Math 111, History of Math, Duchin
Final Exam
Instructions: each question has choices to make. You must clearly circle the ones you want graded;
otherwise, for a problem that says “choose m out of the following n,” I will grade the first m in order.
Note that for many (even most) of the questions, I am not looking for just one right answer, but for you to
demonstrate your knowledge by making relevant and correct statements.
I. Everybody
(1) For one of the following quotations from class, discuss it briefly. This may include who said it, but
more importantly, describe what is meant by the quote, illustrating with an example. Is there a
movement or a school of thought whose philosophy fits well with the quote? Your answer gets four
sentences max.
(a) “If we want to show that a = b by first showing a ≥ b and then b ≥ a, it is unfair... We should
instead disclose the inner reason for their equality.”
(b) “By relieving the mind of all unnecessary work, a good notation sets it free to concentrate on
more advanced problems, and in effect increases the mental power of the race.”
(c) “The essence of mathematics is its freedom.”
(2) Choose two of the following four.
(a) A certain political blog had a February 29, 2008 entry where one writer accused another of
“Ptolemaic epicycles” of reasoning. Explain the historical reference and what was meant by the
blogger.
(b) Mitt has a slip of paper that says π4 = 1 − 13 + 51 − 17 + · · · ; Rudy has a slip of paper that
says π = [3, 7, 15, 1, 292, 1, 1, 1, 2, . . .] Explain the meaning of each expression. Only one of the
two expressions allows you to find an expansion of π to arbitrary degrees of precision, but the other
one makes it much easier to find the decimal expansion of π to five decimal places. Why?
2
(c) These days, the Archimedean principle refers to the statement that for any two positive real
numbers, if the smaller one is added to itself enough times, it becomes bigger than the larger one. In
what form did that appear in Euclid’s Elements? What does that have to do with the foundations
of calculus?
(d) What is an axiomatic definition? Which of the people or groups in the readings have been
associated with axiomatic definitions? Give an example of an axiomatic definition.
II. Reading track
Each of the following statements is false. Choose four of the following seven and replace each of them
with correct, relevant statements.
√
(1) Al-Khwarizmi’s surviving text, YBC 7289, gives very accurate approximations for 3 2 that he dedicated to the Hindu goddess Namagiri.
(2) The original goal of Bourbaki was to refine elementary school mathematics through a movement
they dubbed “The New Math,” in which all of arithmetic is taught algorithmically.
(3) Euler hoarded his results until he could humiliate rival mathematicians with them.
(4) Lakatos argues that the best way to do math is to use the method of proofs and refutations until
theorems can be proven to be perfect.
3
(5) The makers of ancient calendars must have known about the Euclidean algorithm, because otherwise
they could not have produced such accurate calendar corrections (achieved by a carefully arranged
pattern of adding extra hours to a few days per year).
(6) Gauss rejected complex numbers as “merely a façon de parler” that could never be considered to
have solid ontological status.
(7) Srinivasa Ramanujan was “discovered” by Johann Bernoulli, who verified some of his mathematical
theorems by making astronomy-based calculations.
(8) Cardano and Tartaglia published several journal articles together on the general solution to polynomials of degree five.
III. Problem set track
Choose three of the following five.
√
(1) Recall that the Wallis plane gave a solution to x = −b ± b2 − c2 by drawing an appropriate right
triangle on a figure (like the ones drawn on the board). He used this to geometrically interpret
solutions to the quadratic equation x2 + 2bx + c2 = 0. For instance, when b = 5 and c = 4, what are
the corresponding values of x and what is the figure? When b = 3 and c = 4, what are the values
of x and where on the Wallis plane are they located? Compare this to the location in the modern
complex plane.
4
(2) Given a length ` and a length m, explain how to construct `m +
`
2
with straightedge and compass.
(3) For this problem, we write n for the Eyptian fraction 1/n.
First find 37 times 9 using the Egyptian algorithm.
Next: You’re an archaeologist and you find the following calculation on a papyrus fragment;
the boxes represent undreadable parts of the artifact. From context, figure out what values were
supposed to go in the boxes. Using the whole calculation, give an Egyptian fraction expression for
the vulgar fraction 4/5.
1
2
4
5
2
1
4
2
0
0
0
(4) Prove that the union of two countable sets is countable. Write this fact as an arithmetic equation
using the cardinal number ℵ0 .
p
(5) Euler wrote that 12 i ln(−1) is “equal to the side of the square equal to the circle with diameter
one.” Give a step-by-step description of how to interpret this as a modern formula.