The lecture was divided up into three parts. The first part was about Leonardo of Pisa, better known as Fibonacci. Then we discussed Jordanus De Nemore who might have been a woman. Finally we looked at Nicole Oresme who dealt with ratios.
The rabbit problem was very interesting to me because of the similarities to Newton's Method (or the Newton-Raphson Method). In that situaution we are looking for the approximation of the root of an equation, and the equation that we iterate is x_2 = x_1 - f(x_1) / f'(x_1). The answer to the first part is included in the second part. Maybe a better way to say this is that computation of the next iterate requires complete knowledge of the present iterate. It seemed like that this principle was similar to what we used in the rabbit problem.
Wouldn't the bird buying problem by Fibonacci be considered an indeterminate problem because of the fact that there are so many solutions?
Student lecturer Kristin Jensen started off today's lecture by discussing Leonardo of Pisa, also known as Fibonacci. Leonardo wasn't one of history's great mathematicians, but he was one of the best during his time. Ms. Jensen introduced three problems during her discussion of Leonardo. The first involved finding a way to buy thirty birds with thirty coins, given the price of each of three kinds of birds. Rather than use a linear system as we would today, Leonardo devised a clever solution, finding a way to buy three birds for three coins and another way to buy five birds with five coins. He then used multiples of these solutions to make each bring fifteen birds for fifteen coins. The last problem was a rabbit breeding problem, whose solution turned out to be what we now call the Fibonacci sequence.
Ms. Jensen then discussed the mathematician Jordanus de Nemore, who she said was considered the best mathematician of the middle ages. Little is known about the life of Jordanus. Ms. Jensen even commented that he may have been a woman. Jordanus put forth a number of propositions that appear to be more mental exercise than useful discovery.
Finally, Ms. Jensen introduced the mathematician Nicole Oresme. Oresme studied ratios, and advanced algorithms for multiplying and dividing them. These algorithms are the same thing we teach today for multiplying and dividing fractions. Dr. Barsky then finished the discussion by showing how to devise an algebraic equation to represent the Fibonacci sequence. During this derivation, we saw a reappearance of golden ratios.
This derivation was interesting to me because the only way I have ever seen before to derive Fibonacci numbers was either to sit down with pencil and paper and do it, or write a computer program using a recursive function. For obvious reasons, the former is unrealistic if we need, say, the 5000th Fibonacci number. But the second is really not much better when the derivation calls for a number deep into the sequence. A computer has limited memory space, and when you consider that each call of the function requires two recursive calls, which each require two more recursive calls, and so on until we reach the base case, these recursive calls can deplete the memory of the computer eventually.
The presentation given today by Kristin Jensen, after Dr. Barsky's commentary on today in math history, held to the main concept that mathematics during this period of time had a lot to do with algebra. The topics covered consisted of Leonardo of Pisa (better known as Fibonacci), Liber abbaci, the practica geometriae, the Liber quadratorum, the Bird problem, the Lion and the Pit, the Money Problem, the Rabbit Problem, Jordanus De Nemore, Proposition I-1(Jordanus's version and Diophantis's version), Proposition II-18, Nicole Oresme, Oresme's study of ratios, and a return to the Rabbit Problem with Dr. Barsky.
Among the topics covered, the topic of Fibonacci interested me the most. According to Katz, Fibonacci's father had been a Pisan merchant who had business dealings in Bugia and the North African Coast. Here Fibonacci spent a large portion of his childhood learning Arabic and studying mathematics of Islam. Apparently, the Islamic mathematicians had learned much knowledge from India who in turn had learned much knowledge from China, however no substantial proof exists of the latter. Also according to Katz, Fibonacci used many of the methods of al-Kwarizmi, and he often took problems word for word from certain Islamic mathematicians. Katz also mentions that the majority of the problems show Fibonacci's own creativity. I found this to be absolutely correct. We can see a major shift from Islamic mathematics in his problems, such as the Rabbit Problem. Although we would not do algebra this way, as Dr. Barsky had mentioned, Fibonacci makes a step from using geometry to confirm his algebra to using algebra by itself. We are used to manipulating variables, so it astounds me as to how he made his calculations work so he could understand them. We are still speaking of Medieval times here. This time period also marks a point where mathematicians are learning more from each other.
Today's discussion began with a brief mention of Hans Arnold Heilbronn. He was born on this day.
Kristin began talking about Leonardo of Pisa. Today he is known as Fibonacci and is most famous for the Liber abbaci or Book of Calculations. He gained many sources for this book from the Islamic World.
After briefly introducing Leonardo, Kristin presented his problem of buying birds. He asks how to buy 30 birds for 30 coins, if partridges cost 3 coins each, pigeons 2 each, and 2 sparrows for one coin. A solution was presented.
Next we saw the problem of the lion in the pit. A pit is 50 feet deep. If a lion climbs up 1/7 of a foot each day, and drops 1/9 of a foot in the evening, how long will it take him to get out of the pit? Fibonacci uses a version of false position for the solution. We then looked at the money problem. Fibonacci avoided having to directly solve this system of equations through the introduction of a new variable. Kristin then discussed his most famous problem: the rabbit problem. How many pairs of rabbits can be bred in one year from one pair? In listing the sequence of numbers, it was found that each number is the sum of the previous two numbers. This sequence is now known as the Fibonacci sequence.
Kristin talked about Jordanus de Nemore. His algebra is done close to the way in which we do ours today, however the notation does differ. There was not much known of this mathematician. His major work on algebra, De Numeris Datis (On Given Numbers), is an analytic work on algebra. The class was shown proposition II-18.
The last thing Kristin discussed was Oresme's study on ratios, compounding (multiplying) ratios and dividing ratios. It actually just looks like multiplying and dividing fractions.
Dr. Barsky stepped in for the last portion of the class, of course he would never let the class out 20 minutes early (ha ha). I'm glad he stepped in because in calculus I, I will never forget one specific problem that was on my final. All of a sudden, there it was on the chalkboard. It was the problem that had the answer one plus or minus radical two all divided by 5: the golden ratio. It was interesting to find out that the answer was specific to something. This golden rule is buried inside of the Fibonacci sequence.
There were three main mathematicians that Kristin covered: Leonardo of Pisa, Jordanus De Nemore, and Nicole Oresme. Leonardo was better known as Fibonacci. He lived in the late 12th century. He spent his life in Pisa learning mathematics and wrote many books. He was a great mathematician. Jordanus is a very unknown mathematician. He is a contemporary of Leonardo. I was able to share what I had learned about him from my paper. Finally, Nicole Oresme lived from 1320 to 1382. His life was filled with many accomplishments. He studied ratios and wrote Algorismus propotionum and De propotionibus propotionum. He defined ratios as a mutual relation between two quantities of the same kind. They didn't write fractions like we do today, but in a ratio from (a:b). He knew how to compound or multiply fractions together and how to divide them.
The most interesting part of the presentation were some of Leonardo's problem that Kristin showed. I know I would have never got the lion in the pit one right either. Leonardo seemed very advanced for his time. How he set up the money problem shows this. Instead of working with two equations and two unknowns he used substitution to reduce the problem to a sequence of problems each of which only involved one unknown. The rabbit problem was the best. The Fibonacci sequence looks so simple, but finding an equation to represent F(n) was very difficult. It would seem Leonardo was able to find out the number of rabbits up to any amount he wanted, but he had to go month by month. I didn't see anything in the book which said he was able to just be given some month and plug the number into a formula like the one we figured out in class. I don't think they were able to deal with exponents greater than two at the time and our equation had an nth power.
The lecture for October 8th centered on mathematics in medieval Europe and covered the following mathematicians: Leonardo of Pisa, Jordanus De Nemore, and Nicole Oresme. In covering Leonardo of Pisa, Ms. Jensen reviewed his allocation problems (The 30 birds for 30 coins problem), the Lion and the Pit problem, and population problems centering on the reproduction of rabbits. Also, Jensen noted that Leonardo of Pisa was granted a yearly stipend for teaching and community service in 1240, and wrote the book Liber Abbaci. Next, Jensen covered Jordanus De Nemore, who, she noted, was the best mathematician of the middle ages, wrote De Numeris Datis and may have possibly been a woman. De Nemore worked on the beginning stages of Algebra, and his/her work was to show that given certain information, other information was then determined. This is apparent in his/her proposition I-1, where it is stated: "If a given number is divided into two parts whose difference is given, then each of the parts is determined." Finally, Jensen briefly covered the work of Nicole Oresme (1320-1382), who studied ratios in detail. Covered in the lecture were the multiplication and division of fractions (ratios).
Of particular interest to me was the subject of Jordanus De Nemore, mainly for two reasons. First, I was intrigued that we know so little about the personal life of this individual, so little, in fact, that we don't even know his/her gender for sure. One would think that if this person's contributions could last so long as to still be covered 700 years later, people of that age would have left us better information. Also, De Nemore's use of letters to represent arbitrary numbers (as is covered in the text on page 286) while certainly being a very small step in terms of usage at this time, was a necessary step for mathematics to have evolved as far as it has at the present.