Fibonacci (Pisa!)

Leonardo Pisano, is one of the more well known mathematicians from the Middle Ages. He was responsible for spreading the Hindu Arabic number system in to Europe, and then for the Fibonacci sequence. His father, Gugliemo Fibonacci was an Italian merchant who ran a trading post in Bugia, North Africa. He learned the Hindu Arabic number system there, and from there was inspired to learn more about the Hindu Arabic system. He traveled to Egypt, Syria, Sicily, Provence, and Greece learning from the masters of mathematics there.

After ending his travels he returned to Pisa around the year 1200AD, where he was born. There he wrote many important texts about mathematics. He revived ancient skills of mathematics, as well as making major contributions to the mathematics of the time. Fibonacci is best known for his work Liber Abaci published after he returned to Italy in 1202. Liber Abaci is what introduced Europe to Hindu Arabic numbers, and formed the base of our current number systems.

Liber Abaci was not aimed towards specifically high thinkers, or towards other mathematicians, it was aimed towards tradesman and the public to convince them of the benefits of the Hindu Arabic number systems. The book was split in to three sections, the first portion is about the Hindu Arabic numbers, the second is focused on merchants and how the new number system could assist them. These specifically relate to “price of goods, how to calculate profit on transactions, how to convert between the various currencies in use in Mediterranean countries, and problems which had originated in China. (Leonardo – Gap System)”

The third portion of Liber Abaci is what Pisano, most famously remembered as Fibonacci, is famous for. The Fibonacci sequence. The Fibonacci sequence was proposed through the following question:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

The sequence that you end up with when you work it out, is as follows:

          1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …

The first number was originally omitted in the first printing of Liber Abaci.

The Fibonacci sequence is found by taking the sum of the two previous numbers. So, 1+0=1, 1+1=2, 1+2=3, 2+3=5, and so on and so forth.

This sequence is also seen very frequently in nature, and in other sciences.

Sources: 

http://www.gap-system.org/~history/Biographies/Fibonacci.html

http://en.wikipedia.org/wiki/Liber_Abaci

http://en.wikipedia.org/wiki/Fibonacci

                                                                                                   

Rene Descartes

Rene Descartes is important in two fields, math and philosophy. He was one of the earliest modern philosophers to try and defeat philosophical skepticism. In mathematics he provided Isaac Newton and Gottfried Leibniz much of the basis for their calculus. While he is very well known for his philosophical works, I’ll be focusing mostly on his life, and his works on mathematics.

Rene Descartes was born in La Haye en Touraine, France in March of 1596, to Jeanne and Joachim Brochard. His mother died when Descartes was a mere one year old, and his father was a member of parliament in La Haye. At the age of ten he entered the Jesuit College of La Fleche between the years of 1606 and 1614. After his studies there he proceeded to go to the University of Poiters and earn a baccalauréat (allowing for further studies at university) and his licence in law in 1616. This was mainly done because his father wanted him to become a lawyer. In his autobiographical and philosophical publication Discourse on the Method he states that he entirely abandoned the study of letters. He wanted no knowledge other than what could be found through self discovery, or through books around the world. He spent the rest of his youth travelling and visiting leaders and armies of foreign countries and figuring out ways to profit (mentally, not financially) from the experiences. In 1618 he travelled to Holland to serve in Prince Maurice of Nassau’s army. With that army he went to Germany.

Dutch Mathematician Isaac Beeckman was one of the major influences on Descartes. He would pose number problems with him, to stimulate his mind. He wrote his first work of any substance in 1628, entitled Regulae, or Rules for the Direction of the Mind. It was not actually published until 1701.

As mentioned earlier, Descartes provided groundwork for Newton and Leibniz through his Discourse on the method. This is monumental in the fact that this groundwork was simply an example in Discourse, and not a focal point of the work. His most enduring legacy? The Cartesian coordinate system which is most commonly used for graphing equations, and plotting lines. He is also the father of standard notation, which uses superscripts to represent powers and exponents indicating squaring, cubing, or higher powers.

Descartes died of pneumonia in Stockholm, Sweden shortly before the publication of his final manuscript, Passions of the Soul.

Sources:

http://oregonstate.edu/instruct/phl302/philosophers/descartes.html

http://en.wikipedia.org/wiki/Ren%C3%A9_Descartes

http://en.wikipedia.org/wiki/Discourse_on_Method

Nikolai Lobachevsky

Mathematics took a major leap in the 19^th century.  Things didn’t stay as grounded in basics, and moved more towards abstract math. For example, Nikolai Lobachevsky introduced the idea of hyperbolic geometry(also known as Lobachevskian Geometry), in which it disproves Euclid’s Fifth postulatestated by John Playfair as  “At most one line can be drawn through any point not on a given line parallel to the given line in a plane.” The Kazan Messenger  published Lobcahevksy’s paper A Concise Outline of the Foundations of Geometry, but the paper was unfortunately rejected by the St. Petersburg Academy of sciences for publication there.

So, who was Lobachevsky? He was born in Nizhny Novgorod, Russia, the fifth largest city in Russia, to Ivan Lobachevsky, and Praskovia Lobachvskaya. His father was a clerk in a land survey office, and died in 1800, when Nikolai was a mere seven years old. His mother moved the three sons to Kazan near the border of Siberia. While living in Kazan, Lobachevsky went to Kazan Gymnasium (a sort of advanced High School), after finishing there in 1807 he began his collegiate studies at Kazan University. He had the intentions of studying medicine originally, but was highly influenced by Johann Christian Martin Bartels to move towards mathematics. Johann Bartels was at one point in time a professor of Carl Friedrich Gauss, and was still in contact with Gauss.

Lobachevsky graduated in  1811 with a masters degree in physics and mathematics. He proceeded to become an instructor at the university, becoming a full fledged professor in 1822. Lobachevsky was wll renowned as a professor and students said that his lectures “were detailed and clear, so that they could be understood even by poorly prepared students.” Between 1820 and 1826 Lobachevsky was the dean of the Mathematics and Physics departments.  Lobachevsky was later appointed the position of Rector of Kazan University, in which the university flourished with him at that position.

Back to the idea of Hyperbolic geometry, there are historians that believe that Gauss and Lobachevsky had correspondence and that Gauss suggested ways for Lobachevsky to move forward in his math, but there is no proof to this statement. Gauss also worked on hyperbolic geometry, but never published his works, and Janos Bolyai also worked on non-Euclidian geometry, but as Lobachevsky was the first published, he is looked at as the Father of non-Euclidian geometry, and as such has his name attached to it.  

Sources:

http://www-history.mcs.st-andrews.ac.uk/Biographies/Lobachevsky.html

http://en.wikipedia.org/wiki/Nikolai_Ivanovich_Lobachevsky

A History of: Roman Numerals

Quite possibly the only ancient number system left in use today are the Roman Numerals. Most people know nothing about where they originated from though. Until this class, I didn’t either. Shepherds in Italy would use tally sticks, which were made out of bone or wood, and were used to mnemonic learning purposes, and more importantly to keep track of debts. The debts would be made on a piece of bone or wood on both sides of it which would then  be split in half, and the debtor and creditor would each have part for their records. These tally marks were heavy influencers of roman numerals.

The I, which represents 1, does not come from the letter, but from one of the notches on the stick. When you reached the fifth notch on a tally stick you would make a double cut like a ‘V’, which happens to represent 5, and the tenth notch was a cross notch like an “x” which represents 10! This pattern continues throughout the stick. For example, the number 21 would look like this IIIIVIIIIXIIIIVIIIIXI on a counting stick, and then like this as a Roman Numeral XXI.  

One thing that most people don’t know is that the four in roman numerals has two correct ways to be written. It can be written as IIII, or, IV. On most analog clocks, you will see it written not as IV, which most people are taught, but as IIII. Even more strange, is the fact that the nine is written subtractively as IX. There are many theories as to why this happened, but the two most credible are that in the early history of Rome, the god Jupiter’s name was written IVPPITER beginning with IV so it was more common to see IIII used. The other major theory is that Louis XIV, the king of France, preferred IIII to IV, so he demanded that clocks were produced with the IIII and that is simply the way it has stayed.

So, while most mathematical and economical practices are done with Hindu-Arabic numbers now, what do we use them for today? One of the most common places to see them in use is in affiliation with rulers of monarchies and with popes. You will also see them used in major sporting events, or recurring major events. The Olympic games and the super bowl are two examples of their use. 

Sources
http://chart.copyrightdata.com/ch02.html
http://en.wikipedia.org/wiki/Tally_sticks

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