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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{Discussion 1 -- The Real Number System}
\centerline{September 3, 2015}
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{\it Background}
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You are all familiar with decimal expansions of numbers such as 
$${15\over 8} = 1.875$$
Recall that
in some cases, the decimal expansion must be continued indefinitely to get the exact value.  For 
instance, we write
$${1\over 3} = .33333\cdots = .\overline{3},$$
where the overline indicates that the digit 3 {\it repeats forever in the exact decimal expansion
of} ${1\over 3}$.  Another way of saying this is that if we stopped after any finite number
of 3's the number we would have would be an {\it approximation} to ${1\over 3}$ but an {\it underestimate}:
$$.3333 = {3333\over 10000} < {1\over 3}.$$
The real number system is the set of {\it all numbers} that can 
be represented by these decimal expansions (possibly infinite, possibly nonrepeating).
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{\it Questions}
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A)  How could you tell that ${3333\over 10000} < {1\over 3}$ just working with
the fractions (i.e. without bringing the decimal expansions into it)? 
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B)  What does the decimal expansion $1.875$ for the fraction ${15\over 8}$ actually {\it mean}?
(Hint:  $1.875$ is a way of writing this number as a {\it sum of several fractions} with ``special''
denominators.  What is that sum?)
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C)  It is a fact that {\it every eventually repeating decimal expansion represents a number
that can also be written as a single fraction with integer numerator and denominator} 
(a rational number).  Let's see how this works by considering a representative example.  
For instance, consider the number
$$1.\overline{137} = 1.137137137 \cdots$$
where the repeating part of the expansion starts right after the decimal point.  We want 
to write this number in the form ${a\over b}$ for some whole numbers (integers) $a,b$.  
If we start off by subtracting off the non-repeating 
whole number part, we get $1.\overline{137} - 1 = .\overline{137}$.
Call this number $c$:
$$c = .137137137\cdots = .\overline{137}.$$
Note that 
$$c - .137 = c - {137\over 1000} = .000137137\cdots = .000\overline{137}.$$
But this number can also obtained from $c$ by multiplying by ${1\over 10^3} = {1\over 1000}$, so 
$$c - {137\over 1000} = {1\over 1000} c.$$  
(Make sure you see why this works by thinking about shifting the decimal point.) 
Use this equation to solve algebraically for $c$ as a fraction, and then use that 
to express $1.\overline{137}$ as a fraction.
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We can do something similar to this process for any eventually repeating decimal expansion.
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D)  Even if a decimal expansion {\it never} settles down into a repeating pattern, it still 
represents a real number.  The numbers given by {\it nonrepeating decimal expansions}
are called {\it irrational numbers}.  
You have probably seen several examples of important irrational numbers at some point -- 
give a 6 decimal digit approximations of your ``favorite'' irrational number (using a calculator).
Round to the nearest digit in the 6th decimal place.  Is your approximations an
overestimate or an underestimate of the exact value of your number?    
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E)  Do you know a {\it geometrical} way to picture or think about the real number system?
Describe how the connection between numbers and points of the geometric ``model'' of the 
real number system works.  For instance, what choices do you need to make to set up the 
correspondence between numbers and points?
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