MATH 380 Assignment #3

1. Logs, roots, they're all pieces of trees!

1. Show that log₂(10) is irrational by tracking the usual (modern) proof that the square root of 2 is irrational as closely as you can. [5]

2. How might a Greek mathie have thought of log₂(10) and attacked the problem of showing it was incommensurable? [5] (tiny hint: incommensurable with?)

3. If the square root of 2 hadn't been been found first, how likely would log₂(10) have been to become the first irrational/incommensurable to be discovered? [5]

3. Describe some of the tools besides compass and (unmarked) straightedge that Greek and Hellenistic geometers experimented with. What constructions did these make possible that cannot be accomplished with a compass and straightedge alone? [5]

4. Show that the following construction for trisecting an angle using a compass and a ruler with two marks (a distance of r apart) works.

Given that angle(AOB) = theta, draw a circle with centre O and radius r. Suppose this circle intersects OA at X and the line extending BO past O at Y. Slide the ruler around until its edge runs through X, one mark is on the line extending OY past Y, and the other mark lies on the circle. Let D be the point on the line where the first mark is and E be the point on the circle where the second mark is. Then angle(EDY) = theta/3. [5]

Department of Mathematics

Trent University