Download INTEGERS

Integers
INTEGERS
Negative numbers are used often in every day life. For example, in Oregon, temperatures
can reach 3 degrees below 0, or -3° F. Or, if your banking account is overdrafted, you
receive a statement showing a balance of $-12.50 (in debt). Another example is a loss in
value of a certain stock is shown as $-3.4 (down 3.40 per share).
Integers are the set of positive and negative whole numbers.
{…-5,-4,-3,-2,-1,0,1,2,3,4,5…}
Integers are often shown on a number line such as the one below. This helps many
students perform all the operations with integers, such as ordering, adding, subtracting,
multiplying, and dividing.
● COMPARING INTEGERS
Recall the meaning of the following symbols:
<
less than
>
greater than
A number line can help determine the order of two integers. Notice that the numbers get
larger the further right you go. When comparing two integers, the number on the right is
always larger than the number on the left.
Student Practice: Fill in the blank with either a < , >.
1.
2
4.
-2
7.
-50
9
0
-60
2.
-8
3
3.
-5
5.
-7
-3
6.
5
8.
0
9.
-1
4
-12
-5
-4
Integers
● ABSOLUTE VALUE
The absolute value of a number is it’s distance from 0 on the number line. Two
vertical lines around a number are used to denote absolute value a .
Sample Problem:
Solution:
Simplify.
b.  4
a. 6
c. 0
a. 6  6 since 6 is 6 units from 0 on the number line.
b.  4  4 since -4 is 4 units away from 0 on the number line.
c. 0  0 since 0 is 0 units away from itself.
Tip: Many students like to think of the absolute value as simply making any number POSITIVE always.
Student Practice: Simplify.
11.  6
10. 7
12.  3
13.
11
ADDITION OF INTEGERS
• ADDING INTEGERS USING A NUMBER LINE
To add two integers a + b using a number line,
1. Start at a.
2. Then, if b is positive, move to the right. If b
left.
3. The answer is where you end up.
Sample Problem:
Solution:
Add.
a. 4  (2)
is negative, move to the
c.  3  (4)
b.  5  8
a.
-7
-6 -5
-4 -3 -2
-1
0
1
2
3
4
5
6
7
-7
-6 -5
-4 -3 -2
-1
0
1
2
3
4
5
6
7
-7
-6 -5
-4 -3 -2
-1
0
1
2
3
4
5
6
7
b.
c.
Integers
Tip: It may help to think of positive numbers as a GAIN and negative numbers as a LOSS when adding
integers. When you add two positive integers or gains, you have a bigger gain so the result is still
positive. When you add two negative integers or losses, you have a bigger loss so the result remains
negative. When you add a positive integer and a negative integer, or a gain and a loss, your result is the
DIFFERENCE of these values and the sign will be that of which was larger, the gain or loss.
• ADDING INTEGERS WITHOUT A NUMBER LINE
To add to integers a + b , use the following rules:
• If the signs are the same (both positive or both negative), add the
absolute values and keep the common sign.
• If the signs are different (one positive and one negative), subtract the
absolute values keep the sign of the larger number.
Sample Problem:
Solution:
Add.
a.  11  (4)
a.  11  (4)  15
b.  10  3  7
c. 20  (5)  15
b.  10  3
c. 20  (5)
add and keep the sign.
subtract and keep the sign of the larger.
subtract and keep the sign of the larger.
Student Practice: Add.
14. –15 + 5
15. 25 + (–15)
16. –20 + (–12)
17. –4 + 4
18. 1 + (–4)
19. –2 + (–4)
20.  6  5
21.  1  (9)
22. 10  (20)
23. 8  12
24.  3  (15)
25. 9  (9)
Integers
• ADDING A SERIES OF INTEGERS
When adding a series of integers, you can either
• Add integers from left to right,
OR
• Add all positive numbers together and all the negative numbers together,
then add the results.
Student Practice: Add.
26.  15  21  (8)  (12)  3  8
27. 10  (4)  (20)  5  5
SUBTRACTION OF INTEGERS
Before we determine the rule for subtracting, consider the following examples.
b. 12  4 =
a. 10  5 =
10  (5) =
c. 2  10 =
12  (4) =
d.  3  5 =
2  (10) =
 3  (5) =
• SUBTRACTING INTEGERS
One way to subtract two integers is to add the opposite. In other words, change the
subtraction to addition and change the sign of the second integer. Then use the addition
rules.
If a and b are integers, then a  b  a  (b) .
Sample Problem: Subtract. a. 20  15
Solution:
a.
b.
c.
d.
b. 5  15
20  15  20  (15)  5
5  15  5  (15)  10
 5  9  5  (9)  14
 6  (10)  6  (10)  6  10  4
c.  5  9
d.  6  (10)
Integers
Tip: It may help to think of subtraction ALSO AS A LOSS, just like a negative. In this case, 3 – 6
means a gain of 3 but a loss of 6, so it’s -3. And, –5 – 10 means a loss of 5, then a loss of 10 more, so
it’s –15.
Student Practice: Subtract.
28. 5  20
29.  10  8
30. 8  7
31. 4  (3)
32.  12  (10)
33.  1  15
34. 7  12
35.  3  (9)
36. 12  3
37.  7  (7)
38.  10  10
39. 20  (20)
40.  4  24
41.
43.  10  5  8  3  20
 2  (20)
44. 4  (6)  12  17  2
42.  2  3
45.  6  4  20  2  24
Integers
MULTIPLYING INTEGERS
Recall that multiplication is simply repeated addition. In other words, 2 times 4 really
means add 2 four times.
24  2 2 2 2 8
In the same way, -2 times 4 really means add -2 four times.
 2  4  (2)  (2)  (2)  (2)  8
Additionally, 6 times -3 means add -3 six times which results in -18.
This suggests that the product of a negative integer and a positive integer is always
negative.
Now consider the following pattern:
 3  3  9
 3  2  6
 3  1  3
 3 0  0
 3  1 
 3  2 
This suggests that the product of two negative integers is always positive.
• MULTIPLYING INTEGERS
When multipling two integers a  b , remember this:
• If the signs are different, one positive and one negative, the result is
NEGATIVE, regardless which is bigger.
()()  ()
()()  ()
• If the signs are the same, either both positive or both negative, the result is
POSITIVE.
()()  ( )
Sample Problem:
Solution:
Multiply.
a.  3(9)  27
b.  4(3)  12
c.  1(100)  100
a.  3(9)
b.  4(3)
c.  1(100)
(-) (+) = (-)
(-) (-) = (+)
(-) (+) = (-)
+
Tip: Some students like to use the following triangle.
Cover any two and the result is always the third.
_
multiplication
division
_
Integers
Student Practice: Multiply.
1.  3  6
2.  5  (3)
3. (2)(10)
4. 0(8)
5. (1)(8)
6. 4(20)
7.  9(1)
8. 500(2)
9.  6(6)
10. (2)(3)(5)(2)
11. (4)(1)(3)(2)
EXPONENTS
If the base of an exponential expression is negative, then:
• If the exponent is even, the result is positive.
• If the exponent is odd, the result is negative.
Sample Problem: Evaluate:
Solution:
a. (8) 2
b. (2) 3
a. (8) 2  (8)(8)  64
b. (2) 3  (2)(2)(2)  8
c.  4 2  (4  4)  16
c.  4 2
Note: Only the 4 is squared.
Student Practice: Evaluate.
12. (5) 2
13. (3) 3
14. (2) 4
Integers
15.
 62
18.  (9) 2
16. (1) 88
17. (4) 3
19.  (2) 3
20. (10) 2
DIVIDING INTEGERS
• DIVIDING INTEGERS
a
, remember this:
b
• If the signs are different, one positive and one negative, the result is
NEGATIVE, regardless which is bigger.
When dividing two integers
( )
 ( )
()
()
 ( )
( )
• If the signs are the same, either both positive or both negative, the result is
POSITIVE.
( )
 ( )
( )
Sample Problem:
Solution:
Divide.
9
 3
3
12
 6
b.
2
 15
5
c.
3
a.
9
3
b.
12
2
c.
 15
3
a.
+
Tip: Some students like to use the following triangle.
Cover any two and the result is always the third.
_
multiplication
division
_
Integers
Student Practice: Divide.
21.  8  2
24.
 81
9
27.
8
8
22.  2  2
25.
23. 25  5
 100
10
26.
18
2
29.  12  2
28. 21  (7)
SPECIAL CASES
30.
0
3
31.
5
0
a
is undefined for any value of a
0
Now that we have seen ALL operations and rules with integers, let’s practice simplifying some
expressions that consist of mixed operations to help us remember the rules of signs.
Integers
ORDER OF OPERATIONS
Sample Problem:
 12  (3  5) 3  2  11
Simplify:
 12  (3  5) 3  2  11
Solution:
 12  ( 2) 3  2  11
 12  ( 8)  2  11
 12  ( 16 )  11
 12  16  11
4  11
15
Student Practice: Simplify each expression using the proper order of operations. Be careful
with your signs.
1.  2(3)  5
2.  8  (6)  3
3.  7  4(5)
4.  20  10(3)
5.  20  10(3)
6. 7(2)  (10)
7. (30  5)  5  10
8. 10  (3) 2
9.  14  8  (2)  10
11.  4 2  3  (1  3) 2
12.  35  33
10.
 40  (5)(2)
Integers
EVALUATING EXPRESSIONS
 3x 3  4 x  3
Sample Problem:
Evaluate
Solution:
 3x3  4 x  3
if x = -3
 3( 3) 3  4( 3)  3
 3( 27)  4( 3)  3
81  4( 3)  3
81  ( 12)  3
81  12  3
96
Student Practice: Evaluate each expression if x = -3, y = 4, and z = -2 .
13. x  y  z
14. x  y  z
15. x 2  2
16.  x 2  2
17. 3x 2  2 z
18. 2 z 3  y  5 x
Integers
JUST FOR FUN
Imagine if you were asked to find the sum of the first n whole numbers. For example, find the
sum of the first 3 whole numbers…. the first 4 whole numbers… the first 5 whole numbers…
1+2+3
1+2+3+4
1+2+3+4 +5
How about the first 50 whole numbers? Can you generalize a rule that will help us sum up the
first n whole numbers without actually having to add up the numbers? Could this somehow be
useful?
Problem 1: Imagine a room with 60 people. If each person shakes hands ONCE with every
other person in the room, how many handshakes would there be?
Problem 2: A volleyball league has just started at STC on Tuesday evenings. A team called Full
Throttle has won the last 3 consecutive volleyball tournaments. Ever heard of them? Your
instructor is the captain of that team. What does that have to do with this? NOTHING. I’m just
gloating. Now back to the problem. Suppose there are 16 teams in the league and that each team
must play each other ONE TIME. The team with the best record wins the league. How many
games will there be total?