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THE UNIVERSITY OF AKRON
Mathematics and Computer Science
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Lesson 1: Setting Up the Environment
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• Table of Contents
• Begin Lesson 1
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IamD S
N ⊆ Z ⊆ Q ⊆ R ⊆ C
a 3 a 4 = a 7 ( ab ) 10 = a 10 b 10
− ( ab − (3 ab − 4))=2 ab − 4
( ab ) 3 ( a − 1 + b − 1 )=( ab ) 2 ( a + b )
( a − b ) 3 = a 3 − 3 a 2 b +3 ab 2 − b 3
2 x 2 − 3 x − 2=(2 x + 1)( x − 2)
1
2 x +13=0 = ⇒ x = − 26
G= { ( x, y ) | y = f ( x ) }
f ( x )= mx + b
y = sin x
Copyright c 1995–2000 D. P. Story
Last Revision Date: 2/2/2000
 
Lesson 1: Setting Up the Environment
Table of Contents
1. Setting Up the Environment
1.1. The Real Number System, and friends
• The Natural Numbers • The Integers • The Rational Num-
bers • The Irrational Numbers • The Real Numbers
1.2. The Number Line, and relations
• Less than and Greater than
1.3. Distance and Absolute Value
• Absolute Value • Distance between two numbers • The
Midpoint between two Numbers
1.4. Interval Notation
1. Setting Up the Environment
In this lesson, we review some very basic ideas and terminology of
the so-called Real Number System . Do not take this first lesson
lightly, your knowledge of the real number system and its properties
is key to your understanding why certain algebraic manipulations
are permissible, and why others are not. (When you are manipulating
algebraic quantities, you are, in fact, manipulating numbers.)
Terminology is important in all scientific, technical and professional
fields, and mathematics is no different, it has a lot of it. When you
speak or write, you use words; the words you use must be under-
stood by the ones with whom you are trying to communicate. For
someone to understand your communication, the words must have a
universal meaning; therefore, it is necessary for you to use the correct
terminology to be able to effectively and e = ciently communicate with
others. Correct use of (mathematical) terminology is a sign that you
understand what you are saying or writing.
Section 1: Setting Up the Environment
1.1. The Real Number System, and friends
A pedestrian description of the real number system is that it is the set
of all decimal numbers . Here are a few examples of decimal numbers:
1 . 456445
Positive Number with Finite Decimal Expansion
− 3 . 54
Negative Number with Finite Decimal Expansion
1 . 49773487234 ...
Positive Number with Infinite Decimal Expansion
− 3 . 4 × 10 9
Negative Number in Scientific Notation
5 . 65445 E − 5
Positive Number in Scientific Notation
You have been working with these representations of numbers all
your life. You have worked long and hard to master the mechanics of
adding, subtracting, multiplying, and dividing these numbers. More
recently, the calculator makes many of the operations just mentioned
automatic and routine, but the calculator does not diminish the need
to still have a ‘pencil and paper’ understanding of these operations,
nor does it diminish the need to be able to read, interpret and convert
numbers.
Section 1: Setting Up the Environment
Decimal numbers are, in fact, the end product of a series of prelim-
inary definitions. As important as decimal numbers are, let’s leave
them fornow to discuss a series of definitions that lead ultimately to
the construction of the real number system.
• The Natural Numbers
The natural numbers are the numbers
1 , 2 , 3 , 4 , 5 , 6 ,..., 100 ,..., 1000 ,...
and so on ad infinitum . Mathematicians put braces around these num-
bers to create the set of all natural numbers :
N = { 1 , 2 , 3 , 4 , 5 , 6 ,...,n,...}.
(1)
The letter N is typically used to denote this set; i.e., N symbolically
represents the set of all natural numbers.

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