Answer to Question #215884 in Geometry for Mbalenhle Zikode

Question

1.1. Describe, explain and give examples of three important mathematical ideas which Euclid wrote about:
(a) prime numbers (3)
(b) rational numbers (3)
(c) irrational numbers (3)

1.2. Let a white and black straight line cross three times, each time at different angles. Explain the Euclidean idea that opposites angles are the same size (congruent): (2)

Answer 1

1.1 Euclid puts it as follows:

• Euclid prove that there exists straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only and others in square only, with an assigned straight line , then the assigned straight line is rational and those that are incommensurable with it , irrational.
• The square on the assigned straight line is called rational and those areas which are commensurable with it rational but those which are incommensurable with it irrational and the straight lines Which produce them irrational, that is in case the areas are squares , but in case they are any other rectilinear figures, the straight lines on which are described squares equal to them.
• A prime number is a number measured by a unit only, that is, its only proper divisor is 1.
• If N is a prime number, then it is a new prime not among a,b,c,...n because it is larger than all of this.

1.2

The crossing of the lines three times forms a triangle. If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.