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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Discussion 1: Rational and Irrational Numbers}
\centerline{September 1, 2000}
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{\bf Background}
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The purpose of today's discussion is to start to think about 
what properties {\it number systems\/} should have
in order for us to be able to solve equations, analyze functions, 
and ultimately ``do calculus.''  You have used the set of {\it rational 
numbers} or fractions:
$${\bf Q} = \{m/n : m,n\ {\rm integers}, n \ne 0 \}$$
since grade school.  This collection of numbers is 
sufficient (for example) for solving simple linear equations 
of the form:
$$ax + b = c$$
with $a,b,c \in {\bf Q}$.  We can solve for 
$x$, and obtain a rational number as the result:
$$ax + b = c\ \hbox{ implies } x = {c\over a} + {(-b)\over a}\leqno(1)$$
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\noindent
{\bf Discussion Questions}
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\item{A)}  
\itemitem{1)} What algebraic properties of the integers and rational numbers
are we using to solve the equation in (1) and write the solution in this form? 
\itemitem{2)} Show carefully that if $a = m/n$, $b = p/q$, $c = r/s$,
with $m,n,p,q,r,s$ all integers with $n,p,s \ne 0$, then
the solution in (1) can also be written as a quotient of integers
with nonzero denominator.  Give a reason why every step in your
algebraic manipulations is valid, using known properties of fractions.
\itemitem{3)}  How does what you did in parts 1 and 2 
relate to the first five axioms for the real numbers on page 13 of 
the text?  (Note: an ``axiom'' is a property that is assumed to 
be true, without proof, as part of the description or definition
of a mathematical object.) 
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\item{B)}  In this and the next question, we will ``move up'' to 
quadratic equations.  Some quadratic equations with rational number 
coefficients can also be solved in the rational numbers.  For instance,
$x^2 - 5x + 6 = 0$ has solutions $x = 2$ and $x = 3$.
We want to consider a different equation, though:  $x^2 - 2 = 0$.
From algebra, you know the solutions $x = \pm \sqrt{2}$.  The question
is: Are these numbers elements of the set ${\bf Q}$?  
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\itemitem{1)}  You probably know a way to tell whether a number is 
rational by looking at its {\it decimal expansion}.  What is special
about the decimal expansion of a rational number?  (Note:  The
way to prove this fact is usually discussed in the Algebraic Structures class.)
\itemitem{2)}  But can you tell anything by looking at a finite part of the
decimal expansion $\sqrt{2} = 1.4142136\cdots$?  (Note: a finite part is 
all a human being could ever see!)
\itemitem{3)} Here's a different way to try to answer our question.
Suppose $\sqrt{2}$ is in ${\bf Q}$, and  
write $\sqrt{2} = m/n > 0$, with positive integers $m, n$.
We can assume that the fraction $m/n$ is in lowest terms (no common factors 
between $m$ and $n$).  Then
$$(m/n)^2 = 2,\ \hbox{ or }\ m^2 = 2n^2.$$
Can you find integers $m,n$ with no common factors 
satisfying this equation?  Why or why not?  (Think about the 
{\it factorizations\/} of $m$, $n$, then $m^2$ and $n^2$
into powers of prime numbers.)  
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\item{C)}  Here's another way to get approximations to $\sqrt{2}$.
Note that $\sqrt{2}$ solves the equation $x = 2/x$.
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\itemitem{1)} Explain why for all positive $x_0$, $\sqrt{2}$ lies
between $x_0$ and $2/x_0$.  (Sketching the graphs $y = x, y = 2/x$ 
on the same set of axes may be useful!)
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\item{} So, thinking of $x_0$ as a first approximation, 
the {\it average}
$$x_1 = (1/2)(x_0 + 2/x_0)$$
is probably a {\it better\/} approximation to $\sqrt{2}$ than $x_0$
was.  Similarly, $\sqrt{2}$ is between $x_1$ and $2/x_1$, so
$x_2 = (1/2)(x_1 + 2/x_1)$ is probably closer than
$x_1$ was.  We can repeat this as many times as we like to make
$$x_{n+1} = (1/2)(x_n + 2/x_n)\leqno(2)$$ 
for $n = 0,1,2,3,\ldots$.  
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\itemitem{2)} Using (1), construct
the $x_1, \ldots, x_5$ starting with $x_0 = 1$, then restart
from $x_0 = 3$ and find  $x_1, \ldots, x_5$. 
What seems to be happening in either case?
\itemitem{3)} Prove that all $x_n$ constructed with (1) will be rational
numbers if $x_0$ is a rational number.
\itemitem{4)}  If you assume that the $x_n$ are approaching some fixed number
$a$ as $n$ gets larger and larger, from (2), what number does $a$ have to 
be?
\itemitem{5)}  What properties of rational and real
numbers have you used in this 
question?
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{\bf Writeups:}  Group writeups due in class Wednesday, September 6. 
\bye
