Mathematics and Statistics

Mathematics and Statistics

Abstract
This note describes the requirements for the second paper to be submitted in Math
100, Fall 2018. The paper is due in class on Wednesday, November 14.
1 Introduction
In this note you will (thoroughly) rework and improve your first paper, describing the Babylonian method for calculating the approximate square root of a given positive number. The
intended audience is still curious, motivated high school students with no background in
calculus.1
2 Guidelines
2.1 Mathematical guidelines
1. In addition to the requirements outlined in the guidelines for the first paper, you should
include a simple method to evaluate the accuracy of the approximations that does not
require a calculator/computer approximation of the desired square root.
Hint: When two successive approximations xn and xn+1 are close enough to each
other, then the newest one (xn+1) is close enough to pN.
2. The target audience is still intelligent, curious high school seniors or college first years,
with no calculus background. This means that a person with no more than precalculus
or ‘college algebra’ should be able to read and understand your paper. In particular
detailed references to calculus based material like (but not limited to) Newton’s method
1So nothing too fancy, especially, no mention of Newton or his method.
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or limits of sequences should not be included.2 You certainly should not include formal
proofs.
3. You can include a short paragraph about the historical background. Or just mention
it in a line. Your call.
4. References in this paper should be more along the lines of suggestions of where the
interested reader can find more information about the subject, historical or mathematical.
5. The main part of the paper is a clear, well-written description of how the method
works. I.e., that to find a good approximation to pN you (i) start with a reasonable
first ‘guess’ x0 (an integer that is reasonably close to pN for obvious (to the reader)
reasons). Then you (ii) produce a better approximation x1 using the formula
x1 =
1 2
x0 + xN0 :
Then you (iii) produce an even better approximation (than x1) by using the same
formula with x1 as your ‘input’. Then you (iv) convey the idea of ‘etc.’, i.e., that the
previous step can be repeated as many times as you want, and that each iteration
produces a (much) better approximation than the previous one. At this point it is
appropriate to introduce the general formula
xn+1 =
1 2
xn + xNn :
You may want to illustrate your description with a running example, as you go. E.g.,
\The first step is to find a reasonable first approximation, a number x0 that is reasonably
close to pN. For example, if N = 7, then we can start with x0 = 3 because 4 < 7 < 9
and 32 = 9…. Next, to find a better approximation, we take the average of x0 and N=x0
and call it x1. We can write this as a simple formula: x1 = 1 2(x0 + xN0 ). Using this
formula in our example of approximating p7, x0 = 3 and therefore x1 = 1 2 3 + 73
=
83
…”
!! The initial description should be written in paragraph form with a limited number of
mathematical formulas and lots of words in between. The initial description should not
be given in list form. On the other hand, after the initial explanation of the method
has been given, you can certainly summarize it in a short list.
2It is certainly ok to include a passing reference to the fact that one can explore the Babylonian method
more deeply or that certain statements in your paper can be made more precise by applying more advanced
mathematical theory and/or techniques. But fancy technical details should not be included.
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6. You must also include a (brief) explanation (not a proof) of why the method works,
i.e., why it makes sense that each iteration produces an approximation that is better
than the previous one. You can certainly include relevant (and useful) statements
of fact (without proof) as long as they don’t interrupt the flow and aren’t overly
technical. E.g., including the observation that xn > xn+1 > pN once n ≥ 1 is
fine, especially as you can illustrate this in your example(s). Likewise, the fact that
xn – xn+1 > xn+1 – pN, which is relevant to the stopping time question and which
can also be illustrated in your example(s), might also be ok, but its exposition requires
more careful attention to avoid confusing the reader.
On the other hand identities like
xn – xn+1 =
x2
n – N
2xn
or inequalities like
xn+1 – pN < x2 n2–xnN 2 < x2 n2–xnN = xn – xn+1
would be distracting from the purpose of this paper.3
7. Other technical pointers. (i) Please express your approximations as rational numbers
a=b not as truncated decimals. You can convert your rational approximations to decimals to compare to what a calculator gives as pN (this comparison is instructive),
but only for that purpose. (ii) Please don’t go crazy with greek letters or multiple
variables. We only need two letters here: N for the number whose square root we seek
and the sequence of approximations x0; x1; : : : ; xn; : : :. The Greek ξj is pretty, yes, but
also potentially more distracting (to the intended audience) than the more familiar
(latin) xj. In other words, keep it simple. (iii) That also goes for language. This paper
will not benefit from flowery language or excessive use of creative adjectives (or similes
and metaphors). (iv) Also, ‘square root’ is a noun,4 not a verb.
8. You may find it very helpful to swap papers with another student in the class so
that you can proofread each other’s papers. Mostly just to catch typos and obvious
grammatical errors and such, though you can also share constructive, polite criticism,
if you so choose.
9. As I said in class, before you start your rewriting, please go back and study the method
in detail. Get together with a couple of other students and explain it to each other
in as much depth as you can muster. Once you really get it (and remembering what
you found confusing about it at first) will make it much easier to write Paper 2, even
if you don’t include a lot of what you learn.
10. Finally | the second paper is due in class on Wednesday, November 14 (not November
12, as listed in the syllabus, because 11/12 is Veterans’ day, a national holiday).
3Though these may be very useful in the third paper.
4A ‘noun phrase’, technically.
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2.2 Writing Guidelines
1. You must typeset your paper using LATEX and the official Math 100 LATEX-template.
Retain the formatting commands of template { i.e., do not change the font sizes, etc.
2. Don’t go crazy with sections and subsections.
3. The paper should be between two and three pages in length.
4. Follow the writing guidelines discussed in class and those described in section 5.3 of
the textbook and in the writing resources.5 If you rely on any sources, you should cite
them properly using bibliographical citations collected at the end of the paper. See [1]
or Chapter 12 of [2] for a description on how to do this.
5. Your paper will be graded for the correctness of the mathematics that you present,
the clarity and quality of your exposition and the overall structure of the paper. If the
writing is poor, then your grade on the paper will also be poor even if the mathematical
content is correct.
6. Try hard to eliminate typos and grammatical errors. Do not rely solely on your spell
checker. For instance, \Hear is ware they maid there mistake.” has typos that a spell
checker won’t catch.
7. Important: Pay attention to the corrections/suggestions on your first paper, as well
as the general comments made in class.
2.3 Collaboration
You are allowed (in fact encouraged) to collaborate on the mathematical aspects of the
assignment. However, each student must write his or her own paper. Plagiarism, which the
New Oxford American Dictionary defines in part as ‘the practice of using someone else’s work
and passing it off as your own’, is considered cheating and a breach of academic integrity.
2.4 Suggestions
• Plan on writing at least two or three drafts of the paper.
• Have someone else in the class read your paper before you submit it and listen to their
comments (this doesn’t mean that you have to edit your paper to their liking).
• If you have specific questions about the math, using LATEX or writing in general, then
ask in section or during office hours.
• However, neither the TAs nor I will proof-read your paper for you.
5On the papers-page of the course website.
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References
[1] LaTeX/Bibliography Management,
<http://en.wikibooks.org/wiki/LaTeX/Bibliography_Management>
[2] The LATEX Companion, 2nd edition,
Frank Mittelbach, Michel Goossens, Johannes Braams, David Carlisle and Chris Rowley
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