Assignment 10. Incommensurability

For submission

The square root of 2, √2, has the following shrinking property. Take a rectangle the length of whose sides are in the ratio of √2 to 1. Bisect the longer sides to produce two smaller rectangles. These smaller rectangles also have sides whose lengths are in the ratio of √2 to 1.
(This is the reason that the international standard paper sizes, A1, A2, A3, A4, ... have sides whose lengths are in the ratio √2 to 1. Each sheet can be halved to produce two of the next smaller size.)

1. Prove this bisection property.

(Hint: (1/2) x √2 = 1/√2.)

2. Using the figure below, give a geometrical shrinking proof of the irrationality of √2. Use the similar proof given in the chapter as a guide. Assume that the larger whole number p is even.

(Hint: Posit for reductio that
(i) p/q = √2;
(ii) the values p and q in the left figure have no common factor; and
(iii) p has the smallest value possible for (i) to hold.)

For discussion

Not for submission

A. In Cartesian geometry, we cover the Euclidean plane with an x-y coordinate system. Geometric structures arise as algebraic formulae. A straight line is "y=mx+b," for example. Has the Pythagorean goal of founding geometry on numbers been realized here?

B. The analyses above show that √2 is irrational. What about other numbers? How common is it for a whole number to have an irrational square root? Are there proofs for these other cases?

C. The golden ratio φ is reputed to be the most pleasing proportion for a rectangle and, for this reason, appears often in classical art. Its value is roughly 1.618, but the exact value is irrational. It has a simple geometric definition: take a rectangle whose sides are in the golden ratio; remove a square; and the remaining rectangle is still a golden rectangle. This fact enables a "shrinking geometrical proof" of the irrationality of φ.

D. The algebraic expression of this geometrical fact is that (φ-1) = 1/φ. Solve for φ. (You will need the quadratic formula.)