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\title{Defining the Real Numbers}
\author{Dr. Holmes}
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\maketitle
This is an optional set of notes on defining the real numbers in the
same spirit in which we have defined the integers and rationals.
We will define the {\em positive\/} reals assuming that we have already
defined the {\em positive\/} rationals.
We defined the integers by considering equivalence classes of pairs
$(m,n)$ of natural numbers used to represent the results of the
subtraction problem $m-n$ in the integers: the reason the construction
works is that we can say what it means for the integer $m-n$ to be
equal to the integer $p-q$, where $m,n,p,q$ are natural numbers,
entirely in terms of natural numbers.
Similarly, we define the rationals by considering equivalence classes
of pairs $(m,n)$ of an integer and a positive integer used to
represent fractions $\frac mn$. We already know from elementary
school how to express what it means for $\frac mn$ to be equal to
$\frac pq$ entirely in terms of integers ($mn = pq$) and the rest goes
from there.
The new operation we add to get the reals is {\em infinite sums\/}
over the rationals. We stick to infinite sums of {\em positive\/}
rationals in our basic definition because there are technical problems
with infinite sums involving both positive and negative rationals
(they can be undefined in more complicated ways).
We use infinite sets $S$ of positive rational numbers to represent
infinite sums. A sum of finitely many elements of $S$ is called a
``partial sum'' of $S$.
We say that the sum of $S$ exists (is finite) if there is a positive
rational $b$ such that every partial sum of $S$ is less than $b$.
If the sum of $S$ exists and the sum of $T$ exists, we say that the
sum of $S$ is less than or equal to the sum of $T$ if for any partial
sum of $S$ there is a partial sum of $T$ which is greater. We say
that the sums of $S$ and $T$ are the same if the sum of $S$ is less
than or equal to the sum of $T$ (using the definition above) and the
sum of $T$ is less than or equal to the sum of $S$.
The relation between $S$ and $T$ defined by ``the sum of $S$ is the
same as the sum of $T$'' as we have just defined it is an equivalence
relation. (this is not too hard to prove). The equivalence classes
under this relation of the sets whose sums exist can be used to
represent the positive real numbers.
Notice that an individual real number codes an infinite amount of
information (this is not true for individual natural numbers,
integers, or rationals).
Notice that any infinite decimal representing a positive real can be
converted to an infinite set of positive rationals in a
straightforward way -- the infinite decimal representation maps to our
presentation. The terminating decimals cannot be converted to
infinite sums of positive rationals, but every terminating decimal is
equivalent to an infinite decimal: lower the last nonzero digit by 1
and replace all following digits by 9's ($1.5 = 1.49999\ldots$).
There is a technical problem with this definition: it only handles
infinite sums of positive rationals which are all different! This
makes it harder to define addition and multiplication in a natural
way, though it does not make it impossible. It would be natural to
define $[S]+ [T]$ as the equivalence class of the union of $S$ and
$T$, but this is made difficult by the fact that if $S$ and $T$ have
members in common this will cause the sum to come out too small
(because a repeated term in the sum will be lost). There are ways to
get around this, either by defining addition cleverly using our
definition of the positive reals or by changing the definition of the
positive reals in such a way as to allow repeated terms in infinite
sums. For example, if we used sums of infinite sequences of positive
rationals instead of sums of infinite sets there would be no problem
with repeated terms.
To get zero and the negative reals, construct the general real numbers
from the positive reals in the same way we constructed the integers from
the natural numbers.
So the order of construction will be different: from the natural
numbers, build the positive rationals without first building the
integers (ordered pairs of natural numbers under the equivalence
relation of having the same rational quotient). Then build the
positive real numbers as infinite sums of rationals. Then build the
reals as ordered pairs of positive reals under the relation of having
the same difference.
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