Chapter 2

Represent \(125,62,4821\), and 23,855 in the Greek alphabetic notation.

Represent \(8 / 9\) as a sum of distinct unit fractions. Express the result in the Greek notation. Note that the answer to this problem is not unique.

There are extant Greek land surveys that give measurements of fields and then find the area so the land can be assessed for tax purposes. In general, areas of quadrilateral fields were approximated by multiplying together the averages of the two pairs of opposite sides. In one document, one pair of sides is given as \(a=1 / 4+1 / 8+1 / 16+1 / 32\) and \(c=1 / 8+1 / 16\), where the lengths are in fractions of a schonion, a measure of approximately 150 feet. The second pair of sides is given as \(b=1 / 2+1 / 4+1 / 8\) and \(d=1\) Find the average of \(a\) and \(c\), the average of \(b\) and \(d\), and multiply them together to show that the area of the field is approximately \(1 / 4+1 / 16\) square schonion. Note that the taxman has rounded up the exact answer (presumably to collect more taxes).

Thales is said to have invented a method of finding distances of ships from shore by use of the angle-side-angle theorem. Here is a possible method: Suppose \(A\) is a point on shore and \(S\) is a ship (Fig. 2.16). Measure the distance \(A C\) along. a perpendicular to \(A C\) and bisect it at \(B\). Draw \(C E\) at right angles to \(A C\) and pick point \(E\) on it in a straight line with \(B\) and \(S .\) Show that \(\triangle E B C \cong \triangle S B A\) and therefore that \(S A=E C\)

Suppose Thales found that at the time a stick of length 6 feet cast a shadow of 9 feet, there was a length of 342 feet from the edge of the pyramid's side to the tip of its shadow. Suppose further that the length of a side of the pyramid was 756 feet. Find the height of the pyramid. (Assuming that the pyramid is laid out so the sides are due north-south and due east-west, this method requires that the sun be exactly in the south when the measurement is taken. When does this occur? \(^{20}\) )

Show that the \(n\)th triangular number is represented algebraically as \(T_{n}=\frac{n(n+1)}{2}\) and therefore that an oblong number is double a triangular number.

Show algebraically that any square number is the sum of two consecutive triangular numbers.

Show using dots that eight times any triangular number plus 1 makes a square. Conversely, show that any odd square diminished by 1 becomes eight times a triangular number. Show these results algebraically as well.

Show that in a Pythagorean triple, if one of the terms is odd, then two of them must be odd and one even.

Construct five Pythagorean triples using the formula ( \(n\), \(\left.\frac{n^{2}-1}{2}, \frac{n^{2}+1}{2}\right)\), where \(n\) is odd. Construct five different ones using the formula \(\left(m,\left(\frac{m}{2}\right)^{2}-1,\left(\frac{m}{2}\right)^{2}+1\right)\), where \(m\) is even.

Show that if a right triangle has one leg of length 1 and a hypotenuse of length 2, then the second leg is incommensurable with the first leg. (In modern terms, this is equivalent to showing that \(\sqrt{3}\) is irrational.) Use an argument similar to the proposed Pythagorean argument that the diagonal of a unit square is incommensurable with the side.

Show that the areas of similar segments of circles are proportional to the squares on their chords. Assume the result that the areas of circles are proportional to the squares on their diameters.

Consider the quotation from Plato's Republic: "It will further our intentions if it [calculation] is pursued for the sake of knowledge and not for commercial ends." Discuss the relevance of this statement to current discussions on the purposes for studying mathematics in school.

In Zeno's Achilles paradox, assume the quick runner Achilles is racing against a tortoise. Assume further that the tortoise has a 500 -yard head start but that Achilles' speed is fifty times that of the tortoise. Finally, assume that the tortoise moves 1 yard in 5 seconds. Determine the time \(t\) it will take until Achilles overtakes the tortoise and the distance \(d\) he will have traveled. Note that Achilles must first travel 500 yards to reach the point where the tortoise started. This will take 50 seconds. But in that time the tortoise will move 10 yards farther. Continue this analysis by writing down the sequence of distances that Achilles must travel to reach the point where the tortoise had already been. Show that the sum of this infinite sequence of distances is equal to the distance \(d\) calculated first.