HM/MA 004, Assignment 1, due Friday, 13 September 2002, 10:00 AM

(Note on collaboration: If you do the homework in pairs or groups, please list the name(s) of your collaborator(s) on what you each turn in.)

1. (12 points) We've discussed in class the "number and reciprocal" problem from the Babylonian tablet YBC 6967 on pp. 28--29 of the Reader, verifying the procedure both algebraically and geometrically. Provide a corresponding algebraic justification and geometrical interpretation for the second "two square figures" problem (#12) on p. 32.
2. (10 points) Give an algebraic explanation of one of the "ki-la" or "excavation" problems (#21, #22, #24, #25, #26, #27, or #28) on pages 29--31. What is mathematically peculiar about it? Can you explain the cause of this difficulty in a historically plausible way?
3. (8 points) A historian of math I know says he's never seen any evidence for a genuinely practical ancient application of procedures for solving quadratic equations. What do you think of that claim in light of the Babylonian texts in your readings?
4. (8 points) Using Pythagorean-style "figurate numbers" composed of dots, draw and explain pictures that show that---
   1. the square of an even number is even;
   2. the sum of the first n integers 1+2+3+...+n is equal to (n)(n+1)/2.
5. (10 points) A Greek philosopher named Antiphon said that you could determine the area of a circle exactly by inscribing in it a succession of equilateral polygons, each with twice as many sides as the previous one (e.g., a square, an octagon, a hexadecagon (16-sided figure), etc.): "in this way the area of the circle would sometime be used up and a polygon would be inscribed in the circle whose sides, on account of their smallness, would coincide with the circumference of the circle." Based on your readings, what do you think Aristotle would have thought about this argument?
6. (12 points) We've seen that original-urban-civilization mathematics was primarily concerned with procedures for solving problems, rather than theoretical justifications of the procedures. In a rare counterexample, a first-millennium-BCE Chinese text justifies the "Pythagorean theorem" as follows: Cut diagonally a rectangle of length 4 and width 3, and the diagonal between the corners will be 5. Then after drawing a square on the diagonal, surround it with three other identical half-rectangles so as to make a square tile of area 49. The four outer half-rectangles together make two rectangles, of area 24; when this is subtracted from the square tile, the remainder will be 25. Why is this not a rigorous proof? Rewrite the basic procedure of this construction in a more general form to make it into more of a Euclid-style rigorous proof; you need not knock yourself out eliminating every single "rigor loophole", but be sure to warn the reader about the loopholes you left in.