MATH 15, CLS 15 — Spring 2020 Problem set 1, due 24 January 2020 General instructions for the semester: You may use any references you like for these assignments, though you should not need anything beyond the books on the syllabus and the specified web sites. You may work with your classmates provided you all come to your own conclusions and write up your own solutions separately: discussing a problem is a really good idea, but letting somebody else solve it for you misses the point. When a question just asks for a calculation, that’s all you need to write — make sure to show your work! But if there’s a “why” or “how” question, your answer should be in the form of coherent paragraphs, giving evidence for the statements you make and the conclusions you reach. Papers should be typed (double sided is fine) and handed in at the start of class on the due date. 1. What is mathematics? Early in the video you watched, du Sautoy says the fundamental ideas of mathematics are “space and quantity”; later he says that “proof is what gives mathematics its strength.” English Wikipedia tells us “mathematics includes the study of such topics as quantity, structure, space, and change,” while Simple English Wikipedia says “mathematics is the study of numbers, shapes, and patterns.” In French Wikipedia´ we read “mathematics is a collection of abstract knowledge based on logical reasoning applied to such things as numbers, forms, structures, and transformations”; in German, “mathematics is a science that started with the study of geometric figures and calculations with numbers”; and in Spanish, “mathematics is a formal science with, beginning with axioms and continuing with logical reasoning, studies the properties of and relations between abstract objects like numbers, geometric figures, or symbols” (all translations mine). These descriptions don’t all say quite the same thing, but are they consistent with each other? What different impressions of mathematics do they give? Are there activities that would be considered mathematical under one of these definitions but not another? Discuss, in an essay of about 300–500 words. 2. Sources for the history of mathematics. Our knowledge of Egyptian mathematics depends mainly on a couple of hand-written texts on papyrus; for the Ahmes papyrus (or Rhind papyrus), we don’t even have the entire roll, and the pieces we do have are in several different museums. We know Babylonian mathematics principally from a collection of hand-written texts on clay tablets, and, once again, some are broken or otherwise damaged. • Are these sources sufficient to give us the whole picture of Egyptian or Babylonian mathematics? Explain. • What problems arise from the present state of the sources we have? • Nowadays, mathematicians write mathematics on computers, and the first publication of a new mathematical text is often at a web site called the arXiv (arxiv.org, pronounced “archive”). Later, the text may appear in a scholarly journal or a book, which is probably published on line and may or may not also appear in print. Suppose you are a historian of mathematics living in 3020 and looking back a thousand years to the 21st century. What sources will you find? Will they be complete? Will you be able to read them? 3. Explanations, like those in the ancient texts. Multiply 231 and 732. Don’t use a calculator or computer, but write out the multiplication by hand the way you were probably taught in third or fourth grade. Now, explain in words how to do that multiplication, or any multiplication like that one. How do you know you’ve got the right answer? And how do you know the method you used always gives the right answer (assuming you haven’t made a mistake)? MATH 15, CLS 15 — Spring 2020 Problem set 2, due 31 January 2020 1. Reading Egyptian numerals. Write the following numerals in the modern Western system (the Hindu-Arabic system). a. 2|| b. 22 |||| 222 |||| c. 3 22 d. 4322| e. 433|||| f. r ||| g. r 2| h. r 22 2. Writing Egyptian numerals. Write the following numerals in the Egyptian system. a. 37 b. 94 c. 125 d. 1852 e. 2020 f. 1/2 g. 1/10 h. 1/17 Problems 3, 4, and 5 are on the reverse. 3. Egyptian fractions. Here are some ordinary fractions. Write them as an Egyptian would, that is, as sums of unit fractions; you can also use 2/3 if it’s useful. First write as actual fractions, then as over-barred numerals. Write at least two of them using Egyptian numerals, too. For example, given 3/4 you would write 1/2 + 1/4 (actual fraction), then 2 4 (over-barred), then r r (Egyptian style). || |||| Remember you can’t use the same unit fraction twice! a. 5/6 b. 5/8 c. 2/5 d. 2/15 e. 3/10 Optional additional question, a bit more challenging: There is always more than one way to write a fraction Egyptian-style. Write at least one of the numbers above as a different sum of Egyptian fractions. For example, 3/4 = 1/2 + 1/4 but you could also write it as 2/3 + 1/12 or 1/2 + 1/6 + 1/12, or any number of other ways. Is one version “better” than another? Why? (This means you have to decide what “better” means here, of course.) 4. Egyptian calculation. Multiply 231 and 732 using the Egyptian method: pick one or the other number to start with, match it up with 1, double each column, and so on. This is the same multiplication you did on the last problem set; which way is easier? Of course you got the same answer in this example; can you determine if the Egyptian method will always work? Can you say why or why not? 5. Social context of Egyptian mathematics. What does the content of the Ahmes Papyrus (and of the other Egyptian texts) tell us about who does mathematics in Egypt, and about what they do with it? Think about what kinds of problems appear in the sources, but also about what’s in the text other than problems. Discuss, in about 300–500 words. MATH 15, CLS 15 — Spring 2020 Problem set 3, due 7 February 2020 1. Egyptian geometry: volumes. Problem 41 of the Ahmes papyrus asks for the volume of a cylindrical granary whose diameter is 9 cubits and whose height is 10 cubits. Here is how Ahmes works it out. Take away 1/9 of 9, namely 1; this makes 8. Multiply 8 times 8; this is 64. Multiply 64 times 10; this gives 640 cubic cubits. Add 1/2 of it to it; that makes 960, which is the content of the granary in khar. Take 1/20 of 960, which is 48: then 4800 hekat of grain go into the granary. You’ll need to know that a khar is a measure of volume, 2/3 of a cubic cubit. There are 20 hekat to a khar, and as you know hekat is the usual volume measure for grain. What is Ahmes actually doing here? Where does he get the numbers he uses? His method does not actually give the right answer: why not? Is it pretty close? (You’ll need to compare his result to the result you’d get if you use the actual formula for the volume of a cylinder; if you don’t remember it, the Wikipedia article “Cylinder” will tell you.) Consider a similar problem (not in the papyrus): suppose the granary has a diameter of 3 cubits and a height of 6 cubits. Work out the calculation using the same method as in the papyrus, and then give the actual answer. 2. Babylonian numerals: the base–60 system. The following are Babylonian numerals in the conventional transliteration. Express them as ordinary numerals in the modern Western system. a. 1, 2 b. 1; 2 c. 0; 1, 2 d. 1, 4, 7 e. 1, 4; 7 f. 1; 4, 7 g. 5, 34 Here are some ordinary numbers; write them in base-60 in the conventional transliteration. If you want, you can also draw the cuneiform symbols. a. 35 b. 350 c. 65 d. 119 e. 5 1/2 f. 6 2/3 You will not be surprised to find out that there are more problems on the reverse. 3. Babylonian numerals: cuneiform. Write the following cuneiform numerals in the conven- tional transliteration, and then as ordinary numerals. Assume they are all whole numbers. The numeral is 3,5 if it represents a whole number, but the numbers transliterated as 3;5 and 0;3,5 would be written exactly the same way. For that matter, 3,5,0 would also be written this way. Return to the above list and interpret each number as a whole number plus a fraction (like 3;5): transliterate and then write in our system. 4. Babylonian arithmetic: base 60. In our base-10 system, you can look at a number and tell whether it is divisible by 2, or by 5. How do you know? What about base 60? Is there an analogous set of divisibility tests? Can you look at a base-60 num- ber and say at once whether it is divisible by 2? What other possible factors can you notice immediately using a test of the same type? For concreteness, apply your tests to the following base-60 numbers. For example, if you were working in base 10 and you were given 15, you’d say it is divisible by 5, and it isn’t divisible by 2.