In the Footsteps of Mathematics

PI

2015-11-29T19:44:00.002-06:00

By Robin Prokaski

As many mathematicians know, Pi is the ratio between a circle’s circumference and diameter. Many of us know that Pi can be approximated at 3.14. However, it is a never ending number. As of 2013, Pi has been calculated to 12.1 trillion digits (The Math Mystery). Even though Pi is such an integral part of many aspects of mathematics, few truly know the early history of Pi. Furthermore, rarely people understand that it comes from nature, and how abundant in nature it really is.

To start off lets first talk about the history of Pi. To this day, we do not know the first to become conscious of this ratio. However, we seem to think human civilizations could have been aware of this as early as 2550 BC. The great Egyptian pyramids were built between 2550 BC and 2500 BC. It turns out that the height and the width come out to approximately 2 times pi. These were measured in cubits, were approximately 18 inches. However, a cubit was measured by the length of a person’s forearm. Therefore, it varied from one person to another. Early texts reveals the Egyptians found an approximation for pi to be 3.16 (Purewal 2013).

One of the biggest contributors to the development of Pi was Archimedes. He is considered to be the first person to calculate an accurate approximation of Pi. He did this by using area of two Polygons. One of these polygons inscribed the circle, while the other was inscribed by the circle. Essentially, he kept increasing the number of sides of the polygons and effectively got closer and closer to the area of the circle. He continued this process until reaching polygons with 96 sides. This process allowed him to get an approximation between 3.1408 and 3.1429 (Smoller 2001).

One of the text that contains an approximation of Pi that is not as frequently talked about is the Bible. The following verse contains an approximation for Pi: “"And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about"(1 Kings 7:23). From this quote, one would think that their approximation was 3:1. However, cubits were the unit of measurement. As mentioned previously, cubits change from person to person, and the approximation could really have been more accurate than it seems (Purewal 2013). From here, many different mathematicians made contributions for Pi. Now let’s focus on topic that many people are not that familiar with, Pi in nature.

Pi is often thought of just the ratio between circumference and diameter of a circle. In the Documentary “The Mystery of Math” a professor explains the following example: If a sheet of paper is divided into any amount of sections in which r is the length between lines, and a needle, of length r, is dropped onto the paper, the probability that the needle cuts across a line is 2/Pi. Even though there is no circles or diameters of circles, Pi still comes into play. Experts say that Pi is able to show up in the strangest of places because it is one of only concepts to relate a straight object (diameter) with a rounded one (circle). One of the places that comes into play is the path of rivers. As we know, a river is rarely ever straight, but is not circular either. On average, the actual length of the river over the direct distance from start to finish is about Pi. Similar to rivers, any models that deal with waves has Pi in them. Most commonly, these are light and sound waves. The documentary mentioned earlier also explains that Pi tells us which color should appear in a rainbow, and how middle C should sound like on a piano. On the biology side of things, Pi explains how cells grow into spherical shapes such as oranges or apples (Mystery of Math).

However, Pi is found in many other different aspects of nature. One area that many people do not realize pi is involved in is probability.

Pi is part of a hidden interconnected web that relates many different aspects of our world. It is our job as mathematicians to find these hidden connections, to better understand our world and nature.

References

The Math Mystery: Mathematics in Nature and Universe - Documentary. (2015, June 11). Retrieved

November 24, 2015, from https://www.youtube.com/watch?v=8gd-gUlBv_s

Purewal, S. (2010, March 13). A brief history of pi. Retrieved November 11, 15, from

http://www.pcworld.com/article/191389/a-brief-history-of-pi.html

Smoller, L. (2001, February). The amazing history of pi. Retrieved November 24, 15, from

http://www.ualr.edu/lasmoller/pi.html

Topology and the Seven Bridges of Konigsberg

2015-11-29T18:26:00.003-06:00

By Courtney Correa

Topology is the mathematical study of the properties that are preserved through deformations, twisting, and stretching of objects. (wolfram.com) Tearing of the objects in topology are not allowed, and an example of two shapes that are topologically equivalent are a circle and an ellipse. Topologists study the shapes and their properties that remain the same when stretched or compressed. Leonhard Euler is credited with the discovery of topology of networks and in 1735 his work in this field was inspired by the Seven Bridges of Konigsberg problem. In Konigsberg, which is now modern day Kaliningrad, Russia, a river ran through the city and created a center island. After the island, the river split into two parts and the people of Konigsberg built seven bridges for people to use to get around the city. (mathforum.org)

Euler presented the question of if it was possible to cross all seven bridges one time, and be able to access each area of land beside each bridge. He concluded that this is not possible, and he analyzed the geometric position of each bridge and piece of land to prove this. With a point or node representing each piece of land, and an edge representing each route to a bridge, he counted four nodes and seven edges. Euler looked at how many vertices each node had and noticed that the nodes could be odd or even. An odd node has an odd number of vertices and an even node has an even node of vertices. The problem with the seven bridges was that all of the nodes were odd, and this meant that it did not matter where you started or finished, you would end up getting stuck and not crossing one of the bridges.

When analyzing the Seven Bridges of Konigsberg, Euler created what is known today as a topology of networks. (mathforum.org) A network represents points and lines that when connected, they create odd or even vertices. A network can be considered traversable if one can use a pencil to trace the shape without lifting up the pencil and without going over a side more than once. Euler’s creation of the graph of the seven bridges is not traversable because the four vertices are odd. Euler’s graphical representation of the Seven Bridges of Konigsberg hinted at the discovery of topology because of the fact that the most important components of the problem were the number of bridges and their endpoints, not their exact positions. Topology is not concerned with the shape of objects, and the seven bridges problem is a great example of this because we are not focusing on the actual layout, we are focusing on the graph.

Videos:

https://www.youtube.com/watch?v=2iovbcPwAro

https://www.youtube.com/watch?v=Kw6g31HFMDA

Websites:

http://mathforum.org/isaac/problems/bridges1.html

http://mathworld.wolfram.com/Topology.html

Exploring the Cosmology of Math

2015-10-16T21:58:00.004-05:00

Exploring the Cosmology of Math

by Jessica Salgado

Mathematics have proven to be the best way to describe how the world works. The biggest question in math still seize to be unanswered. Where does math come from? In the documentary “The Great Math Mystery”, explores the question of why does math do a great job in explaining the world we live in today by asking and analyzing math in nature.

The documentary starts off by drawing attention to the fibonacci sequence. The Fibonacci sequence is a set of numbers that are produced by adding the two last numbers. A person first start off with 0,1 and then adding both of the numbers up which results in the answer 1. The result is then added to the sequence. So now the sequence is 0,1,1. Then after that you add the last two numbers of the sequence which results in number two. Then result is added to the sequence. The sequence then looks like 0,1,1,2. This pattern continues on and has been seen to appear a lot in our world. For instance, the Fibonacci sequence seems to appear in places such as, the number of pedals in daisy’s, the number of spirals in a sunflower, the bracts of a pinecone, and tree branches. The number of pedals on each daisy’s seems to sound like a random number but this type of flower seems to favor Fibonacci numbers.

So why then does nature seem to favor the Fibonacci sequence? Does nature know how to do mathematical operations? I do not think so. Is the sequence arbitrary sequence? I do not think so either.

Furthermore, in the documentary they mention river sinuosity, which is the curves of the river. The importance of the curves of the is that many mathematicians have theorized that in most rivers the average approximation of a river’s bendiness is pie. They measure its bendiness by obtaining the length of the river divided by the direct route. When a river is completely straight the river’s sinuosity is 1. To better understand the gist of river sinuosity I watched a video called “Pi me a river” that perfectly describes this theory.

This video explains the dynamic of the theory. River sinuosity is a theory without any concrete applications. But with this in mind, many great mathematicians like Sir Isaac Newton discovered gravity by assuming that an invisible force such as when an apple falls down. He believed that the same force that holds the planets in order is the same invisible force on the moon-gravity. Without having an sort of application Sir Isaac Newton discovered gravity.The sinuosity of a river seems to not have any sort of applications, but is a recurring measurement that appears in rivers. Is this a arbitrary thing? How is it possible that math is an exact approximation to understand how things in nature works? What is the essence of math? What is math’s origin? So this leads me to ask, if math is not arbitrary then is math discovered or invented?

As the documentary progresses, there seems to ask whether math is discovered or invented. If math were to be discovered it would not make sense to state that a person discovered something that has already been there or to explain something that has already been there. For now, let’s say math is invented. Which brings me to my next topic. If math were to be invented, then is math something created by something, someone, or a thing? In the documentary, Max Tegmark is a professor at MIT, who believes that math is like a video game and that the foundation of math built in our brains. A video game is a set of algorithms, properties, and equations of things that are all mathematical which is put in a software that defines the game. Tegmark believes that their are a lot of similarities between this world and video games because math does not only describe reality in some parts but it is the essence of reality. He believes that math can go as far and be our physical reality. Like a digital photo as a person tries to view the picture closer and closer, only pixels are seen. Tegmark believes that our reality is like this and that math is invented by math and that is all too it. But a huge counter argument against this claim is that If math were to be built in our brain then why do we sense that when we have solved a mathematical problem it seems that we found an answer that was not there but it was already there. Discovered instead of invented.

Furthermore, in the documentary a scientist named Liz Duke lemur center in North Carolina does varies experiments with lemurs. She compares the lemurs reaction when choosing between two different quantities of food. She notices that every time the lemurs always pick the amount that has the most. Animals, such as lemurs, have a notion of what math is, without understanding what it is. If math were to be invented from our minds what is its extent? In the documentary a speaker, Mario Livio, states how is it possible that “The product of human thought does so well in defining the universe,” with this in mind is the product of the human thought ineffective-having no significant results? Throughout time many mathematicians have theorized and made equations based on assumptions and experiments that has given them results that have patterns, without explaining or having any applications to back up their points. I watched a video called “The Unreasonable Effectiveness of Mathematics” where Mario Livio describes the strange way in which math seems to work without having any sort of applications, it just works until many years later someone figures out or gives these applications needed.

In brief, we use math everyday, we see math everyday, math is everything, so how could it possibly be ineffective? Math is something significant to us but produces results that describes the world around us. How is it possible that the length of every river to the ocean is approximately pie? Why does nature seem to favor the Fibonacci sequence? Does nature know how to do mathematical operations? How do animals, such as lemurs, have a notion of what math is, without understanding what it is? If math were to be invented from our minds what is its extent? Although people have created this definition of what math is, it does a very good job in explaining the world we live in. In the documentary “The Great Math Mystery”, explores the question of why does math do a great job in explaining the world we live in today by asking and analyzing math in nature. It seems that the math itself is a mystery that has an answer to everything, but not for itself.

Numbers in an Amazonian Tribe

2013-12-09T20:31:00.001-06:00

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:

http://www.apa.org/monitor/feb05/fuzzy.aspx

Lucky Ducky

2013-10-15T09:51:00.000-05:00

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:

Patterns in Card Shuffling

2011-12-06T22:28:00.001-06:00

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.

The History & Method of Gelosia

2011-12-03T17:20:00.003-06:00

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 14^th 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 15^th 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.

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

Life: The Game of Chance

2011-11-14T20:28:00.000-06:00

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

2011-11-14T20:21:00.002-06:00

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.

Happy 11/11/11

2011-11-11T11:11:00.001-06:00

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
....

Timothy Gowers Speaks at CMI (year 2000)

2011-04-04T15:36:00.001-05:00

Timothy Gowers received the Fields Medal for research on functional analysis and combinatorics in 1998. In the year 2000, he was the Keynote Speaker at the Clay Mathematics Institute where he spoke about various applications of mathematics in other areas such as computer science, finance and engineering. His speech is remarkably clear and easy to follow. You can watch his speech here..

History of Math Contest

2010-11-06T13:24:00.007-05:00

The Mathematical Association of America has its eighth annual undergraduate student writing contest in the History of Mathematics. This contest is open to all undergraduate students. Deadline for submission is March 31st. More information (including 2010 winners’ papers) is available on the HOM SIGMAA website at http://www.homsigmaa.org/.

Fractal Geometry

2010-10-21T16:06:00.004-05:00

Benoît Mandelbrot, the father of fractal geometry, died last Thursday at the age of 85. We will learn about fractal geometry in class when we get to the modern era.

You can read more about it here.

Archimedes Palimpsest

2010-10-12T09:26:00.009-05:00

Palimpsest comes from the Greek word palimpsestos meaning "scraped again". It is a manuscript that has another text written over it. Around 1200 A.D., a Christian monk turned Archimedes' book into a new prayer book. In October 1998, it was sold for $2 million at an auction. You can watch the documentary that we saw in class about the story again here: part 1 , part 2 and part 3.

Since 1998, a group of scientists have been working on the recovery of the writings of the greatest mathematician of the ancient era. You can find up to date information here.

As well, you can find a digital copy of the recovered Archamides Palimpsest after the prayers are removed on google books.

Dangerous Knowledge

2010-10-04T22:23:00.016-05:00

“Dangerous Knowledge” is a fascinating documentary by the BBC about the story of four masterminds, Goerg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing. What do these four have in common?

Cantor created "set theory". Cantor’s Theorem says that there exist infinitely many infinite sets of different cardinalities. The theory was nonsense and shocking to some of the mathematicians of his time and it created a lot of opposition. Some philosophers saw his theory as a challenge to the uniqueness of “God”, the absolute infinity. Cantor is also known for the Continuum Hypothesis which states that there is no set with cardinality between the cardinality of natural numbers and of real numbers. (We will go over Cantor’s Theorem and the Continuum Hypothesis in detail in class). Cantor could neither prove nor disprove the Continuum Hypothesis. In 1940 and 1960 respectively, Kurt Gödel and Paul Cohen showed that, in fact, one cannot prove or disprove this hypothesis using the set theory axioms. Gödel is best known for his Incompleteness Theorem which states one cannot find a complete and consistent set of axioms for all mathematics. This gave a negative answer to Hilbert’s second problem. If you would like to know more about Cantor, Gödel, Boltzmann and Turing, watch this 90 minute documentary!

Numeral Systems

2010-09-17T20:24:00.003-05:00

In the last few weeks we learned about various numeral systems from Babelonian to Hindu/Arabic numerals. In this interesting video you can watch a summary and more ...

Is there a meaning to Beckham's wearing jersey number 23?

2010-09-08T10:31:00.003-05:00

If the title of the post interested you enough to find out, read the short essay by Marcus du Sautoy, professor of mathematics and a soccer fan. 23 seems to be the number of choice for other sport celebrities and world figures like Michael Jordan or Pope John XXIII. Nature also gave us 23 pairs of chromosomes. And let's not forget that according to historical records Julius Caesar was stabbed 23 times, which most likely was a pure coincidence and not a deliberate counting by the assassins. So, what's so special about number 23?
________________________________________
Beckham in his prime

See also an interactive resource to learn through games all about prime numbers.

Why do we need to study the history of mathematics?

2010-08-25T14:43:00.001-05:00

If you answered instinctively -- Because I need the course credit for my major -- I recommend that you view this captivating lecture (you might also download the audio only) from the famous Gresham College in the U.K. Professor Robin Wilson in his lecture Keep Taking the Tablets on the history of ancient Egyptian and Mesopotamian mathematics explains eloquently:

First of all, mathematics is part of our heritage – as central as literature and painting – and we need to study a culture’s mathematical artefacts for the same reasons as we study its cave paintings, its buildings or its literature: indeed, it’s been claimed that mathematics is one of the two oldest professions in the world, and so we certainly need to investigate its origins.

Secondly, mathematics is not just the tidied-up God-given subject that we find in our textbooks. On the contrary, it reveals itself as a fundamentally human endeavour – whether we’re interested in the counting of sheep, the collection of taxes, or the measurement of time. On the other hand, if we’re primarily interested in the mathematics itself, then our understanding may be significantly increased if we can explore the context from which it developed – whether as a practical problem arising in technology or astronomy or the biological sciences, or whether it arose out of a desire to solve a mathematical puzzle or develop a mathematical idea.

And finally, going back to the original sources can provide new motivation – our reasons for studying a subject now may differ markedly from the original purpose in studying a problem. Perhaps bored schoolchildren would find quadratic equations more fun if they’re told that the methods originated in Iraq some four thousand years ago, and if they’re encouraged to decipher the solution themselves by studying the markings on a clay tablet.

You can download the video, audio, lecture notes and print the transcript
____________________________________________
Keep Taking the Tablets. Robin Wilson. 2004

Five Numbers

2010-08-12T23:33:00.003-05:00

Here you can listen to Simon Singh at BBC Radio who talks about histories of five special numbers. He talks about 1, 2, 6, the number that defines the universe and the first Ramanujan number! LISTEN...

The Mathematician who didn't really exist!

2010-08-12T23:33:00.001-05:00

Here you can listen to author and scholar Amir Aczel who talks about Nicolas Bourbaki, the genius mathematician who didn't exist! Listen to Amir Aczel at NPR...

The Birthday Problem

2010-08-12T23:21:00.000-05:00

Stanford professor Keith Devlin at NPR program explains about the birthday problem. "The birthday problem asks how many people you need to have at a party so that there is a better-than-even chance that two of them will share the same birthday" Here you can listen ...

Math in Poetry: Pi

2010-05-30T17:02:00.000-05:00

_____________________

poem by Wislawa Szymborska, Polish author and Nobel Prize Winner in Literature

Source: Wisława Szymborska, Monologue of a dog, 96 pp., Foreword by Billy Collins, New Poems translated from the Polish by Clare Cavanagh and Stanisław Barańczak, Harcourt,Inc.,New York, Toronto, London, 2005, ISBN 0-15-101220-2

Ancient Egyptians Mathematical Papyrus

2010-05-30T15:16:00.001-05:00

One of the most famous artifacts that survived from the ancient Egypt is the Rhind Mathematical Papyrus dating to 1550 B.C. now in the collection of the British Museum.

It contains 84 different mathematical problems - such as how to distribute 100 loaves of bread among a workforce in different ratios. Such skills were constantly needed by scribes - the administrators of the Egyptian society. This papyrus was placed in its owner's tomb as a sign of his highly educated status.

You can listen to the 15 minute podcast, see the interactive version of the object and read more details on the history and significance of this papyrus on the British Museum website titled, A History of the World in 100 Objects.

The Unintended Consequences of Math

2010-05-30T15:16:00.000-05:00

Scientific discoveries often follow the law of unintended consequences when new applications, not predicted at the time of the discovery, often lead to drawbacks or even troubling uses decades or centuries later. In this excellent audio program from BBC Radio three academic experts chat about such unintended consequences of mathematical discoveries. Some of the examples are:

The cubic equations which led, after 400 years, to the development of alternating current - and the electric chair.

The centuries-old work on games of chance which eventually contributed to the birth of population statistics.

The discovery of non-Euclidean geometry, which crucially provided an 'off-the-shelf' solution which helped Albert Einstein forge his theory of relativity.

The 17th-century theorem which became the basis for credit card encryption

Solving the integrals equation

2010-05-30T15:15:00.000-05:00

Here is a little refresher of your calculus knowledge ---a step-by-step presentation on how to solve the integrals equation using Pierre de Fermat method.

But this is only the first part of the presentation. Read the second article here.
At the end of the article the author Chris Sangwin from the University of Birmingham has a little test for you:

If this article has inspired you to do some calculus, here is a problem for you: