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Name: KEY MthEd/Math 300– Williams Fall 2011 Midterm Exam 1

Closed Book / Closed Note. Answer all problems. You may use a calculator for numerical computations. There is a handy list of Greek on the last page.

Section 1: Fill in the name of the Ancient Greek who best fits the description. One point each.

1. Proved the Law of the Lever from axioms. 2. Worked on the problem of squaring the , and succeeded in squaring a lune. 3. Developed the , a theory of proportions, Eudoxus and an astronomical model involving nested . 4. Was the first Greek mathematician and philosopher, supposedly Thales studied mathematics in Egypt. 5. Wrote a definitive treatment of conic sections. Apollonius 6. Used the method of exhaustion to approximate π as being Archimedes 1 10 between3 7 and 3 71 .

7. Wrote the Mathematical Collection, which came to be known as the Greatest Collection, and which encompassed astronomy and trigonometry. 8. Compiled and organized the mathematical knowledge of his time into what became the most published mathematics book ever. 9. Leader of a group that studied the geometric properties of whole numbers (.e.g. and triangular numbers). 10. Was the first woman mathematician of which we have record. 11. In his , he studied indeterminate equations and Dophantus developed a symbol system for equations. 12. Alexandrian mathematician who wrote commentaries on earlier Pappus, but also Theon works. or Hypatia. Section 2: Match the following descriptions with the correct ancient culture by circling B for Babylon, E for Egypt, or G for . There may be more than one correct answer for some questions. One point each.

13. No original sources for studying their mathematical history are G available. 14. Flourished mathematically between 600 BC and 400 AD G 15. Seemed to be concerned with the problem of trading bread and E beer of various strengths. 16. In at least one source, provided a very accurate approximation to B 2 . 17. Were aware of and used the . BG 18. Provided demonstrative proofs for mathematical propositions. G

19. In most sources, used an approximation of   3 to determine the B of a circle.

2 E 1 20. In a number of sources, used the formula dd to determine 9 the . 21. Used a number system based on groups of 10 EG 22. Multiplied by doubling and adding. E 23. Divided by multiplying by reciprocals. B 24. Seemed to rely on reciprocal tables do to do some of their B arithmetic. 25. Dealt extensively with the notion of incommensurabililty. G 26. Used the notion of seked as a measure of steepness. E 27. Had trouble dealing with the fraction 1/7 in their number system. B 28. Distinguished between magnitudes and numbers G 29. Wrote all fractions (except 2/3) in terms of unit fractions. E 30. Could solve quadratic relations using . BG Section 3: Problems and Short Answers. Five points each.

31. Complete the following Egyptian division of 392 by 12, by A) marking the correct rows with an “X”, and B) adding up the appropriate entries in the left hand column.

112 224 448 896 16 192 32 384 X 1/3 4 2/3 8 X 32 2/3

32. Find 356 ÷ 3 in the Babylonian style by completing this multiplication:

5 , 56 ; ×0 ;20 18 40 140 1 58; 40 33. Describe the method of exhaustion. Give the names of two mathematicians who used or developed this method.

Eudoxus and Archimedes worth one point each; I also gave a point for Euclid.

Finding an unknown area or volume of a figure or solid by successively approximating it with figures (or solids) of known area/volume which were inscribed within it, and which could be modified to better approximate the area / volume step by step, eventually “exhausting” the area or volume of the unknown figure/solid. It was sometimes combined with circumscribed figure or solids, and sometimes combined with a reductio ad absurdum argument to prove the result was correct.

Note: I took a point off for being too specific, e.g. just talking about finding the area of a circle. I also took a point off for not capturing the idea of successively changing the inscribed figure to better and better approximations.

34. Describe what is meant by two magnitudes being incommensurable.

There is no third magnitude which measures, or divides evenly into, each of the two magnitudes.

Usually the only problem with this was some confusion over whether numbers were incommensurable (which of course, from a Greek viewpoint, they are not, because every pair of rational numbers has a common measure which is the unit fraction with their common denominator). 35. What is meant by geometrical algebra? Where do we read about it in Greek mathematical works? Give an example of a theorem of this kind.

I gave two point for expressing the idea that geometrical algebra was the use of geometry theorems to accomplish what we today would generally accomplish using algebra.

I gave one point for noting that it was found in Elements, Book II

I gave two points for giving an example. Valid examples are Propositions II-4, II-5, II-6, II-11, II-14, all given on pp. 63-64 of Katz. Others were also possible, such as the construction of the square root of x given x.

36. Describe briefly what the “method” was in Archimedes’ The Method.

There were several important parts to this, including:

1. It is a method of comparing two or volumes 2. It involved slicing the areas or volumes into segments or disks 3. It involved using the properties of the figures to... 4. ... compare distances and lengths along a line used as a lever. 5. It uses the Law of the Lever along with 3 and 4 above to get relative “weights” of the segments and disks. 6. It assumes the same relationships found in 5 apply for the whole area or volume, taking into account centers of gravity.

Of these, 2, 3-4 and 5 are most critical. Points were awarded based on how many of these points you made. You didn’t have to hit all 6 to get full credit. 37. Give three reasons ’ work in his Arithmetica might qualify him to be the Greek “Father of Algebra.”

1. He used symbolism. 2. He stated general rules for dealing with subtraction (i.e. “negatives,” but not really negative numbers). 3. He stated general strategies for reducing equations to forms with known solutions. 4. He gave solutions to equations, without recourse to geometry. His solutions were numeric, not geometric.

Any three of these four would work. Typically I deducted a point if you said something that just didn’t make sense.

38. Describe the Babylonian tablet called Plimpton 322 and briefly say why it is significant.

Description (4 points): It was a tablet with several rows and four columns. One column was just the row number. A second column gave the hypotenuse of a right (or the largest of a Pythagorean triple). A third column gave the smaller leg of the (or the smallest of the Pythagorean triple). The fourth column gave the square of the ratio of the hypotenuse and the longer side (the side not given); or alternatively, the square of the ratio of the largest to the middle of the Pythagorean triple. This corresponds to the square of the secant of the opposite the smaller leg.

(Optional but interesting and the source of my confusion – if you take the arccsc of the square root of the numbers in that fourth column, they correspond to growing steadily from about 45 to 58 degrees.)

The significance (1 point) is that it showed that 1) the Babylonians knew about Pythagorean triples and probably knew how to generate a large number of them, and/or 2) they new about right-triangle trigonometry in some form. Section 5: Slightly longer questions; 10 points each.

39. Regarding Euclid’s Elements:

a. Briefly describe the mathematical content of the Elements (include at least 5 major topics).

Plane geometry; inscribed and circumscribed; geometric algebra; ratios and proportions of magnitudes and of numbers; similarity; polygons and their construction; number theory, including primes, composites, etc.; ; polyhedra; method of exhaustion. (5 points for five of those.)

b. Why was this work so important? What “footprint” did it leave in today’s mathematics?

Some combination of:

1. It brought together most of the mathematics known at the time of its publication, supplanting all other sources. 2. It introduced an axiomatic approach or axiomatic system. 3. Many mathematicians used it as a basic text; they learned mathematics from it and learned what mathematics was like from it. 4. It became “the” pattern for how to do mathematics; we still use the basic structure of postulates/definitions/ theorem/proof today. 40. Briefly describe the sources we have for our knowledge of the mathematics done in each of the following early cultures. Be sure to address whether the sources are original, and if not, describe how close to the originals they are.

A. Egypt

The Ahmes or Rhind Papyrus and the Moscow Papyrus, along with a few others (2 points). These were mostly original, although some were copied from earlier manuscripts, they still come directly from the time of Ancient Egypt (1 point).

B. Babylon

A few hundred clay tablets, mainly problem tablets and table tablets (2 points). They were mostly original. Again, even though they may have been copies of other manuscripts, they were from the Babylonian period (1 point).

C. Greece

The sources we have are copies of copies at best, dating from hundreds of years after the Greek civilization. Many of them are Arabic or Latin translations. (3 points)

Everyone got one point for showing up to the exam.

41. Below are the counting words of the Minica Huitoto language, spoken in Colombia. Some information about the meanings of words is provided below as well.

1. daa 12. eíbamo mena 2. mena 13. eíbamo daámani 3. daámani 14. eíbamo fígoménarie 4. fígoménarie 15. eíbamo hubeba 5. hubeba 16. eíba enéfebamo da 6. enéfebamo daa (lit: ''one finger [is marked] 17.eíba enéfebamo mena on the other hand'') 18.eíba enéfebamo daámani 7. enéfebamo mena 19.eíba enéfebamo fígoménarie 8. enéfebamo daámani 20. eíba nagáfebe (lit: ''both feet'' ) 9. enéfebamo fígoménarie 21. eíba nagáfebe caipo onóbaimo da 10. nágafeba (lit: ''both hands [are marked ]'') 25. eíba nagáfebe caipo onóbai hubeba 11. eíbamo da (lit: ''one toe on the foot'' ) 30. eíba nagáfebe caipo onóbai nágafene

A. What can you suggest about the origins of counting from this language and other languages we have studied in class?

Many counting systems are derived from counting body parts, which leads naturally to grouping by 5's, 10's, and 20's. (5 points)

(If you didn’t discuss counting systems in general, but instead this one in particular, I docked you a point.)

B. What cycles are present in this counting system? Provide a few sentences of justification for your answer.

There are certainly 5 cycles in this system, since they keep referring to groups of 5 (hands and feet). The fact that 30 references 20 leads me to believe that there are 20- cycles, too. It is difficult to tell without more information whether there are any 10- cycles.

I pretty much gave everyone 5 points on this, as long as you mentioned 5 cycles and didn’t display other misconceptions. A Handy List of Greek Mathematicians

Apollonius Archimedes Diophantus Euclid Eudoxus Heron (or Hero) Hippocrates Of Hypatia Pappus Ptolemy Pythagoras