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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Discussion 1: Algebraic and Order Properties of Rational and Real Numbers}
\centerline{February 2, 2011}
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{\it 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.  
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{\it Discussion Questions}
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A)  Linear Equations. Consider a simple linear equation $ax + b = c$
with $a,b,c \in {\bf Q}$ or ${\bf R}$ more generally.  We can solve for 
$x$ by elementary algebra:
$$ax + b = c\ \hbox{ implies } x = {c\over a} + {(-b)\over a}\leqno(1)$$
Now of course the $a,b,c$ here are covered by the algebraic Axioms 1 - 5 on page
12 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{1)} Which of the algebraic Axioms for real numbers 
are used in solving the equation in (1) and writing the solution in this form? 
\item{2)} Show carefully that if $a = m/n$, $b = p/q$, $c = r/s$, are {\it rational
numbers} 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.
(See Theorem 1.3.1 on page 13.)  
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B)  The {\it absolute value} of a real number $x$ is defined
by 
$$|x| = \cases{x & if $x \ge 0$\cr  -x & if $x < 0$.\cr}$$
If $x,y$ are reals, then $|x - y|$ can also be interpreted as
the distance from $x$ to $y$ along the number line.  
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\item{1)}  Find all solutions of the equation $|x - 2| + |x + 6| = 12$.  
\item{2)}  What is the smallest $c$ for which $|x - 2| + |x + 6| = c$ has a solution?
Prove your assertion about this $c$.  
\item{3)}  Let $\max(a,b)$ be the larger of the two real numbers $a,b$, and similarly
$\min(a,b)$ be the smaller.  (If $a = b$, then that single value is both the max and the min.)
Show that 
$$\max(a,b) = {a + b\over 2} + {|a - b|\over 2} \quad {\rm and} \quad \min(a,b) = {a+b\over 2} - {|a - b|\over 2}.$$ 
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\item{C)}  In this and the next question, we will ``move up'' to 
quadratic equations.  
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\item{1)}  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 $x = 3$.
Show how we know this, using the algebraic Axioms.  (Hint:  If $a,b$
are real numbers with $a\cdot b = 0$ and $a\ne 0$, then why must
$b = 0$?)
\item{2)}  Now consider the quadratic equation $x^2 - 2 = 0$, with
solutions $x = \pm\sqrt{2}$.   What is special about the decimal expansion
of a rational number.  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!)
\item{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?  (You may use without
proof the fact that if $m$ is an integer and $m^2$ is even,
then $m$ must also be even.)  
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\item{D)}  Here's another way to get approximations to $\sqrt{2}$.
Note that $\sqrt{2}$ solves the equation $x = 2/x$.
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\item{1)} Explain why for all positive $x_0$, $\sqrt{2}$ lies
between $x_0$ and $2/x_0$ -- either 
$x_0 \le \sqrt{2} \le 2/x_0$ or $2/x_0 \le \sqrt{2} \le x_0$.
  (Sketching the graphs $y = x$ and $y = 2/x$ 
on the same set of axes may be useful!  Both of the possibilities
above occur for some $x_0$.  Say when each is true.)
\item{2)} Thinking of $x_0$ as a first approximation, 
show that the {\it average}
$$x_1 = (1/2)(x_0 + 2/x_0)$$
also lies between $x_0$ and $2/x_0$.  
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\item{}  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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\item{3)} Using (2), 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?  (We will return to
this example in a couple of weeks!)
\item{4)}  Which of the Axioms for real
numbers and other results from section 1.3 in the text
have you used in this question?
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{\it Writeups:}  Group writeups due in class Monday, February 7. 
\bye
