Download a -a additive inverse and

MATH 20053
CHAPTER 5 EXAMPLES
Sections 5.1 & 5.3
DEFINITION OF ADDITIVE INVERSE: For every number a there exists a unique
number, -a, the additive inverse of a, such that: a + (-a) = 0 and –a + a = 0.
Ex. A) Use the definition of additive inverse to find the additive inverse of –10.
Ex. B) Use the definition of additive inverse to find the additive inverse of zero.
DEFINITION OF INTEGERS: The set of integers is the set formed by the union of
the whole numbers and the additive inverses of the natural numbers.
Z = { . . . , -3, -2, -1, 0, 1, 2, 3, . . .}
Ex. C) Demonstrate that: –2 + (–4) = –6.
PROPERTIES OF INTEGERS
1. Closure for addition and subtraction.
2. Closure for multiplication.
3. Commutative for addition.
4. Commutative for multiplication.
5. Associative for addition.
6. Associative for multiplication.
7. Distributive property of multiplication over addition.
8. Additive identity.
9. Multiplicative identity.
10. Additive inverse.
DEFINITION OF ABSOLUTE VALUE: The absolute value of an integer is the
distance of that integer from the origin. NOTATION: | a |
MATH 20053
CHAPTER 5 EXAMPLES
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Ex. D) Find the values of:
(a) |5|
(b) |–11|
(c) |0|
(d) |x|
ANOTHER LOOK AT ADDITION:
1. To add two numbers with the same sign, add their absolute values and give
the sum their common sign.
2. To add two numbers with different signs, find the difference of the absolute
values and give that difference the sign of the number with the larger absolute
value.
SUBTRACTION OF INTEGERS: For any two integers a and b, a – b is the sum of a
and the additive inverse of b. a – b = a + (–b)
Ex. E) Find a way to demonstrate that: –5 – (–2) = -3.
Ex. F) One week, the daily temperatures in degrees Celsius for a certain town were:
–18, –13, 5, 2, –10, –8, and –7. (a) What was the average temperature for the
week? (b) What was the average change in temperature during the week?
Practice Problems for Section 5.1
1. A submarine has a depth of –4300 feet, and dives down 1400 feet. What is its
new depth?
2. Sarah had a bank account. (a) She first deposited $100. The first week, she
wrote checks for $15, $30, and $12. Then she deposited $50. How much is
currently in her account? (b) The next week, Sarah’s bank assessed a $15 penalty
because her balance was under $100. She deposited $40 and wrote a check for
$17. What is her current balance? (c) During week 3, Sarah protested the penalty,
because her contract with the bank stated that there was no minimum balance. She
next wrote checks for $19, $32, and $14, and deposited $75. At the end of the week,
the bank reversed the $15 penalty. What is Sarah’s bank balance at the end of the
third week ?
MATH 20053
CHAPTER 5 EXAMPLES
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Sections 5.2 & 5.3
Ex. G) Complete the table using repeated subtraction:
5·3
15
4·3
12
3·3
9
2·3
6
1·3
3
0·3
0
-
1·3
2·3
3·3
4·3
5·3
Ex. H) Complete the table:
-
-
-
-
5· 3
4· 3
-
-
-
-
-
2· 3
1· 3
-
0· 3
-
12
-
3· 3
-
15
9
6
3
0
-
1· 3
-
2· 3
-
3· 3
-
4· 3
-
5· 3
MULTIPLICATION OF INTEGERS: The product of two integers, where a and b
are whole numbers:
1.
–a × (–b) = a × b
2.
–a × b = b × (–a) = – (a × b)
MATH 20053
CHAPTER 5 EXAMPLES
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Ex. I) In a lab, the temperature of various chemical reactions was changing by a
fixed number of degrees per minute. Write an equation that describes each of the
following: (a) The temperature at 8:00 A.M. was –5°C. If it increases by d degrees
per minute, what will the temperature be m minutes later? (b) The temperature at
8:00 A.M. was 0°C. If it dropped d degrees each minute, what was the temperature
m minutes before?
DIVISION OF INTEGERS: If x and y are integers and y ≠ 0, then x ÷ y = z if and only
if z · y = x.
Ex. J) Simplify the following expressions:
2
3
2
3
(a) -x · x
(b) (-x) · x
("x)3
("x)4
("x)3
#
("x)2
(c)
!
(d)
x
"x4
!
Practice Problems
for Section 5.2
1. A weather station kept the following records of the low temperature each day. On
Monday the temperature was below zero, and on Tuesday it was 8°F colder.
Wednesday the reading was twice as low as for Tuesday. Thursday the temperature
was 4°F higher than on Wednesday. On Friday the temperature warmed up to
–14°F, which was half of Thursday’s temperature. What was the recorded
temperature on Monday?
2. If the current temperature is 9°F, what was it 2 hours ago if it has been
decreasing 6°F each hour?
3. If the temperature is now 12°F, what was it 6 hours ago if it has been increasing
4°F each hour?
MATH 20053
CHAPTER 5 EXAMPLES
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