Notesale: Turn your study into money

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CHAPTER 00
ALGEBRA REFRESHER
0
...
For the most part, your prior exposure to this material is assumed
...
Therefore, we urge you to devote whatever time necessary to these
sections in which you need review
...
2

AXIOMS OF THE REAL NUMBERS
Certain fundamental principles, which we assume to be true, govern the system of
real numbers
...

Axioms of Equality
Let a, b and c be real numbers
...

a = a (reflexive)
2
...

If a = b and b = c, then a = c (transitive)
4
...

If a = b and c = d, then a + c = b + d and ac = bd
(addition and multiplication)
Although the axioms of equality may appear to be rather trivial, they are quite
powerful and vital
...

Axioms of Addition
If a,b and c are real numbers then,
1
...

a + b = b + a (commutative)
3
...

There exists a real number 0, called “zero”, such that a + 0 = a for each
number a (additive identity)
5
...
(additive inverse)
Axioms of Multiplication
If a, b and c are real numbers, then
1
...

ab = ba (commutative)
3
...

5
...
1 = a for each number a
...
a-1 = 1 (multiplicative inverse)

Division:
If a and b are real numbers and b ≠ 0, then a divide by b, written a | b ,
a
1
is defined by
 a(b 1 )  a
...

If a, b and c are real numbers, then
a(b+c) = ab + ac and (b+c)a = ba + ca (distributive)
Note that by the definitions of division and the distributive laws, we have
ab
1
1
1
a b
 (a  b)
 a
...



c
c
c
c
c c
This important result does not mean that
a
a a


bc
b c
This error is very common
...
3

OPERATION WITH SIGNED NUMBERS
Listed below are various properties of signed numbers which you should study
thoroughly and make sure that you understand them
...

All denominators in the following are different from zero:
1
...


a – (-b) = a + b

3
...
a

4
...


–(a + b) = - a – b

6
...


– (-a) =

8
...
0

9
...
0 = 0
= -(ab) = a(-b)
2

10
...


a
b

12
...


a
b

14
...


a
a

 1

when a  0

b
a   b , a  0
a

16
...


a
bc

18
...


b c

1 a

...


a c

b d



ad  bc
bd

21
...


a b

c c



ab
c

23
...

b d



ac
bd

19
...
4

24
...


a
b
c

26
...

b c





ad
bc

EXPONENTS AND RADICALS
The expression
x
...
x
is abbreviated x3 and in general, for a positive integer n, xn represents the product
of n x’s
...

x n  x
...
x
...
x

n factors
1
xn

1
x
...
x
...


x n

3
...
00 is not defined
...

The principal nth root of x is that nth root of x which is positive if x is positive,
and is negative if x is negative and n is odd
...

We denote the principal nth root of x by the symbol
and n is the index, and
is the radical sign
...


xm
...


x0

=

1 if x  0

3
...


1
x n

 xn

4

5
...


xm
xm

 1 if

x0

7
...


(xy)n

= xn
...


x
 
y

1
n m

x

xmn

n

xn
yn



1

11
...
n y

n

x

12
...


x

15
...




xn

10
...
5

n



 x



xm

m

n

 x

Note the following pitfalls:
n
(1)
ab  n a
(2)

xy

ab



a b



2

n

a



n

b



n

b

ab

ORDER PROPERTIES
Let a,b and c be real numbers
...

Either a < b, b < a or a = b (trichotomy)
2
...

3
...


5

4
...


0
...

If a < b and c < 0, then ac > bc
...

Suppose that a < b
...
7

ABSOLUTE VALUE
Definition:
 a; if a  0
 
 a; if a  0
Other characterization:
a

a 

a2

Geometric interpretation:



(a) 2

|a|
- distance between 0 and a
|a - c| - distance between a and c

Properties of absolute values:
1
...


|-a| = |a|

3
...


a
a

b
b
|a - b| = |b - a|

6
...


|a| - |b| ≤ |a - b| ≤ |a| + |b|

8
...


Note the pitfall:

a 2 is not necessarily equal to a, but

7

a 2 = |a|



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