« August 2007 | Main | February 2008 »
October 14, 2007
TMA 1.1
This was a gobbet question concerning a problem from ancient Egyptian mathematics.
‘SB’ indicates a reference to the source book, i.e. Fauvel and Gray, and “SB1.D1” means extract D1 from chapter 1. In this case the extract comes from A. B. Chace's 1927 translation of the Rhind Papyrus.
A quantity and its 1/2 added together become 16. What is the quantity?Assume 2.
\ 1 2 \ 1/2 1 Total 3. As many times as 3 must be multiplied to give 16, so many times 2 must be multiplied to give the required number.
\ 1 3 2 6 \ 4 12 2/3 2 \ 1/3 1 Total 5 1/3. 1 5 1/3 2 10 2/3 Do it thus:
The quantity is 10 2/3 1/2 5 1/3 Total 16.
(untitled)
Problem 25 of the Rhind Papyrus consists of the statement of a numerical problem requiring an unknown quantity to be determined; a series of calculations that arrive at an answer; and a final calculation demonstrating that the value found satisfies the problem statement. It is very similar in form—both the nature of the problem and the method of its solution—to Problem 24 (SB1.D1).
The problem to be solved is stated as: “A quantity and its 1/2 together become 16. What is the quantity?” In modern terms we might call the quantity x, and say “x + 1/2 x = 16. What is x?”.
A modern way of solving this would be to manipulate the equation until it gave an answer: re-expressing x + 1/2 x as 1 1/2 x, and then dividing both sides by 1 1/2. However, the approach taken by the scribe is to make a “guess” at the answer (2), and then use a comparison between the total yielded by the guess and the one given by the question in order to adjust the guess and so get the correct answer (10 2/3).
To describe the approach in modern terms we might call the guess x′,
and say that the scribe relies on
| (x + 1/2 x) ÷ (x′ + 1/2 x′) | = | x ÷ x′ |
| x | = | x′ × (x + 1/2 x) ÷ (x′ + 1/2 x′). |
| x | = | 2 × 16 ÷ (x′ + 1/2 x′) |
| = | 2 × 16 ÷ 3 | |
| = | 2 × 5 1/3 | |
| = | 10 2/3. |
The scribe performs the calculations in these lines (using the normal Egyptian method for multiplication and division) in the order given. The principle I said the scribe relied upon is expressed: “As many times as 3 must be multiplied to give 16, so many times 2 must be multiplied to give the required number.”
Assuming 2 is not a “good guess” in the sense of being close: the method doesn't require this. It looks rather that it was chosen so that the total given by the first step of the calculation is an integer. Dividing by a non-integer would have been cumbersome using Egyptian notation for fractions: had this been easy the problem could have been solved more directly.
The final check most likely served to guard against simple mistakes in the calculations, and maybe also against the wrong method being used.
Chace (SB1.D6) and Toomer (SB1.D7) differ as to whether the Rhind Papyrus shows mathematics pursued for its own sake or is a textbook for scribes. Problem 25 looks more like a textbook than research. The problem is rather easy: possibly too easy to justify the method used for its solution; certainly too easy to require that every step of the calculation be set down. Rather, it is the method of solution that is the point of the text, and the particular problem given is being used to illustrate it. (The final check illustrates good practice.)
The text shows that the Egyptians were comfortable with number as an abstract notion: the problem deals in quantities, but the text does not specify of what; and the intermediate result 5 1/3 is dimensionless. More abstractly still, the scribe was able to consider a quantity without knowing what is was.
However, although this text seems to be for explaining a method for solving a whole class of problem, it does so purely in terms of a particular example. There is no evidence that the scribe could have described the method being used in general terms, let alone investigate its validity and scope of applicability
This limited level of abstraction fits with the relative lack of development in Egyptian mathematics. The Egyptians could devise and codify fairly abstract methods for solving concrete problems, but did not take the next step of pursuing abstract problems.
Thoughts
There's not very much to say here. At the time I did wonder if my account of why the Egyptian method worked was a bit unnecessary: however, I'd decided (rather arbitrarily) that equations shouldn't be included in the word count, so cutting that section wouldn't have given me much scope for doing anything else extra.
The method used was later called “the rule of single false position”, with there being a rule of double false position for slightly more complicated problems. ‘Later’ meaning the Middle Ages, when these terms appeared in treatises on commercial arithmetic (such as the works of, uh, at least one of Fibonacci and Luca Paccioli). The fact that this method had a name implies that it was still in use then, although I don't know how long it continued to be used. I suspect that double false position would have had a longer life than single false position. I know that double false position was described in Robert Record's popular English arithmetic The Ground of Artes of 1543.
Both single and double false position can both be thought of in terms of a straight-line graph, where the question is to find an input that will produce a give output. The problems soluble with single false position are the ones where this line passes through the origin, which is why only one guess needs to be made. Double false position can cope with problems where this is not the case, but requires two guesses to be made.
The main benefit of single false position is that it avoids dividing by a fraction; so I imagine that it would lose its appeal as reasonable notation for vulgar fractions exists. Whereas double false position avoids some mildly more involved algebraic re-arrangement, and so might continue to be convenient even when other ways of solving such problems are available.
Posted by robin2 at 07:11 PM | Comments (0)
MA290
I've recently completed my first Open University course: MA20 Topics in the History of Mathematics. I remember watching some of the TV programmes for this—which are supplied on DVD these days—when I was a kid. I had already picked up the course book—The history of mathematics: a reader eds. John Fauvel and Jeremy Gray— a number of years ago in a book sale, and so when I found at that this was the last year that the course was running I really had to sign up.
I found doing the assignments (TMAs) an interesting experience. I've always found writing—even short things—a fairly painful process. Here the problem was compounded rather by the assignments having word limits. However, there were points where I thought I was getting the hang of it, and I started to get a sense of the meaning of the word ‘copy’ as a mass noun: not to do with quantity of words so much as a certain detachment, slabs of verbiage to be shoved around.
I'm going to put up my TMA answers here, as I want to mull over a couple of them some more.
Posted by robin2 at 05:38 PM | Comments (0)