Explorations in Math Project 2

By:Stephanie Primm

My Sandwich Haiku

I eat my sandwich

Now I eat the rest.

Keegan Adrian Winslow's Story

Keegan Adrian Winslow needs to have a SSN, ISBN, money order, credit card, VIN, zip code + 4 with check number, UPC, and a soundex code. She needs her SSN and zip code + 4 with check number to fill out her application for a new job. She needs her soundex code for the U.S. Census Bureau. She needs a credit card and a money order to pay for things. She needs a UPC to easily look up an item at the grocery store. She needs her VIN because her car was stolen and she needs to give the number to the police for them to track it down. She needs an ISBN to look up a book at the library. She also needs to know Morse code for a test.

She was born in Pineapple, Alabama on November 13, 1983. The area numbers for Alabama are 416-424. Her SSN is 421-22-1359. The highest group number issued is 55. The serial numbers are assigned in increasing order. A SSN is invalid if any field is all zeroes (no field of zeroes is ever assigned) or the first three digits are above 770.

Her soundex code is W524. This is what she filled out on her census form.

Keegan has now moved to Tyngsboro, Massachusetts. Her zip code is now 01879-1329 with a check digit of 9. 0 is the area that she lives in, 18 is the mail distribution point, 79 is the town she lives in. 9 is the check digit number because 0+1+8+7+9+1+3+2+9=40. The answer has to be divisible by 10.

Keegan needs a credit card to buy the soda that she wants with the UPC she has to find it. Her credit card number is 4653 972106 23247 with a check digit of 5. This is valid because if you take the odd positioned numbers and add them up(4+5+9+2+0+0+0+7=31), multiply it by 2(31*2=62), add three(62+3=65), add that to the even positioned numbers(65+6+3+7+1+6+3+4=95), then add a number so that it equals a number with a zero at the end(95+5=100), you have a valid credit card number and a valid check digit.

The soda that Keegan wants has a UPC of 049000011340. The last 0 is the check digit. This is determined when you add together the odd positioned numbers and add them up(0+9+0+0+1+4=14), multiply it by 3(14*3=42), add this to the add that to the even positioned numbers(4+0+0+1+3=8), add this to the sum in step two(42+8=50) and since this number ends in zero already, 0 is the check digit number.

Keegan needs a money order to pay off a speeding ticket that she got before her car was stolen. Her money order is 5436048597 with a check digit of 6. If you add the numbers and divide them by 9(5+4+3+6+0+4+8+5+9+7=51, 51/9=6) the remainder is the check digit.

Keegan needs a VIN to have the police help her find her stolen car. Her VIN is 3C3EL40H5YT209210. If each letter is turned into a number, then multiplied by a number in the order of 8…1 then starting over at 9, then the answer divided by 11 gets the remainder of 5. This is the check digit.

At the library, Keegan looks up a book. The book has an ISBN of 0-6158-2605-4. These numbers each multiplied in a decreasing order of 8-2 add up to 183. The check number, 4, added equals 187 which is evenly divisible by 11.

She needs to know Morse code to pass a test in her math class that she is taking in college. Morse code is a type of code that uses a series of dits and dahs which are a type of communicating. It has been used in wars and was used on the Titanic to call for help. Her test says to translate “Have a nice day” into Morse code. She got “…. .- …- . .- -. .. -.-. . -.. .- -.-- “as her Morse code. She aced the test.

References:

My Timeline

*About 100BC: Chinese mathematicians use negative numbers for the first time.
*594: The decimal notation for numbers less than one were first used in India.
*628: Zero was first used to help solve quadratic equations, sum series, and compute square roots.
*1200: al-Hassar first used the modern day notation for fractions.
*1202: Leonardo of Pisa used teh first variable to represent a number.
*1220: Leonardo of Pisa first used the symbol for square root.
*Between 1356 and 1361: Nicole d' Oresme used the first symbol representing the addition symbol, meaning "and" and the subraction symbol meaning"to take away".
*1463: Benedetto of Florence first used a variable for an unknown. He used the Greek letter rho.
*1556: Nicolo Tartaglia first used the ( ) symbols for parenthasis as grouping symbols.
*1557: Robert Recorde was the first to use the equals sign(=) as the sign for equality in an equation.
*1628: William Oughtred first used the modern day notation for multiplication(x).
*1655: John Wallis used the first symbol for infinity.
*1659: Johann Rahn first used the the obelus (÷) as the modern division symbol.
*1706: William Jones was the first to use pi to represent the ratio of the circumference to the diameter of a circle.
*1777: Leonhard Euler first used i to stand for imaginary numbers.

References:

Who Am I?

My real name is Leonardo Pisano. I changed my name because I thought that Leonardo was too common. Sometimes I even used the name Bigollo, which may mean good-for-nothing or a traveler.

I was born in Italy in 1170, but I went to school in North Africa, where my father represented the merchants of the Republic of Pisa. I traveled with my father while writing books on my mathematical discoveries for 25 years, until year 1200 when I returned to Pisa.

I wrote many books on my mathematical findings. I was the one who discovered the number sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... which I named after myself. This sequence is that each number is the sum of the two preceding numbers. These numbers are found everywhere in life, you just have to look for them.

I first discovered that square numbers could be constructed as sums of odd numbers. I did this by using the n2 + (2n+1) = (n+1)2.

After year 1228, I kept out of the limelight and nothing was documented. I died around year 1240 in Pisa.

References:

Albert Einstein

On March 14, 1879 in Ulm, Württemberg, Germany, one of the most famous mathematicians and scientists ever known was born. In 1886, he began his school career in Munich. He attended violin lessons at the age of six until he was thirteen. He was also home taught his religion of Judaism. In 1894, Einstein was left behind in Munich while his family moved to Milan.

In 1895, Einstein took an exam to let him study to receive a diploma as an electrical engineer at the Eidgenössische Technische Hochschule in Zurich. Einstein ended up failing this exam and ended up at a secondary school in Aarau. He thought that by doing this he would get a better chance in attending the Eidgenössische Technische Hochschule.

From 1902 until 1906, Einstein had a job as a technical expert third class. In 1906 he was promoted to technical expert second class. During his time here, Einstein wrote some papers. In 1905 he wrote On a new determination of molecular dimensions, which he received a doctorate from the University of Zurich. Later on in 1905 he wrote his second paper which covered his theories of relativity. In this same year he proposed that mass and energy were equivalent.

In 1908, Einstein took on another job. After he wrote his Habilitation thesis Consequences for the constitution of radiation following from the energy distribution law of black bodies, he became a lecturer at the University of Bern. This job led to him accepting the job as professor of physics at the University of Zurich. Then in 1911, Einstein took the job as a full professor at the Karl-Ferdinand University in Prague.

After studying his theory of relativity and revising it many times, Einstein moved back to Zurich from Prague in 1912. This is when he took up a chair at the Eidgenössische Technische Hochschule in Zurich. He finally got his dream of attending this school.

In 1914, Einstein was offered many positions back in Germany. He took the research position in the Prussian Academy of Sciences and he took a chair at the University of Berlin. Einstein didn’t accept the position of directorship of the Kaiser Wilhelm Institute of Physics in Berlin, which was also offered to him.

Between 1915 and 1920, Einstein traveled around Europe and lectured at many different conferences. He went to the United States in 1921 to lecture and to help raise funds for the future Hebrew University of Jerusalem. In this year, 1921, Einstein received a Nobel Prize for his work on the photoelectric effect in 1905. He later received the Copley Medal of the Royal Society in 1925 and the Gold Medal of the Royal Astronomical Society in 1926.

In 1935, Einstein applied, and was granted, full residency of the United States. In 1940, Einstein was able to be named a citizen of the United States, but chose to stick with his Swiss citizenship instead.

In 1952, Einstein was offered the role of second president of Israel after their president died. Einstein declined, but felt awful because he caused offense to Israel.

Einstein died on April 18, 1955 in Princeton, New Jersey in the United States. His body was cremated that night in Trenton, New Jersey.

References:

Math in Art

Math is found everywhere in art. Math even makes art. Math is not only about formulas but about patterns and symmetry too. In an article about Leonardo DaVinci, there are many explanations of where math is used in art, and how it is used.

In the renaissance, to receive a master’s degree in the arts, math courses needed to be taken to pass. The courses needed were geometry and arithmetic. These courses were needed to be able to compute the ratios for perception paintings.

Leonardo DaVinci, a famous artist, used math a lot in his art. He used to study multiplication and proportion tables to help him in his art. He became obsessed with math and literature, even though he never attended school. He had many notebooks filled with math problems and quotes. He had the quotes "There is no certainty in sciences where one of the mathematical sciences cannot be applied ...” and” Let no one enter who is lacking in geometry” in his notebooks. He did not know how to write, but he self taught himself everything he knew about math. He was fascinated by ellipses and ellipsographs. He even invented many compasses to help himself with his art.

He used perspective in a lot of his paintings. This is how the painter made paintings equal in proportion to real life. He used math to make a proportional compass which helped him to measure real life things and downsized them in his paintings to the same proportion.

There are also atmospheric perspective paintings. This shows the mathematics used to make things in the distance look smaller compared to closer objects in the painting.

DaVinci used fractals in many of his paintings. He was very intrigued by the shapes and formations made by fractals. He said that they “were extremely useful in arousing the mind.”

DaVinci made many drawings that showed knot work. This is part of the mathematical field called topology.

DaVinci used the golden mean in his art work also. In his drawings of the platonic solids the golden men could be found.

The Vitruvian Man was a good example of the golden mean in art. It shows the ratio of the whole body to the length of the navel to the feet. It also shows the ratios of the other parts of the body that show the golden mean.

Even though DaVinci had never been formally taught math, he managed to teach himself and apply it to his work. He became very gifted in the areas of mathematics.

References:

The Golden Mean

The golden mean, also called the golden ratio and more recently Phi, is a huge part of art, nature and math. There are many different shapes of the golden mean. There is the golden spiral, the golden rectangle, the golden triangle and the golden ellipse.

The golden mean is 1.61803. This is derived from taking a line segment and dividing it so that the ratio of the whole piece is equal to the ratio of the larger segment to the smaller segment.

The golden rectangle is supposed to be the most visually appealing of all rectangles. Tests done show that people will unconsciously choose postcards, pictures, mirrors and packages with the proportions of the golden rectangle over other shapes.

The Fibonacci numbers help to create the golden spiral. The spiral is made of small squares with the area of the Fibonacci numbers. The middle square to the outer squares go in the order of 1, 1, 2, 3, 5, 8, 13, 21, 34 and so on.

The golden mean is also thought to be shown in the great pyramid. The ratio of the slant height to half of the base is supposed to be equal to Phi. The Parthenon is also supposed to contain the golden mean. There are theories that it is the golden rectangle, but some theorists don’t agree with this. They say that there are parts of the Parthenon that would stick out of the golden rectangle so that it couldn’t possibly be containing the golden mean.

One of the most commonly found places of the golden mean is the human face. The ratio of the length of the mouth to the width of the nose is one place it can be found. The golden mean is also found from the height of the human body to the height of the navel from the feet. It is said that the most beautiful people most closely fit the golden mean ratios on their bodies.

This can also be connected to animals. The animals that more or less fit the golden mean ratios are hawks, horses and fish.

The golden mean has a wide range of things in life that it helps to make the shape of. If looked for, the golden mean is probably right in front of you, even part of you. Even though there have been many studies on the golden mean, there are still many people who think that some theories are wrong and are still trying to find something that really does most closely show the golden mean.

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Fibonacci Numbers

The discoverer of the Fibonacci Numbers is Fibonacci himself. Fibonacci’s real name is Leonardo Pisano. He changed his name because he thought that Leonardo was too common.

The sequence of the Fibonacci Numbers is 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, and so on. To get the next number, you add the first two and the answer is the next number. There is an infinite number of Fibonacci Numbers with any chosen number as a factor, so basically every number is the factor of a Fibonacci Number.

Fibonacci came up with these numbers in a tournament given by the emperor of Pisa, Frederick II. The problem was “Beginning with a single pair of rabbits, if every month each productive pair bears a new pair, which becomes productive when they are 1 month old, how many rabbits will there be after n months?” Fibonacci came up with the answer of “Imagine that there are xn pairs of rabbits after n months. The number of pairs in month n+1 will be xn (in this problem, rabbits never die) plus the number of new pairs born. But new pairs are only born to pairs at least 1 month old, so there will be xn-1 new pairs. The equation he came up with for the Fibonacci Numbers is xn+1 = xn + xn-1 .

Fibonacci Numbers occur everywhere in nature. They occur in the spiral of a shell, the number of bones in the human body, the branches of a tree, pine cones, leaf arrangements, the petals of a flower and even the seeds of sunflowers show the spiral pattern for Fibonacci Numbers.

The Fibonacci spiral is a collection of squares with the areas of Fibonacci Numbers, set in a spiral. Each square has a side as long as the sum of the two previous squares sides. The Fibonacci rectangle is a rectangle that has two longer sides that are the same Fibonacci Numbers and two shorter sides that are the same Fibonacci Numbers.

Tree branches can grow in ways that are Fibonacci Numbers. One branch can turn into two, two into three, three into five and so forth.

So as you see, Fibonacci Numbers are all over, you just have to look for them. Fibonacci Numbers aren’t just random numbers that Fibonacci came up with, they are a real sequence used in nature and math.

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