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College Algebra—Winter 2005
Sections P.1&2
Section P.1--The Real Number System
Our real numbers consist of several classes of numbers that can be grouped as in the
following diagram. Notice that I have included all of the real numbers within the complex
numbers.
Natural numbers are those you learned before going to school: {1, 2, 3, 4, 5, 6, 7, 8, 9,
10, ...}
Numerals
1
2
3
4
5
6
7
8
9
10
English
one
two
three
four
five
six
seven
eight
nine
ten
Spanish
uno
dos
tres
cuatro
cinco
seis
siete
ocho
nueve
diez
German
eins
zwei
drei
vier
funf
sechs
sieben
acht
neun
zehn
French
un
deux
trois
quatre
cinq
six
sept
uit
nouf
dix
Chinese
yi
er
san
si
wu
liu
qi
ba
jiu
shi
Swahili
moja
mbili
tatu
nne
tano
sita
saba
nane
tisa
kumi
Language?
Whole numbers include all of the natural numbers and zero.
Integers contain all of the natural numbers, zero and all of their negatives.
Rational Numbers consist of fractions where the numerator and denominator are integers.
The denominator is never zero! The decimal expansion of a rational number either
1
 0. 5 or repeats like 13  0. 3333. . .
terminates like
2
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College Algebra—Winter 2005
Sections P.1&2
The set of Irrational Numbers is infinitely bigger than any of the above sets. Numbers
like 2 , 3 , and  are all irrational.
Complex numbers include all of the above plus those numbers that involve i where
i2  −1 . So 3 − 4i , log−2 , and sin−1 2 are all complex and not real.
All real numbers can be place on a number line. We usually write the number line with 0
somewhere in the middle and positive integers spaced evenly to the right and negative
integers spaced evenly to the left.
We use the symbols:
a  b if " a is less than b " when a lies to the left of b on the number line.
a  b if " a is greater than b " when a lies to the right of b on the number line.
Two numbers are called "opposites" if one number is the same as another except that they
have differing signs. I personally dislike the term opposite when used to describe
numbers and will not use it very often.
The "absolute value" of a number is its positive distance from the origin 0 .
|−5|  5 and |5|  5.
Rules for algebraic operations
We have attached names to various rules that we have made concerning multiplication
and addition. These properties become very important as you move on to upper division
algebra. I shall use them freely throughout the rest of the quarter. You may wish to refer
back to this first lesson as a review. Expect one or more these properties to be on the next
test!
Commutative Property states that the order of multiplying or adding is not important:
Example
Multiplication
Addition
23  32
23  32
Associative Property states how we multiply or add three numbers:
Multiplication
Addition
Example 2  3  4  2  3  4 2  3  4  2  3  4
Identity Property Talks about multiplication by one and addition by zero.
Multiplication
Addition
Example 1  3  3  1  3 0  3  3  0  3
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College Algebra—Winter 2005
Sections P.1&2
Zero Property of Multiplications Talks about multiplication by zero.
Multiplication
Addition
Example 0  3  3  0  0
Inverse Property Talks about multiplying to get an answer of one, or adding two
numbers to get an answer of zero.
Multiplication
Example
1
3
3  3
1
3
Addition
 1 −3  3  3  −3  0
Distributive Property States how we combine multiplication and addition in the same
problem.
Example: 2  3  4  2  3  2  4
If you wish to appear erudite, you may refer to these properties as the "field properties of
the real numbers."
Convention: Always do operations in the following order.
1. Perform operations inside grouping symbols first. These include parentheses (),
ab
brackets [], braces {}, and the fraction bar cd .
2. Perform exponential operations next
3. Then do multiplication and division from left to right
4. Finally do addition and subtraction from left to right.
Some people like to remember this using the mnemonic: "Please excuse my dear aunt
sally." Really we should write this in a table:
1
2
3
4
, ,  , fraction bar
a 2 exponents
, 
, −
Please
Excuse
My Dear
Aunt Sally
Interval Notation
We use the following notation very freely:
Closed interval: 1, 2  x|1 ≤ x ≤ 2
Open interval: 1, 2  x|1  x  2
An combinations of these, e.g. 1, 2  1  x ≤ 2 and 2,   x|x ≥ 2
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College Algebra—Winter 2005
Sections P.1&2
Set theory Notation
We will not place a great emphasis on set theory notation. However you should be
familiar with the notation in the text. We do use the union (  ) symbol frequently:
−, 2  3, 4 to represent all numbers less than two or between 3 and 4 inclusive.
Question 9: Given that A  −3, −2, −1, 0, 1, 2, 3 and B  −2, 0, 2, 4, 6 , find
AB
A  B  −3, −2, −1, 0, 1, 2, 3, 4, 6
Question 33: Write without absolute value notation: |3|  |−4|
3  4  12
Question 52: Graph the set: −4, 0 ∩ −2, 5
)
0
[
(
-4 -2
]
5
Since we are looking for the intersection, we just want the part of the graph between −2
and 0.
[
-2
)
0
Section P.2--Integer and Rational Exponents
Review the properties of exponents. You should be very familiar with the basic laws
from Intermediate Algebra or Algebra II.
b m  b n  b m n
bm
 b m −n
bn
b m  n  b m n
ab n  a n b n
n
a n
 ab n
b
n
n
n
n
n
n
Remember a  b ≠ a  b and a − b ≠ a − b
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College Algebra—Winter 2005
Sections P.1&2
From these you can quickly deduce that
b 0  1 as long as b ≠ 0
b −n  b1n
b
m
n

n
bm
Radicals are regarded as simplified if
1. the nth radical n x contains no perfect nth power of a number,
2. the radical does not contain a fraction, and
3. the radical does not occur in the denominator of a fraction.
Furthermore don't leave negative exponents in an answer.
2
Not! Questions 1 & 2: Evaluate −5 2 and −5
−5 2  − 25
−5 2  25
2
−4x 2 y 3
2xy 2
Question 29: Simplify
First distribute the exponents.
3
−4  2 x 4 y 6
2 3x 3y 6
16x 4 y 6
8x 3 y 6


Now do the arithmetic first with the numbers and then with each letter in turn.
2x
Question 76: Simplify 3 −250
Notice that 5 3  125 and 250  125  2 so we can break up the radicand by
3
−125  2
and the cube root of −125 is −5 so
3
−250  −5 3 2
4
3
3
Question 85: Simplify 4 32y  3y 108y
Notice that each term appears different so we can't just collect like terms. So simplify
each radical:
4 3 32y 4  3y 3 108y 
4  2y 3 4y  3y  3 3 4y 
8y 3 4y  9y 3 4y 
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College Algebra—Winter 2005
Sections P.1&2
17y 3 4y
6
2 5 2
Question 109: Simplify
Here we multiply by the conjugate of the denominator: 2 5 − 2
6

2 5 2
6
2 5 2
2 5 −2

2 5 −2
6 2 5 −2
20−4
12
5 −1
16
3



5 −1
4
Notice no radical in the denominator, no perfect squares, and no fraction under the
radical.
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