Introduction of co-ordinates

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 co-ordinates in space using the Euclidean framework. One important thing to note is that use only the parallel postulate--congruence (hence distance and angle) play no role in the introduction of co-ordinates.

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).

Figure 1: Addition

Figure 2: Multiplication

Exercise 1   Show that the following (usual) rules of arithmetic hold; in other words the points on a line form a field.

1. The commutative law for addition: a + b = b + a.
2. The commutative law for multiplication: ab = ba.
3. The associative law for addition: (a + b) + c = a + (b + c).
4. The associative law for multiplication: a(bc) = (ab)c.
5. The distributive law: a(b + c) = ab + ac.
6. The identity for addition: a + 0 = a.
7. The identity for multiplication: a ^. 1 = a.
8. For any a there is a point (- a) so that a + (- a) = 0.
9. For any non-zero a there is a point (1/a) so that a(1/a) = 1.
10. 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.

1. Use the Archimedean Property to show that for any real number x there is an integer n so that n $\leq$ x < n + 1. This integer is called the integer part [x] of x.
2. Show that the sequence x_n = [10ⁿx]/10ⁿ is a non-decreasing sequence.
3. Use the Least Upper Bound property to conclude that x_n has a limit y.
4. Using the principle of the excluded middle show that y = x.

Finally, we choose four non-coplanar points in space and designate them o, e₁, e₂ and e₃. The point o is called the origin the line through o and e₁ (respectively e₂ or e₃) is called the x-axis (respectively y-axis or z-axis). 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 co-ordinates in decimals as well.

Exercise 4   Show that a line in the plane is the locus of all points with co-ordinates (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.