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Let us assume that we have the usual axiomatic framework of Euclidean
geometry. We will show that the points on a line can be given
arithmetic operations and identified with the ``usual'' decimal
numbers. Moreover, we can introduce coordinates in space using the
Euclidean framework. One important thing to note is that use only the parallel postulatecongruence (hence distance and
angle) play no role in the introduction of coordinates.
We are given some gadget that can draw the line joining two
points and the line parallel to a given line through a point outside
it. Such a gadget is a ruler with a roller. (Alternatively you can use the xfig program).
In the following constructions (see Figures 1 and 2)
we have numbered the lines and points in the sequence in which they
are obtained. Assume given a pair of points 0 and 1. We can define
addition and multiplication for points a and b on the line l
joining 0 and 1 by the constructions given below (the final point
in each construction is the sum or product of the the two original
points).
Exercise 1
Show that the following (usual) rules of arithmetic hold; in other
words the points on a line form a field.
 The commutative law for addition: a + b = b + a.
 The commutative law for multiplication: ab = ba.
 The associative law for addition:
(a + b) + c = a + (b + c).
 The associative law for multiplication:
a(bc) = (ab)c.
 The distributive law:
a(b + c) = ab + ac.
 The identity for addition: a + 0 = a.
 The identity for multiplication:
a^{ . }1 = a.
 For any a there is a point ( a) so that a + ( a) = 0.
 For any nonzero a there is a point (1/a) so that a(1/a) = 1.
 If O' and 1' are two other points then give a natural
correspondence between the points of the line l' joining 0' and
1' and the line l so that the arithmetic structure is
preserved.
In addition, we can use the notion of order on the points of a line to
define an order in our arithmetic by saying that a number lies between
two other numbers if the corresponding points have the same relation.
In particular, we say that a > 0 if a is between the points 1
and 0 or if 1 is between a and 0 or if a is 1.
Exercise 2
Show in addition that
if a > 0 and b > 0 then a + b > 0 and
a^{ . }b > 0.
The following two important axioms are due to Archimedes (but only one
carries his name):
Axiom 1 (Also known as ``Big step  Little
step'')
If x > 0 (is a Little step) and y > 0 (is the Big step) then
there is a natural number n (the number of little steps) so that
y is less than nx.
The second axiom is perhaps even less ``obvious'' but is essential.
Axiom 2 (Least Upper Bound)
If
A_{n} is a sequence of points so
that for all
n,
A_{n + 1} lies between
A_{n} and
D for some
fixed point
D (i. e.
A_{n} move towards
D but do not reach it).
Then there is a point
B which is the ``limit'' of
A_{n}. In other
words,
A_{n + 1} is between
A_{n} and
B for all
n and if
C is
any other point so that
A_{n + 1} lies between
A_{n} and
C then
B lies between
A_{n} and
C for all
n (see figure
3).
Figure 3:
The Least Upper Bound

Exercise 3
We introduce the decimal representation of a real number as
follows.
 Use the Archimedean Property to show that for any real number
x there is an integer n so that
n x < n + 1. This integer is
called the integer part [x] of x.
 Show that the sequence
x_{n} = [10^{n}x]/10^{n} is a
nondecreasing sequence.
 Use the Least Upper Bound property to conclude that x_{n} has a
limit y.
 Using the principle of the excluded middle show that y = x.
Finally, we choose four noncoplanar points in space and designate them
o, e_{1}, e_{2} and e_{3}. The point o is called the origin the
line through o and e_{1} (respectively e_{2} or e_{3}) is called the
xaxis (respectively yaxis or zaxis). By drawing lines
parallel to the axes we can produce for any point a unique triple of points
(x, y, z) one on each axis which uniquely determine the point in space.
By the above method we obtain the coordinates in decimals as well.
Exercise 4
Show that a line in the plane is the locus of all points with
coordinates (x, y) such that ax + by + c = 0 for some constants a,
b and c so that a and b are not both zero. Also show the
converse.
Next: Conic sections
Up: Prerequisites
Previous: Prerequisites
Kapil H. Paranjape
20010120