Wednesday March 26
Discuss assignment for this week - make note of project and portfolio.
How to know the mathematics is "right'?
Proof starting with Euclid.
Review: Proposition 1 Book 1
Euclid's tools. Proposition
1 and Proposition
Models for possibilities and impossibilities.
A model for the plane geometry of Euclid:
The basis for this model is understanding numbers!
Points: (a,b) where a and b are rational numbers.
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)2 + (y-B)2 = C where A,B, and C are rational numbers and C >
Basic Facts: If I can build a model for the properties - the properties are consistent ( no contradictions or absurdities)
If I assume that something is possible (or exists) and arrive at a
contradiction or an absurdity, there is no possible model and that
something does not exist.
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 r2=3.
r = a/b where a, b are positive natural numbers.
Then rb=a and r2b2=a2. Or 3b2=a2 .
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 a2: The number of factors of a2and 3 b2 that are 3 is fixed.
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.