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Home , Percentage, Unit fraction

Oakland University MTE 3118 Project #7 Funny and DUE: Start of class on Tuesday, October 31

You should work on this project in groups of two or three, with each of the participants contributing approximately equally. Hand in one stapled-together write-up, with all of your names on it. There are a total of 100 points. This project will probably take substantially more time than some of the other projects; there is a lot to do here. You will need a good scientific with a square root button, a button, and a reciprocal (1/x) button. You can buy one at a dollar store for $1.

1. Continued fractions A continued is an expression of the form shown in this example:

1 3 + 1 5 + 1 1 + 1 1 + 2 Because of the way the fraction bar implies grouping, it should be clear how to simplify this very complicated looking fraction, namely from the bottom up. (a) (5 points) Simplify the expression shown here and write the answer as a simple fraction (i.e., as one natural number over another). It will be an improper fraction, of course. The rules for continued fractions are that each numerator is to be the number 1, and each number in front of a plus sign has to be a positive , with the exception of the very first number (the whole number part), which can be 0. Every positive can be put into form. We will do an example in class to see how. Basically, the idea is just to perform the indicated by the fraction to obtain a and remainder, then invert the remainder and repeat until the process terminates. The key point is that continued fractions are another way of representing rational numbers, just as ordinary fractions and decimals are ways of representing numbers. Each representation method has advantages and disadvantages, so there is no one “best” method. (b) (6 points) Find the continued fraction representation of the rational number 0.123 (that is, the fraction 123/1000). Make efficient use of your calculator if you use it, or do this with paper and pencil. (It goes without saying that for all parts of this assignment you need to show your work and explain things well.) Once we have a continued fraction, we can chop it off before any of the plus signs, to obtain approximations to the entire quantity. In the display shown above, for example, if we stop after 1 16 the 5, then we obtain 3 + 5 , or 5 . If we stop after the first 1 (before the next plus sign), we 19 35 obtain 6 . If we stop after the second 1, we obtain 11 . This is a sequence of better and better approximations to the value of the whole continued fraction (i.e., the answer you got to (a), above). (c) (6 points) List all the approximations to 0.123 obtained from the continued fraction you found in (b). Write them both in fractional and form, and note how fast they converge to 0.123.

1 In some senses (which can be made precise), these approximations provide the “best” possible fractions to approximate a given number. Here is an application. On November 18, 1997, on the NASDAQ stock exchange 2028 stocks fell in price and 1417 rose. A radio newscaster who wants to give a brief, easy-to-understand indication of the between these numbers might say something like “Decliners led advancers today by a margin of 5 to 4” or whatever the approximate ratio really is. The true ratio of course is “2028 to 1417”, but those numbers are far too big to be useful in an oral report. (d) (7 points) Find what you think is the most appropriate ratio to report for this data. Choose from the first few continued fraction approximations of 2028/1417, and give an argument as to why your choice of numbers is neither “too small” (not accurate enough) nor “too big” (numbers hard for listeners to comprehend) but “just right”. Another way to report this information would be for the newscaster to say something like “of the stocks that changed price today, 59% declined and 41% advanced” (verify that these are the right figures for our data). Which do you think is the better way to report this, in the ratio form you just found or this form? Why? Now comes the fun part. If we start with an irrational real number, rather than a rational number, and try to find the continued fraction for it, the process will never stop, and the fraction will just keep going down the page forever! That’s okay, though, just as we get infinite decimal representations of certain real numbers (like π = 3.14159265 ...). The natural numbers appearing before the plus signs are analogous to the digits in the decimal expansion. They determine the exact value of the real number when you take all of them (i.e., continue it forever), and by stopping at any place along the way, you get an approximation to the exact value (just as, for instance, 1 0.333 is an approximation to 3 = 0.333333 ... = 0.3, and 3.14 is an approximation to π). We will do an example in class to see how to get this infinite continued fraction representation. √ (e) (9 points) Find the continued fraction decomposition of 28 = 5.291502622 ... to about 12 levels, and find the nice pattern that develops. (It is a theorem that the continued fraction form for the square root of a nonsquare whole number always eventually repeats, just like the decimal expansions of a rational number repeats.) In doing this work, you must be careful not to reenter any values into your calculator! If you do, then you will almost certainly get a wrong answer due to round-off errors. Have your calculator start with what it thinks is the square root of 28; it will probably display fewer decimal places than it actually keeps internally. As always, show your work. (f) (9 points) Find the continued fraction decomposition of π = 3.14159265358979 ... to six or seven levels, and see whether any nice pattern develops. Notice that the “digits” in the continued fraction, i.e., the numbers in front of the plus signs, can be more than one digit long—you will find a 15 here, for instance. (Again, be careful not to reenter any values into your calculator.) Then find the first five rational approximations to π obtained from this continued fraction [HINT: One 355 of them is 113 ], writing them both as an improper fraction and as a decimal to as many decimal places as your calculator will give you. Notice how fast or slowly the answers converge to the true value, and whether they are too small or too big. Comment on this convergence and pattern. A search of the Web (using Google) of “continued fractions” (with the quote marks) will lead you to some good references to help with this project (such as a way to check that your answers are correct). You might want to use Wolfram Alpha (www.wolframalpha.com) for checking your work, a resource that we will explore in a future project. As always, feel free to come for help.

2 2. Farey fractions Everyone knows that you don’t get the sum of two fractions by adding their respective numerators 2 3 5 and denominators. For example, 5 + 7 6= 12 . On the other hand, you do get something; the question we want to explore is, what do you get? Let us call this the “Farey operation”, named after the person who investigated it. (John Farey (1766–1826) was actually a British geologist, not a mathematicaican.) It is sometimes called “baseball ” for a reason we will discuss in class. We will think of constructing new fractions by this process, on a day-by-day basis. 0 1 Suppose we start on Day 1 with the simplest two fractions, namely 1 and 1 . On Day 2 we take the neighboring fractions we had on Day 1, and perform this add-the-numerators-and-add-the- 1 denominators operation on them. This gives us one new fraction, namely 2 . On Day 3, we again perform the Farey operation on neighboring fractions that have been formed up to that time: when 0 1 1 1 1 2 we do it to 1 and 2 , we get 3 ; when we do it to 2 and 1 , we get 3 . You should check that on 1 2 3 3 Day 4 we get 4 , 5 , 5 , and 4 . Remember to apply the Farey operation only to previously formed fractions that are next to each other on the number line. The fractions we get are called Farey fractions. (a) (8 points) Continue this process through Day 7, and display the results in a neat form. You will want to record the fractions you get, in order, and you should plot them carefully (to ) on a number line (a calculator will come in handy here). Note that they are not uniformly spaced. (b) (8 points) You will no doubt notice that the Farey operation applied to a pair of positive fractions results in a fraction strictly between these two. Prove algebraically that this is always so. (c) (8 points) There are other things you should notice about all the Farey fractions you get. You are not asked to prove these things, but in fact they are true. For one thing, they are all in lowest terms. For another, you eventually get every fraction between 0 and 1. Also, on any given day, you get fractions with various denominators, and fractions with a given denominator appear over a span of days. Look at the sequence f1, f2, f3, ... , where fi is the fraction that appears on Day i having largest denominator (actually there are two such fractions—take the one that is 1 2 ≤ 2 ). For example, f4 = 5 . Find a pattern to them. Extend the list beyond f7 by following the pattern, and find the value (to five or six decimal places) that fi converges to (gets close to) as i increases.

3. Egyptian fractions (15 points) The ancient Egyptians were fascinated with the possibility of writing a fraction as the sum of unit fractions (i.e., those with numerator equal to 1), all of them different. So for example, we can write 119/240 as 1/3 + 1/10 + 1/16 (check this to make sure I’m not lying to you!). It can also be written as 1/3 + 1/7 + 1/51 + 1/28560); use your calculator to get decimal approximation to convince yourself of this one, too. It’s easy enough to write down a bunch of unit fractions and find their sum, but can we go the other way? If I asked you to find some distinct unit fractions that added up to 119/240, how could you do it? We won’t deal with the problem in quite this level of generality or difficulty. Instead, we’ll look at the special case of writing a as the sum of two other unit fractions. The simplest example is 1/2 = 1/3 + 1/6. Note how we’ve taken a special case of the problem at hand in two ways: first, we’ve restricted the fraction we are trying to decompose (requiring that it itself be a unit fraction, rather than allowing it to be any old fraction); and we’re asking for just two summands, rather than an arbitrary number of summands.

3 You job is to explore this problem. Find lots of solutions. Try to say something in general about how many different solutions there are for a given unit fraction. For example, we can write 1 1 1 1 1 = + = + . 10 11 110 12 60 Are these the only ways to decompose 1/10, or are there more? Are there an infinite number of ways to decompose 1/10, or just a finite number? Try unit fractions other than 1/10, too. Is it always just a finite number of decompositions or might there be infinitely many in the case of some unit fractions? If the former, can we make that finite number as large as we please by taking the right denominator of the fraction we are trying to decompose? Write up what you figure out, in a nice, coherent, readable, understandable essay, with lots of examples. You may want to include a little algebra, to make things easier to talk about. Think of it as explaining the discoveries you have made to another person at your level of mathematical ability (or to a very bright 11-year-old, if you’d rather), who hasn’t considered the problem before. It is not against the rules of this assignment to look at outside sources (books, journal articles, Web pages, etc.) that may discuss the problem. If you do, however, you must still write up your essay in your own words, and you must include a citation to the sources you used.

4. Base Six “decimals” Just as we can use base six to represent whole numbers, using place values that are positive powers of six rather than positive powers of ten, we can use base six to represent real numbers in exactly the same way that we use base ten. Because the word decimal has the notion of ten-ness in it—the d-e-c is the same Latin root as in such words as decade (ten years) and decimeter (a tenth of a meter) and December (the tenth month of the year)—we’d better use a different word, and one obvious choice is “heximal”. Each column to the right of the heximal point represents the next 1 1 1 negative power of six. So, for example, 0.13 is 1 · 6 + 3 · 36 = 4 . You will see in doing this problem that determining which fractions have finite expansions will depend on the base.

1 1 1 1 (a) (9 points) Write all the fractions 1 , 2 , 3 , ... , 21 (I’m using base six numerals here, so there should be thirteen values in your list) in their heximal equivalents. You probably want to do this using long division with paper and pencil. (That is, do long division in base six .) As a check, you should find that five of these thirteen numbers have infinite repeating heximals, and the other eight have finite heximal expansions. Also, the length of the repeated portion is always a of the number of positive less than but relatively prime to the denominator (which is one less than the denominator if the denominator is prime), and in one of these thirteen numbers it’s actually quite long. Make sure in the case of the infinite repeating ones to show what gets 1 repeated; for example, you will find that 14 = 0.03333 ... = 0.03. [Here’s another hint for double- checking your work: There’s a rule by which we can determine whether the heximal for a given fraction (assumed to be in lowest terms) will be finite or infinite, very similar to (and analogous to) the corresponding rule for base ten. See whether you can discover the rule.] (b) (10 points) The decimal expansion of π starts 3.14159265358979 ... . Because it is irrational, the expansion neither stops nor repeats, and the same is of course true about its heximal expansion. Compute the heximal expansion of π accurate to at least eight heximal places. (There are several different ways to do this; you need to figure out at least one of them. This may be the hardest part of the project—spend some time trying to figure it out on your own, and come to me for help if you need it. Explain what you’re doing and why this gives the right answer.) Check your answer (by converting it back to base ten) to make sure it’s correct.

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