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Monday, December 9, 2013

Numbers in an Amazonian Tribe


Posted by Gabriela Sanchez 

When a child is asked to count a handful of pebbles, he or she would; however, when the smartest member of the Amazonian tribe, Pirahã is asked to count that handful of pebbles, he or she would just give a blank stare. It is because the Pirahã tribe lacks words for "precise quantities or the action of counting"; however, they quantify objects approximately with words that are similar to our language such as "few" and "many". 
Peter Gordon, PhD, a psychology professor at Columbia University Teachers College, who studies the tribe, gave seven Pirahã tribe members nine objects, and asked them to make a group out of an equal number of nuts. None of the members used exactly nine nuts. Gordon commented on his finding, stating, "The finding suggests that language, especially number-words, plays a critical role in people's comprehension of quantity". In other words, the members of the tribe were unable to to comprehend the quantity that was being asked of them because they do not have a number-word. 
Cross-cultural research supports the idea that, although distinct cultures develop unique counting words, they still share similar properties. For instance, they still appoint a " ... unique word for each counted object, and using the last number-word to stand for the entire group of objects". In other words, despite being different, cultures still share a common core, the ability to use natural numbers. 
It is argued that the Pirahã tribe never needed to know how to count because they never traded with the outside world. They were able to communicate with themselves by a single gesture. For instance, their gesture would indicated, "These nuts for that chicken". Furthermore, Gordon stated the Pirahã tribe never needed to count because their trading system was not as elaborate as other cultures, "... especially those that use currency [and] require the ability to label specific quantities".
When Gordon conducted another experiment on the Pirahã tribe, where he placed one to ten nuts on the table, and asked the each participant to place an equal number of batteries on the table as well, the participants got a hundred percent accuracy when matching sets of up to three batteries. However, the accuracy rate dropped to about 75 percent at four batteries. By the time the participants got to nine batteries, none of them got the answer correct. 
Finally, despite not having a number-words, one can still make approximate comparisons.  Gelman, who researches how children develop number concepts, believes that people are born with an internal representation system for numbers that operates regardless of language. Hours-old babies, and animals such as chimpanzees and pigeons, do recognize change in arrays. Gordon, on the other hand, states, "the lack of number-words seems to preclude the ability to entertain concepts of exact number. There may be other ways to learn and represent exact numbers, but in the normal course of human learning, language is the route we take". In other words, yes, we are able to learn and represent exact numbers; however, the most common and popular course is language. 
The article is quite interesting to me because the basics of math is counting. It is the first thing that one is taught when it comes to math. The article was written in 2005, so that means this news is still fairly new. We are not talking about a tribe back centuries ago. This is just eight years ago. It is understandable that because they never trade with other tribes, just among each other, that that is why they never needed to learn to count; however, it is astonishing that they never encountered a situation where they needed exact quantities. Now, let it be clear that they do not appreciate exact quantities because they lack the language for it, or as it called in the article, "number-words". It's highly intriguing that old civilizations such as the Egyptians did amazing work because they had a numbering system, and this tribe is in the 21st century and lacks a numbering system. 
Also, counting is helpful when it comes to food. Take rations for example, if the tribe is running low on food, they would need to know how many people will eat, thus how many of each of the ingredients they need to have. They cannot just make a large amount of food and have it go to waste. At the same time, I also thought about how when cooking, my mother doesn't use a measuring cup or spoon. My mom just knows, especially after so many years of doing so, that maybe that is how the tribe works. 
Furthermore, knowing how to count saves time. For instance, if one has walk miles to pick fruit for the tribe, then it's important to know how to count. One could count how many hungry people there are and then just pick that amount. Not knowing how to count could mean not bringing enough fruit for everyone and having to make that extra trip or two. 
In addition, I feel like they don't value quantity, but quality. For example, the article read, "Because they don't trade with the outside world, they can simply indicate by gesture that they would like to exchange, for example, this basket of nuts for that chicken". That being said, they don't care that it is a numerous amount of nuts and one chicken. They believe that the chicken is worth numerous amount of nuts. If they had currency like we do, they would give value to money and not to what they are trading.
Peter Gordon, PhD, a psychology professor at Columbia University Teachers College, who studies the tribe, stated that tribes word "'hói,' might be more accurately translated as 'about one'". I like precision and exactness. I dislike when I go to the store for my mother, and she tells me that she needs about five or six things of something. I like it better when she tells me she needs five things, not "about" so many things. Furthermore, it's astonishing that other cultures have their own unique term for a number, but this tribe does not have the ability to use natural numbers. 
Finally, it's intriguing that "language, especially number-words, plays a critical role in people's comprehension of quantity". Not many people think to associate numbers with words. Letters, yes, but not words. Many people think that math is just numbers, but after reading this article, we learn that without words, we cannot comprehend the basics of math: quantity, counting. 

Reference:

Tuesday, October 15, 2013

Lucky Ducky


By Brittany Whitfield
Watching math movies are sometime boring, so I went on a search for a math movie that was both entertaining and educating. What I found was a corky little movie with a Disney title: Donald Duck in Mathmagic Land (1959). This movie is about 27 minutes long and worth the watch. We find Donald on a journey through math explorations from the history of Pythagoras to conic section, as well as many other topics. The film applies mathematic concepts to common use items.
Topics shown in the video:
·         Pi
·         Pythagoras’ Contributions to Mathematics
·         Origin of Musical Scales
·         Special properties of Pentagrams
·         Golden Sections
·         Golden Rectangle
·         Golden Ratio
·         Archimedes Spiral
·         Ancient Architecture
·         Math in Art
·         Math in Nature
·         Mathematics of games
·         Mental math
·         Geometry
·         Conic Sections
·         Infinity
The blog ends with the quote:
“Mathematics is the alphabet with which God has written the universe.” –Galileo
Although this movie may not dive into the concepts deeply, its brief mention of concepts revitalizes a youthful interest in mathematics that I began to feel I lost. Sometimes, in the upper levels of math, the need to explore certain concepts deeply wears on the student mind; however, Donald Duck in Mathmagic Land lightens the rigor, if only for a moment. It is a fun film to watch even if it is geared toward adolescents. So relax for a while and watch Donald explore math topics in a light way. I walked away from the video repeating to myself, “What luck for the Duck.”
To view the video, click on the link to youtube below:
 

Tuesday, December 6, 2011

Patterns in Card Shuffling


 By Tyler Ethridge  

    When shuffling cards, one may think that the cards become random after one shuffle, let alone many shuffles; however, this is not always the case. The greatest magicians that do card tricks have most likely learned a type of shuffling known as “perfect shuffling.” Perfect shuffling is the technique of cutting the deck exactly in half and shuffling so that the cards are alternated perfectly. That is to say that one card from the left hand is released, followed by a card from the right hand and that process is repeated until the entire deck is released. This may not mean anything to us now but after a bit of mathematics we can see that this will develop a pattern.
From Claymath website
   
The standard 52-card deck will return to its original order if it is shuffled using a perfect shuffle 8 times. Different sized decks have a different number of shuffles before they return to their original pattern. Decks of number n where n is a power of 2 return to their original form faster than other decks. To see this more clearly, picture a deck with the number of cards being 2n. Starting with the top card, label each card 0, 1, 2, 3,…, 2n-1. When a single perfect shuffle takes place, a card of position i will change to be in position 2i modulo (2n -1) (modulo gives the remainder in a division problem). After a second perfect shuffle, a card changes to 4i modulo (2n-1). After a third shuffle, it changes to be in the 8i modulo (2n-1) position. The deck restores itself when 2k modulo (2n-1) yields a result of 1.

    In the standard 52 card deck, 2n=52 and 2n-1=51. The various results of 2k modulo 51 are 0, 2, 4, 8, 16, 32, 13, 26, 1, where k is 1 through 8. We see that 28=256. We then calculate 256 modulo 51 and find that it equals 1. This means that each card has moved back to its original spot in the deck as it was before any shuffling took place. It is evident that a little bit of math combined with the technique of perfect shuffling can provide a large advantage in any card game when you are the dealer.

    Bibliography

Diaconis, P. (2006). Mathematics and magic tricks.

Saturday, December 3, 2011

The History & Method of Gelosia


By Danielle Zalesny

The method of Gelosia is used in the application of multiplication. As always in math, there are several ways to go about solving a problem. Today, we are accustomed to solving multiplication of multi-digit problems through the long multiplication method. However, another method widely was used, and even popular today, is the method of Gelosia. This method differs from long multiplication because it distinctly breaks down the multiplication and addition into two steps. It is also a lot less dependent on place value. This organizational method of multiplication also allows for numbers to be multiplied in a visual way, using a lattice-looking diagram. Today, in elementary school, this method, commonly known as the grid method or the box method, is usually shown as an introductory approach to solving multi-digit multiplication problems. Children go on to learn the traditional method of long multiplication once they are comfortable with this method. However, the method of gelosia proves to be extremely reliable. This method aims to show the steps of multiplication (multiply, carry, add) clearly and help people understand how numbers work.

As for the history of the method, Gelosia originated in India, as did much of our arithmetic. It first appeared in Hindu works. In India, this ancient multiplication approach was called the “quadrilateral.” From India, this method spread to ancient Chinese arithmetic in 1593. It also spread to Persia and the rest of the Arab world. It was especially popular in Arab and Persian works. In Arabia and Persia it was called the “method of the sieve” or “method of the net.”

To continue, this method eventually reached Italy, where it appeared in manuscripts in the 14th and 15 centuries. The method of gelosia appears in the first printed arithmetic book which was printed in Treviso, Italy in 1478. The name ‘gelosia’ was actually given to this method in Italy since the grid resembled the lattices or gratings that covered the windows of buildings. These gratings, which are called “gelosia” in Italian, were put on the windows of buildings in order to protect high class Italian women from the public view. In the old Byzantine custom of Italy, wives and daughters of Venetian nobles were kept sequestered in their homes. Another more popular name for this method is the Lattice method.

Furthermore, this method then reached Europe by way of Italy. It was introduced to Europe by Fibonacci, whose 1202 treatise Liber Abacii (Book of the Abacus) explained his work on arithmetic and number theory. The method remained popular for quite some time in Europe. It was widely accepted because of how organized it was. However, it eventually fell out of practice because it was difficult to print the lattice diagram with the printing methods used during the 15th century in Europe. Also, long multiplication took over as the dominant way of solving multiplication problems.

Additionally, the method of Gelosia is also how in 1617, John Napier’s, Napier’s bones evolved. Napier used an approach based on what is known today as the expansion of squares. Napier’s bones are a series of rods made from bone, wood, or ivory. The rods incorporate all of the possibilities in a Gelosia table. These rods are used to perform operations like multiplication and division, along with square and cube root problems.

As for the operation of this method, Gelosia is completed as follows:

First, a large square is drawn. The number of digits in each number you are multiplying corresponds to the number of squares you need to divide the large square into. For example, if one was multiplying 45 by 39, one would need to construct a large square divided into four squares. The two by two square box is the foundation for the grid-like method of Gelosia. The number of boxes varies depending on how many digits are in the numbers being multiplied.  An example of a three by three box to multiply three digit numbers, as shown on the Wolfram MathWorld website, is shown below:


http://mathworld.wolfram.com/LatticeMethod.html 
Next, each square is diagonally split in half, with the diagonals extending outside the box. This crisscross look gives the appearance of a lattice looking diagram. To continue, the first number is written across the top, while the second number is written down the left hand side.

To begin finding a product, the first left upper hand box is looked at. The numbers on the corresponding row and column of the box are the ones that are multiplied. For instance, when multiplying the numbers in the diagram above, 948 by 827, the upper left hand box would correspond to the numbers 9 and 8. These two numbers are multiplied, and the partial product would be placed into the box. However, being diagonally spilt helps to establish place value in the box. In this case, 72 would be written with the 7 on the left side of the diagonal, representing the tens place and the 2 on the right side, representing the ones place. This same method is then performed for each of the remained boxes.

Once the multiplication is completed in each of the boxes, the operation of addition is performed. The diagonals in each box create diagonal stripes in the diagram. These diagonal stripes are added starting with the right one. The numbers that are included in the diagonal stripe are added together. The sum is written at the bottom of the diagonal stripe, outside the box. This is done for each diagonal stripe in the diagram. The sums of each stripe will be written from right to left across the bottom and then up along the left side of the box. The numbers, read from going down the left side and across the bottom, make up the product to the answer. The lattice arrangement of this method also effectively takes care of place value, with the right diagonal stripe being the ones column, followed by the tens, hundreds, etc.


LatticeMethod4
From: http://mathworld.wolfram.com/LatticeMethod.html
In some cases, like with multiplication problems with larger numbers, adding along the diagonal stripes will result in a number bigger than nine. In this case, the sum of the diagonal stripe, for instance if the sum was 13, is written with a three outside the box, and the one being carried over to the next diagonal stripe and added along with the numbers in that diagonal stripe.

Overall, this method, although the explanation seems complex and lengthy, is actually very simple to understand and use. The organizational element of the lattice arrangement allows for large multiplication problems to become a lot of small basic ones.

Interestingly, another aspect of this multiplication method that is extremely beneficial is the ability to multiply numbers with decimals. In this case, a problem involving a multi-digit decimal number, for example 12.3 X 1.24, allows for the decimal point position in the product to be easily determined. The process for solving problems with decimals is the same as multiplying whole multi-digit numbers, as explained above. The only difference is in the beginning when the numbers are written along the top and down the right side of the box. Although the digits are still written as before, the decimal point is drawn on one of the corresponding lines of the grid-like diagram. The two lines that match up the decimals, on the column and row, have a diagonal line in which they meet. This diagonal line becomes the point at which the decimal is placed in the product of the problem. This method successfully and easily locates the decimal position point in the product.

Overall, the method of Gelosia, is a simple and fascinating technique used to demonstrate the understanding of number sense. From ancient India all the way to the present day, this method allows users of it to visually focus on the steps of multiplication and turn complex multi-digit problems into smaller simpler multiplication problems.

Bibliography
Len, G. (n.d). Lattice Method—From Wolfram MathWorld. Wolfram MathWorld: The Web’s Most Extensive Mathematics Resource. Retrevied November 30, 2011, from http://mathworld.wolfram.com/Latticemethod.html

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
....