

This 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), which is equal to b. Hence the terms are a and b; and the rule follows.

 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  xy  when

a)   x = 5,  y = −2.    5 − (−2) = 5 + 2 = 7

b)   x = −5,  y = −2.    −5 − (−2) = −5 + 2 = −3

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 technique is usually more skillful.

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 will 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

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