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Monday, November 14, 2011

Life: The Game of Chance

By David Etherton

     A writer named Dale Carnegie once said, “Take a chance! All life is a chance. The man who goes farthest is generally the one who is willing to do and dare.” When you think about it, he is absolutely right. Everything we do in life is by chance. Whether you are starting a business or starting a rock band, running in a marathon or running for president, playing poker or playing slot machines, everything we do involves the chance that we may not succeed. It may be for this reason that a whole subfield of mathematics has been dedicated solely to chance; this subfield is known as probability theory.

     Probability theory has existed for at least 350 years, although many believe that it has been around much longer than this (Burton, 2011, p. 439). This field originated from two different subjects: analyzing data to determine death & insurance rates (statistics), and how to win at games that are solely based on chance (probability) (Burton, 2011, p. 439).

     To begin, the man who first laid the groundwork for the field of statistics was John Gaunt (Burton, 2011, p. 440). Gaunt, who worked as a merchant in London, was the earliest person to use a large set of data in making a set of statistics (Burton, 2011, p. 440). In 1662, he created a work that tried to categorize the causes of death among females and males in London (Burton, 2011, p. 440). When he saw that his work was not being utilized to its full potential, however, he decided to take matters into his own hands (Burton, 2011, p. 440). To put his work to good use, Gaunt collected over 50 years of data, in fact, and then analyzed it to see the mortality rates for people at different ages (Burton, 2011, p. 440).

     Now, for the field of probability, we need to go a little bit further back in time. It is said that people started playing games of chance (gambling) over 5000 years ago, and that the first dice were invented around 3000 BC in Iraq (Burton, 2011, p. 443). The invention of playing cards is a little bit hazy, as it is said that the Chinese, Egyptians, and Indians may have all created it (Burton, 2011, p. 444). Much later on during in the 16th century, a man by the name of Girolamo Cardan wrote a book called Liber de Ludo Aleae, which not only defined probability, but was also the first source to relate probability to games of chance (Burton, 2011, p. 445). It is not Cardan, however, who is said to have revolutionized probability theory. In 1654, a man named Chevalier de Méré was concerned with questions of gambling, so he sent his inquiries to Blaise Pascal (Burton, 2011, p. 454). Pascal, in turn, wrote Pierre de Fermat to help him with these problems (Burton, 2011, p. 454). They worked together on the probability involved in throwing dice and, as a result of their combined efforts, these two people are known to have developed probability theory (Burton, 2011, p. 455).

     After all these years, probability and statistics hold a fine place in mathematics, and in my opinion, it is because there are many so different ways to apply this topic to the real world, and, over the years, probability and statistics have evolved quite a bit and are now the focus of many careers. An example of one of these careers would be an actuary, who gets paid to determine how much risk is involved in various situations, how much money could be lost or made during these situations. Another example would be statisticians (big surprise, right?), which collect data and analyze it by various means. There are many careers that revolve around probability and statistics, and one thing worth mentioning about many of these jobs is that they can pay very well. According to the Bureau of Labor Statistics (2009), actuaries made a median salary of over $84,000 in the year of 2008, and some actually made over $160,000. The Bureau of Labor Statistics (2009) also states that statisticians were making a median salary of about $72,000 in 2008, with some easily breaking the six-figure mark.

     For more information about what these types of careers entail, you may want to visit the following websites.

http://www.amstat.org/careers/

http://www.beanactuary.org/

Bibliography 

Bureau of Labor Statistics, U.S. Department of Labor, Occupational Outlook Handbook, 2010-11 Edition, Actuaries, on the Internet at http://www.bls.gov/oco/ocos041.htm (visited November 10, 2011).

Bureau of Labor Statistics, U.S. Department of Labor, Occupational Outlook Handbook, 2010-11 Edition, Statisticians, on the Internet at http://www.bls.gov/oco/ocos045.htm (visited November 10, 2011).

Burton, D. M. (2011). The development of probability theory: Pascal, Bernoulli, and Laplace. In
The history of mathematics (Seventh ed., pp. 439-496). New York, NY: McGraw-Hill. (Original work published 1999)

Chance quotes. (n.d.). Retrieved from http://www.brainyquote.com/quotes/keywords/chance_2.html

The Abacus and its Development


By:  Joe Hruban
A Russian Abacus at the Ashmolean Museum, Oxford

            The Abacus was an advantageous tool to use to count numbers and do simple operations.  It helped count, add, subtract, and a little bit of multiplication and division.  It assisted the user in calculations and was used by many different cultures.  This gave it many different forms as well.  However, the history of its development begins in Sumer.

The Abacus first originated in Ancient Sumer sometime in the mid 2000s B.C.  It is possible that the Babylonians used it to make calculations, however it is not for certain.  We start to gain a little more certainty of the abacus when we get to Egypt.  The Greeks mentioned that Egyptian used counters or pebbles to help with calculations or just counting.  The Persians begin to use the abacus after this, yet, it isn’t until the Geeks that we actually have proof that the abacus was used.  With the Greeks, tables with some sort of counters in them have been discovered.  The Romans adopt this from the Greeks.  The Chinese and Russians also used the abacus and made further changes to it.  W. W. Rouse Ball (1912) states that “this instrument was in use among nations so widely separated as the Etruscans, Greeks, Egyptians, Hindoos, Chinese, and Mexicans,” to show the popularity and vast spread of this instrument.  This expansive distribution of the abacus leads to its development and modifications.

The abacus went through many changes and modifications as it progressed from place to place.  It first started out as a wooden board with grooves cut into it.  These grooves were made to hold pebbles or counters of some kind.  A variation of this form was a table covered with sand with grooves in it that were made by someone’s fingers.  However, it was the same concept.  To represent a number, one would look at the grooves and the pebbles to understand place and value, respectively.  The number of pebbles or counters in the first groove signified the value of the one’s place.  The number of pebbles in the second groove represented the value of the ten’s place, and so on.  Once you reach ten in any of the grooves, you clear that groove and put one pebble in the following groove.  One must keep in mind that whichever groove was considered the first groove depended on which way the culture that was using it read; in other words, left to right or right to left.  This model of the abacus was just the template for another form, the swan-pan.

The swan-pan was a form of the abacus where there were parallel wires within a wooden frame.  This is the form that we know today when we think of an abacus.  The swan-pan of the Greeks and Romans did not have a top to the frame.  This was so they could place or remove their counters on the wires.   They created two margins on their abaci: one with four counters and one with twelve counters.  This signified the addition of fractions with denominators of four or twelve.  Their abaci were made to represent numbers of up to one-hundred million.  Although this abacus suited the needs of the Greeks and Romans, the Russians decided to modify it further. 

The Russians made some changes to the abacus as well to suit their needs.  They elongated the wires and put nine or ten beads on each wire and closed the frame.  This eliminated the need to add or remove beads from each wire and since the wires were longer, all they had to do was slide the desired beads up or down to represent the number they needed.  Gerbert added to this by coloring the beads different colors and by marking them one through nine.  This made using the instrument more organized and slightly quicker.  The Chinese saw further developments that could be made.

The Chinese and Japanese made a slightly different model.  There model had a top row with two beads on each wire and a bottom row with five beads on each wire.  To represent a number, one would slide the beads up towards the divider.  The two beads above the divider represent a value of five (in decimal notation) so if one would want to represent the number seven, the beads in the top row would be one up and one down and the beads in the bottom row would be two up and three down.  This made performing the operations of addition, subtraction, multiplication, and division, much faster.  Ball (1912) stated “I am told that an expert Japanese can, by the aid of a swan-pan, add numbers as rapidly as they can be read out to him.”  The abacus was a great aid to the Japanese as well as many other civilizations.

The abacus helped civilizations in making calculations until the invention of the conventional calculator.  It went through many changes with each civilization that used it but benefitted all.  The fact that the idea of it stayed relatively concrete throughout every civilization it was a part of further solidifies its notion of an effective tool.   

Bibliography
Ball, W. W. R. (1912). A Short Account of The History of Mathematics. London, England: Macmillan & Co., Ltd. 123-126.      

Friday, November 11, 2011

Happy 11/11/11


Happy 11/11/11!! Here are a few interesting facts related to such numbers!

11×11 = 121
111×111 = 12321
1111×1111 = 1234321
11111×11111=123454321
111111×111111=12345654321
....

1 / 11 = 0.09 09 …
2 / 11 = 0.18 18 …
3 / 11 = 0.27 27 …
4 / 11 = 0.36 36 …
5 / 11 = 0.45 45 …
6 / 11 = 0.54 54 …
7 / 11 = 0.63 63 ...
....

11-3×3=2
1111-33×33=22
111111-333×333=222
11111111-3333×3333=2222
....