Quiet Millenium, Worksheet 1
Chinese Mathematics
MATH 4367, Spring 2012
(None of this material requires mathematics beyond the high school curriculum.)
1. For each of the cultures in Eves chapter 7, Chinese, Indian, Arabian, describe one mathematical
contribution (beyond that of the Greeks.)
2. An ancient Chinese text has the following problem. “There is a circular field with circumference
181 bu and diameter 60 13 bu. Find the area of the field.” What would have been the answer given
by this text?
3. (Magic squares)
(a) Complete the 9 x 9 magic square begun in the notes.
(b) Find or construct a 5 x 5 magic square.
4. (From the Nine Chapters, referred to in Burton, pp. 253-4.) Solve the following problems:
(a) A number of people are going together to purchase some goods. If each person pays 8 then
they have paid 3 too much; if each person pays 7 then they are still owe 4. How many people
are there and what is the cost of the goods?
(b) A bamboo shoot of height 10 chi’ih breaks and the upper end leans over and touches the
ground 3 ch’ih away from the root of the shoot. Find the height of the break.
(c) (Broken bamboo problem from Eves, p. 240, problem 7.4 (b).) A bamboo stalk is 18 feet high
but it breaks and the top portion bends over and touches the ground 6 feet from the base.
Where did it break?
(d) Monthly interest is 3 % (over 30 days.) How much interest should there be on 750 qian if it is
loaned out for 9 days?
5. An ancient Chinese problem (Swetz p. 329) roughly translates as follows: there is a common well
belonging to five families. Each family has ropes they have cut to a certain length. If we take 2
lengths of rope from the first family the remaining part (to reach the bottom of the well) equals 1
length of rope of the second family. If we take 3 ropes from the second family we need one rope
from the third family to reach the bottom of the well. If we take 4 ropes from the third family we
need one rope from the fourth family to reach the bottom of the well. If we take 5 ropes from the
fourth family we need one rope from the fifth family to reach the bottom of the well. If we take 6
ropes from the fifth family we need one rope from the first family to reach the bottom of the well.
How deep is the well? (Let’s find the answer, say, in terms of the rope lengths used by the first
family.)
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Quiet Millenium, Worksheet 2
Mathematics of India
MATH 4367, Spring 2012
(None of this material requires mathematics beyond the high school curriculum.)
1. (Algebra rules)
(a) What is the “rule of 3”? And how did the Greeks solve that problem?
(b) What is the “rule of 9”? Why does it work? (Note that the rule of 9 is very different from the
rule of 3!)
(c) What is the “rule of 11”? Why does it work?
2. Use the rule of nine to reduce the following integers modulo 9.
(a) 13542777423
(b) 7497199844
(c) 44754044019
(d) 27005287547557717993
3. Use the rule of eleven to reduce the following integers modulo 11.
(a) 13542777423
(b) 7497199844
(c) 44754044019
(d) 27005287547557717993
4. Use both the rule of nine and the rule of eleven to check the following claims:
(a) 4567 × 8899 = 40640633.
(b) 7497199844 × 44754044019 = 335530011837615933036
5. (From Eves, p. 240, problem 7.4 (a)) “Two ascetics live on a cliff at height h = 100, whose base is
distance d = 200 from a neighboring village. One climbed down the cliff and walked to the village.
Another, being a wizard, flew up a height x and then flew directly to the village. The distance
traveled by the two ascetics was the same.”
(a) What is the height x?
(b) Generalize your answer to arbitrary heights h and distances d.
6. (From Eves, p. 241, problem 7.7)
(a) Show that a quadratic surd cannot be the sum of a rational and another quadratic surd.
√
√
√
√
(b) Show that is a + b = c + d where b and c are quadratic surds and a and c are rational
then a = c and b = d.
q
q
p
√
√
√
(c) Prove Bhaskara’s identity (Eves, p. 226): a ± b = (a + a2 − b)/2 ± (a − a2 − b)/2.
p
√
(d) Express 17 + 240 as the sum of 2 quadratic surds.
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7. (Pell’s equation)
(a) Use the solution (x, y) = (3, 2) to Pell’s equation x2 −2y 2 = 1 to generate five positive solutions
to this equation. (Use the method in section 3.2 of Ken’s notes on the Quiet Millenium and
compute (3, 2) ? (x, y) where (x, y) is a successive list of solutions beginning with (3, 2).)
√
(b) Use your work in part (a) to compute rational approximations to 2. Then compare your work
here with problem 1 of Worksheet 3 on the Greek Age.
(c) Use the solution (x, y) = (2, 1) to Pell’s equation x2 −3y 2 = 1 to generate five positive solutions
to this equation.
√
(d) Use your work in part (c) to compute rational approximations to 3.
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Quiet Millenium, Worksheet 3
Mathematics of Arabia and Europe
MATH 4367, Spring 2012
(None of this material requires mathematics beyond the high school curriculum.)
1. Use the method of double false position with starting values x1 = 0 and x2 = 2 to approximate the
solutions to the following equations:
(a) 2x2 − 5x + 3 = 0
(b) x4 + 8x3 + 3x = 12
2. Use Khayyam’s method (from Ken’s notes) to find all positive roots to x3 + 2x + 8 = 5x2 .
3. Can you modify Khayyam’s argument to find negative roots?
4. For each of the equations, below, do the appropriate substitution to turn the polynomial on the
left-hand side into a “depressed” polynomial.
(a) 2x2 − 5x + 3
(b) x3 − 6x2 + 11x − 6.
(c) x3 − 6x2 + 2x + 3
(d) x3 − 6x2 + x + 5.
5. Finish your work in equation part (a), above, finding the solutions to the equation by completing
the square.
6. For each of the polynomials in problem 4, guess a rational solution and then use this to find all
solutions to the polynomial.
7. Find a rational solution for the following polynomials. Then use this solution to find all the roots.
(a) Viete and Cardano-Tartaglia examined this polynomial: x3 + 63x − 316.
(b) Viete and Cardano-Tartaglia also examined this polynomial: x3 − 63x − 162.
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