Proof starting with Euclid.

Euclid's tools. Proposition 1 and Proposition 2

Models for possibilities and impossibilities.

A model for the plane geometry of Euclid:

The basis for this model is understanding

Lines: Sets of points (x,y) that satisfy an equation Ax + By = C where A,B, and C are rational numbers and not all are 0.

Circles: Sets of points (x,y) that satisfy an equation of the form (x-A)

Basic Facts:

Examples: (i) There is a rational number (fraction) which will measure the hypotenuse of an isosceles right triangle with a unit length for the side. (ii)There is a rational number (fraction) which will measure the side of the hypotenuse of a right triangle with a unit length for one side and a hypotenuse of length 2 units.

Proposition: The square root of 3 is not a rational number.

Proof: Suppose r is a rational number and r

r = a/b where a, b are positive natural numbers.

Then rb=a and r

Count the number of 3 factors of the right hand side: Even .

Count the number of 3 factors of the left hand side: Odd .

The Fundamental Theorem of Arithmetic ( Euclid VII) applied to a

Thus we have a contradiction and so the square root of 3 is not a rational number. End of Proof.

This result connected to the rational model for geometry leads to the conclusion that Euclid's proof was not correct. The rational geometry will satisfy all of Euclid's stated axioms, yet will not have the point needed to construct and equilateral triangle on the segment detemined byu the points (0,0) and (2,0) because the point of intersection of the two circles would lead to a statement that the square root of 3 was a trational number.

I will update this later- (entered on 4-7-2014)

For now you can find more discussion in the presentation on Philosophy and Proof at http://flashman.neocities.org/Presentations/Philosophy_Proof_Euclid.pdf

The remainder of the class was an introduction to coding with alphabet- numerical codes using the activity sheet.