 Home
 INTERMEDIATE ALGEBRA
 Course Syllabus for Algebra I
 MidPlains Community College
 FRACTION OF A WHOLE NUMBER
 Systems of Linear Equations
 MATH FIELD DAY
 Course Outline for Finite Mathematics
 Calculus
 Algebra Final Examination
 Math 310 Exam #2
 Review of Trigonometric Functions
 Math 118 Practice test
 Precalculus Review
 Section 12
 Literal Equations
 Calculus Term Definitions
 Math 327A Exercise 2
 Public Key Algorithms II
 Maximizing Triangle Area
 Precalculus I Review for Midterm
 REVIEW OF A FIRST COURSE IN LINEAR ALGEBRA
 Math 6310 Homework 5
 Some Proofs of the Existence of Irrational Numbers
 ALGEBRAIC PROPERTIES OF MATRIX OPERATIONS
 Math 142  Chapter 2 Lecture Notes
 Math 112 syllabus
 Math 371 Problem Set
 Complex Numbers,Complex Functions and Contour Integrals
 APPLICATIONS OF LINEAR EQUATIONS
 Week 4 Math
 Fractions
 Investigating Liner Equations Using Graphing Calculator
 MATH 23 FINAL EXAM REVIEW
 Algebra 1
 PYTHAGOREAN THEOREM AND DISTANCE FORMULA
 Georgia Performance Standards Framework for Mathematics  Grade 6
 Intermediate Algebra
 Introduction to Fractions
 FACTORINGS OF QUADRATIC FUNCTIONS
 Elementary Algebra Syllabus
 Description of Mathematics
 Integration Review Solutions
 College Algebra  Applications
 A Tip Sheet on GREATEST COMMON FACTOR
 Syllabus for Elementary Algebra
 College Algebra II and Analytic Geometry
 Functions
 BASIC MATHEMATICS
 Quadratic Equations
 Language Arts, Math, Science, Social Studies, Char
 Fractions and Decimals
 ON SOLUTIONS OF LINEAR EQUATIONS
 Math 35 Practice Final
 Solving Equations
 Introduction to Symbolic Computation
 Course Syllabus for Math 935
 Fractions
 Fabulous Fractions
 Archimedean Property and Distribution of Q in R
 Algebra for Calculus
 Math112 Practice Test #2
 College Algebra and Trigonometry
 ALGEBRA 1A TASKS
 Description of Mathematics
 Simplifying Expressions
 Imaginary and Complex Numbers
 Building and Teaching a Math Enhancement
 Math Problems
 Algebra of Matrices Systems of Linear Equations
 Survey of Algebra
 Approximation of irrational numbers
 More about Quadratic Functions
 Long Division
 Algebraic Properties of Matrix Operation
 MATH 101 Intermediate Algebra
 Rational Number Project
 Departmental Syllabus for Finite Mathematics
 WRITTEN HOMEWORK ASSIGNMENT
 Description of Mathematics
 Rationalize Denominators
 Math Proficiency Placement Exam
 linear Equations
 Description of Mathematics & Statistics
 Systems of Linear Equations
 Algebraic Thinking
 Study Sheets  Decimals
 An Overview of Babylonian Mathematics
 Mathematics 115  College Algebra
 Complex Numbers,Complex Functions and Contour Integrals
 Growing Circles
 Algebra II Course Curriculum
 The Natural Logarithmic Function: Integration
 Rational Expressions
 QUANTITATIVE METHODS
 Basic Facts about Rational Funct
 Statistics
 MAT 1033 FINAL WORKSHOP REVIEW
 Measurements Significant figures
 PreCalculus 1
 Compositions and Inverses of Functions
Archimedean Property and Distribution of Q in R
1.2 Archimedean Property and Distribution of Q in R
Definition: We say the set S is dense in R if for
every a < b the intersection
S ∩ (a, b) ≠ Ø. (In other words, there is an element of S between every
two elements of
R.)
• The main goal of Section 1.2 is to show that the rationals are dense in the
reals. To achieve
this goal, the book first proves The Archimedean Property.
Theorem 1.5 The Archimedean Property: For
any positive number c there is a natural number n in N such that n > c. (Equivalently, for any positive number ε, there is a natural number n such that 1/n < ε .) 
• Proof: That the two versions are equivalent is as simple
as noting that when ε = 1/c then
the same n works for both. However, we must now prove one of the two versions of
the claim.
Note that the first version is the same as saying N is not bounded. We will
assume that it has
an upperbound and reach a contradiction. If it is bounded above, then the
completeness
axiom tells us that b = sup is a number in R with the property that n < b
for every
natural number n. But then, since n + 1 is also a natural number, we know n + 1
< b.
This is the same as n < b − 1 for every n, which means that b − 1 is an
upperbound for
N, which contradicts the minimality of b.
• The next thing the book proves is a fact that may seem completely obvious to
us: “Theorem
1.8 For any number c, there is exactly one integer k in the interval [c, c +
1).” Note that it
takes about two pages to prove this result! Merely for the sake of time, we will
accept this
fact here without discussing the proof, but you should look at the proof if only
to see that it
can be proved using the axioms explicitly.
• Now, we will use the previous two results to prove the denseness of the
rationals:
Theorem 1.9: If a and b are real numbers such that a < b then Q ∩ (a, b) ≠ Ø. 
• Proof: We need to find m, n ∈ Z such that
. We use Theorem 1.5 to get n
with . By theorem 1.8 there is an integer m
in [nb − 1, nb). Thus:
nb − 1 ≤ m <nb
b − 1/n ≤m/n < b (dividing by n)
a <m/n< b because a = b − (b − a) < b − 1/n.
• Similarly, we can show that the irrationals are dense in
R. This takes two steps...can we do
them together?
Show that the product of an irrational
number and a rational number is irrational.
Show that there is an irrational number in
(a, b) by using the fact that there is a rational
number in
1.3 Inequalities and Identities
• This section contains a number of formulas that will
prove useful. Most important for us will
be:
Theorem 1.11 Triangle Inequality:
where

• Proof: Since c ≤ d is equivalent to −d ≤ c ≤ d,
we can add together the formulas
−a ≤ a ≤ a and −b ≤ b ≤ b to get this.
• Why is it called the “triangle” inequality?
• Other formulas and definitions here should be equally familiar to you from
other classes, such
as:
and
Homework: 1.2 # 1 / 1.3 # 7 