Points on a Line
This activity belongs to the GeoGebra book GeoGebra Principia.
Let's continue with our construction process of the structure. Now we will define the four elementary operations.
- Addition: To obtain A + B, we reflect O in MAB obtaining a new point on r.
- Subtraction: To obtain A − B, we add A + B'.
- Multiplication: We create the product by constructing similar triangles, obtaining a new point on r.
- Division: To obtain A/B, we multiply A x B–1. Division is not commutative.
- Order. The symmetry I' O I allows us to define a ORDER RELATION:
A ≤ O :⇔AI' ≤ AI A ≤ B :⇔A − B ≤ O
Structure. Based on everything aforementioned, the set of points on the line r, endowed with the operations of addition and multiplication as defined, constitutes a similar structure ("ordered field") to that of ℝ (real numbers). In fact, we can establish a bijection (isomorphism) between both structures:(r, O, +, ×) → (ℝ, +, ×)
correspond each point P on r to the real number –OP if P- Note: Let us mark that we do not delve into the more intricate question of how to geometrically construct all the points on the line (completeness of the real line). We assume that every point corresponds to a number and vice versa. However, if we wish to restrict ourselves to points constructible with the mentioned operations, we can establish an isomorphism of those points (no longer spanning the entire line) with the field of the constructible numbers.
The points on a line are not the only geometric objects we can endow with the structure of a field. We can apply the same concept to any other set of objects that share the same definition, in which there is only one free point residing on a line. In the following constructions we will show some examples.
Author of the construction of GeoGebra: Rafael Losada.