Sunday, June 26, 2016

Blog Post 6: History of Zero

Zero is a unique and central role in mathematics. In math 310 for example, the zero element of each ring,or the additive identity, is what helps determine if a set is or is not a ring. When added to itself or another number, the number does not change. Also when multiplied by any number, that number becomes zero. It is also a placeholder in many number systems. In fact, most ancient civilizations had some sort of characteristic that represented a zero in their numeral systems.

The Babylonians originally did not have a symbol for zero, therefore their notation is hard to interpret. Today, for example, we would find it difficult to tell the difference between 14, 104, and 1,004 without the place holder of zero. Instead the Babylonians left a space in the middle of numbers or at the end to distinguish the zero place holder. Thus, to us who are not use to reading Babylonian numerals, determining place value is difficult. Eventually however, they did come up with a symbol that represented the space between the number, yet they did not have a concept of zero as an actual number.

Around 650 zero became common in Indian mathematics. Mathematicians such as Brahmagupta, Mahavira, and Bhaskara used zero in mathematical operations. Brahmagupta, for example, explained that zero subtracted from itself is zero and zero multiplied by any number is zero. His only mistake was that of dividing by zero. The Bakshali manuscript may be the first document to ever have zero used in a mathematical purpose.  Around 665 the Mayans also developed the number zero. However, it was more isolated where as the Indian concept of zero spread to surrounding civilizations such as the Arabs, Europeans, and Chinese.

Mayans used a base 20 numeral system. They however had three symbols, a shell shape, that represented zero, a dot, representing one, and a stick representing five. They used zero as a place holder. For example in the number 402, they would have 4 dots in the 100's place, a shell in the 10's place, and 2 dots in the one's place. They often used zero when it came to keeping track on their Long Count calendars. 

The Chinese were estimated to begin the use of zero as a place holder somewhere between the 1st and 5th century. They used counting rods for calculations and according to A History of Mathematics, the rods "gave a decimal representation of a number, with an empty space denoting zero." Zero was treated similarly to the way the Babylonians treated zero. It was more of place holder, unlike the Indians who developed the numerical zero. The oldest surviving mathematical text from the Chinese containing a symbol of zero was the was from 1247, the Mathematical Treatise in Nine Sections.

In 773, zero had reached the Middle East. The first to work on equations with zero was the famous mathematician Al-Khowarizmi. He worked on equations that equaled zero, where algebra was invented. By 879 zero was written very similar to how we do now of days, an oval shape, however, he wrote it smaller than the other numbers.

The Europeans, in the 11th century, began to use zero in operations such as addition and multiplication. Voyagers from Arabia were the first to bring texts of Brahmagupts and his colleagues. Fibonacci built on Al-Khowarizmi work in his book Liber Abaci. Fibonacci's developments with zer quickly spread via Italian merchants and German Bankers. Accountants were able to determine when the books were balanced based on when the positive and negative amounts were equal to zero.  In the 13th century, manuals of calculation, such as multiplying, adding, and extracting roots, became common in Europe.

Hossein Arsham, a mathematician at the University of Baltimore, writes, "The introduction of zero into the decimal system in the thirteenth century was the most significant achievement in the development of a number system, in which calculation with large numbers became feasible. Without the notion of zero, the modeling process in commerce, astronomy, physics, chemistry, and industry would have been unthinkable. the lack of such a symbol is one of the serious drawbacks in the Roman numeral system."

Adding, subtracting, and multiplying by zero are now relatively common operations. Zero was not always agreed on and confused many great minds. The concept was not always concievable. Many though why do we need a symbol to represnt nothing. However, in the day and age we are in now, zero is just as common as any other integer. Developing zero has been one of man kinds most significant accomplishments.

Work Cited

Saturday, June 18, 2016

Post 5: Archimedes' Spiral

Many think of a spiral as a smooth curve that coils around and around until it reaches a central point. The spiral of Archimedes was first recorded in his book On Spirals in 225 B.C. He came up with the equations as r = a+bx. Where x represents theta. a represents the starting point of which the spiral rotates and b represents the distance between each spiral.

The most commonly seen Archimedean spirals are that of tightly wound springs, edges on rolled up rugs, and decorative pieces on jewelry. More practical uses are used in machines such as the sewing machine. The machine transforms rotary motion to linear motion. The Archimedean spring is particularity known for being able to respond to both torsional and translational force.

The spiral has the property that any ray from the origin intersects multiple points on the spiral with constant distance between each point. That distance is said to be 2pib if it is measured in radians. The idea is that the spiral has two arms one when theta is positive and one when theta is negative. These arms connect at the origin or center of the spiral. Only one however is shown, the other is not and if was would look like the mirror image of the showing arm across the y-axis. Below I have listed a link that will take you to a interactive graph that allows you to adjust a and b to see how the formula works.

We know that because there is a linear relation between the radius and the angle, the distance between each coil will be constant. Therefore, for this to be true, we know that the arms get bigger by a constant rate as well with each turn. To find the entire length of the spiral, we need to use the equation r = a+bx, where r is the distance from the origin, x represents theta. represents the starting point of which the spiral rotates and represents the distance between each spiral. The total angle turned in the spiral is 2pib, therefore the distance between each loop is 2pib divided by the number of turns.

Sunday, June 12, 2016

Blog Post 4: Review of The Math Book

The Math Book by Clifford Pickover was certainly a fascinating read. It's tough to say if a non math background reader would find all of the discoveries from the book interesting. However, as have a math emphasis, I would say I enjoyed reading, what I like to call, "fun facts" from this book. This piece was not trying to explain every detail about the most significant math discoveries throughout mankind, rather it gave: names, dates, and a brief overview about the discovery. Because it didn't go into depth about each discovery, I believe that it was an easy read, even for those who would not call themselves math oriented. I would call this book the Wikipedia version of mathematical history.

Pickover explicitly states in the prologue of the book that he is not trying to justify and explain each discovery, rather just let the reader know when, why, where, and how the discovery played a significant role in mathematical history. Because his goal was to gloss over each discovery, I think the book was well written and was especially significant for a teacher to read. I believe the research behind each discovery was in depth enough to allow the reader to gain more than just a date and who created it. It also was very broad and not specific to just theorems. It had a lighter side as well. For example their were multiple game theories and other neat facts such as the discovery of the probability of which a monkey could type a specific sentence based on how many keys were on a keyboard and how many letters were in the sentence. The text also discussed important devices created such as the abacus and first hand held calculator. This book did a nice job of balancing serious with light and fun.

I would definitely recommend this book to anyone interested in mathematics, but more importantly to any teacher who teaches math. If the teacher has a problem with justifying mathematics to his/her students, this book would be the first I recommend to them. For example if a student questions the importance of e, I would tell the teacher to look it up in The Math Book. From their the teacher could recite facts such as the purpose of it's discovery, who came up with it, and how it is used. I think because a teacher could give a student a little background knowledge about almost any topic covered in K-12, the student would find reasoning and a connection with the topic. Therefore, I would suggest this book to any teacher who is looking to make learning mathematics more meaningful for students.

Because I read this book cover to cover, I did find this book a little dry. However, imagine if you were to read 250 Wikipedia pages on mathematical discovers, it probably wouldn't be entertaining either. Therefore I can't really make a comment on how interesting or entertaining the book really is. I found certain pages more interesting than others, however, that would vary on who the reader is regardless. I think the book was simple enough to read that even for those who are not mathematically aware could read with ease. Depending on what the writer wanted, I feel that some pages did not have enough information for the reader to fully understand the discovery. I am uncertain that that was the goal of the writer however. He did leave work cited and where to find more information if the reader was interested for each subject. Thus, I believe the breadth of information for most subjects was appropriate with room for more discovery if the reader chose to follow on. Therefore, the book in my opinion, the book would receive a 3.5 out of 5 stars. It was a useful text and was not written in a way that made the reader feel uneducated. It was at the perfect level to discuss complex ideas in an appropriate language.

Wednesday, June 1, 2016

Blog Post 3: What is Math?

Mathematics is something unique. It cannot be define specifically, rather it should be a set of ideas. You could say mathematics is problem solving or abstract thinking. Or you could take a very different approach and say it is completely something else. In my opinion there is not one definition, it depends on how one views it. Mathematics is a system put in place to better understand numerical facts about the universe. Now I know that this could be argued by many, and I would like to hear everyone's argument. For my personal purposes however, mathematics' purpose for me, is to serve as a system that puts complex ideas and numbers into a more observable, practical, sense.

Math is something incredible, it can be an infinite amount of things. It can be theories, equations, ratios, and even games. It falls under a realm of problem solving. Math is not one specific thing, rather it is very difficult to define because it is considered to be many different acts. The most specific definition of math that I can create is: Mathematics is an abstract way to explain quantity and space. It is an art  involved in almost every aspect of our lives. It keeps time, helps us calculate movements, create technology, and many other every day tasks. Without math, we would have no clues to why and how we exist. Math helps us predict simple things such as when we should leave for work, how much we need to pay the cashier, and other minuscule things we take for granted. Simple tasks such as these are basic arithmetic. For example, to pay the cashier, we need to add the dollar amounts in our pockets to be as close to, without going whole dollar bills over. For calculating when we need to leave, we take what time we want to be at our destination, and subtract how much the drive time takes to get to our destination to determine when we need to leave by. 

To do mathematics could mean many things. Examples from above are calculating when to leave for work or paying the correct amount of money. However, fundamental mathematics is what we teach to younger children to help aid them to be able to do every day activities. Teaching numbers, solving simple arithmetic, identifying shapes, and many other simplistic math we teach to students is doing math. However, there is also very complex reasoning as well. Applied mathematics however can be interpreted completely differently. Applied mathematics, for example, is what engineers of physicists do to calculate formulas. They apply math to help them determine forces and many other things linked to their work. 

Famous mathematicians and even high school aged students do mathematics. For example, high school students are taught theorems based on work from mathematicians. Students are only adding their own knowledge to the proofs and are also adding evidence that the proof works. Finding theories, applying theories, and generating proof or reasoning behind problem solving is mathematics. Discovering how and why the world ticks the way it does is mathematics. Math and science are so closely related to one another that they both feed off of one another. Doing math is a very hard question to answer.

My definition of math is not the only one. Leopold Kronecker said "The integers came from God and all else was man-made." In my opinion, it sounds like the tools we work with were given, but to what we do with them is considered mathematics. So a broad definition from Kronecker might be that mathematics is what we do with integers. According to Elaine J. Hom from livescience,com, "Mathematics is the science that deals with the logic of shape, quantity, and arrangement." She goes on to say similar ideas to mine that, "math is all around us." She explains that it is a part of our every day life and that it includes the technology we use, buildings we create, other art such as tessellations, money, applied mathematics such as engineering, and even sports. Hom has a very similar outlook of mathematics as I do.

Because I am planning on going into education, I wanted a definition from a person with similar interests. Therefore I found informal writing from Wendy Petti, a 4th grad math teacher on She asked students and teachers of all ages to reflect on what math is. An elementary school volunteer she interviewed said "math is more than a subject we learn in school."
A first grade student said "Math is you. Math is me. Math is everything we see!" A 13 year old student replied with "Math is the entire world simplified on a piece of paper... Math is ingeniousness morphed into a tiny simple forumula so we can harness its fantastic powers."  Lastly a 12 year old stated, "math is the universal language of the world,." According to just about anyone who is surveyed about what mathematics is, math is more than just something we study in a classroom, it is what the world is made out of, it's what makes the world turn. Other experts have various answers as well. Professor Dave Moursund, states that mathematics can be broken into 3 overlapping areas. 1. Mathematics is a human endeavor. 2. Mathematics as an academic discipline. 3. Mathematics as an interdisciplinary language and tool.

To sum up what mathematics is, mathematics is something that can be studied, created, used as a tool or language, used in our everyday lives, and how the world works. Mathematics is very difficult to define because it can be used in many ways. The best way to define mathematics is to say that it is complex body that requires logic and reasoning to describe the world around us.

Saturday, May 28, 2016

Blog 2: Liu Hui and the Nine Chapters on the Mathematical Arts

Liu Hui wrote the book of The Nine Chapters on the Mathematical Arts. Lui Hui goes through and breaks down the methods and examples and provides reasoning behind his work. His work seemed to provide general methods rather than deducing in a logical order such as Euclid's Elements. This piece was the main mathematical text for learning in many countries over many centuries.
In class we went over the problems in chapters, 6:12; 7:1; and 7:18.

The first problem reads, A good runner can go 100 paces while a poor runner covers 60 paces. The poor runner has covered a distance of 100 paces before the good runner sets off in pursuit. How many paces does it take the good runner before he catches up the poor runner.

The second problem reads, Certain items are purchased jointly. If each person pays 8 coins, the surplus is 3 coins, and if each person gives 7 coins, the deficiency is 4 coins. Find the number of people and the total cost of the items.

The third reads, There are two piles, one containing 9 gold coins and the other 11 coins. The two piles of coins weigh the same. One coin is taken from each pile and put into the other. It is now found that the pile of mainly gold coins weighs 13 units less than the pile of mainly silver coins. Find the weight of a silver coin and of a gold coin.

Below I have inserted a picture of the scratch work I did in class.

In the first problem, I first created a chart to kind of find a rough estimate of where the good runner would catch the poor runner. After seeing the range I needed I could make sure that my x value made sense. Following, I came up with the formula 60x+100=100x to find how long it would take the second runner to catch the first. The 60x represents the amount of paces the poor runner runs per time value, the +100 represents the amount of the head start, and the 100x represents the amount of paces the good runner would need to catch the second runner. Therfore, to find the total amount of paces for the good runner needs to run to catch the poor runner, I needed to solve for x. I did so by subtracting 60x from both sides to get 100=40x. Then I divided both sides by 40 to get 2.5 hours. Lastly I determined how many paces were in 2.5 hours for the good runner to get 250 paces.

In the second problem, I took both amounts and made equations. If each person paid we knew that we had 3 leftover, thus we have 8 cents per person with 3 cents left over, same follows for when the amount is split that everyone pays 7 cents. They are 4 cents short. Therefore we know that these are equal to one another so we set the equations equal to one another. We therefore find that there are 7 people total and that collectively they spend 53 cents.

In the last case we are given two weights that are equal to one another. We know that we unbalance the equation therefore we add 13 to the left side representing the difference of 13. We are able to determine the ratio of the gold by getting x by itself, therefore we can plug this into our x value to find that silver weighs 29.25 and gold weighs 36.75.

Each individual problem was solved algebraically using variables that represent numbers that can be altered. We were able to discover solutions by setting equations equal to one another in order to determine what value the variable needs to be in each case. Through the mathematics shown, we see how creative this new kind of mathematics was to create a functional generic system that can be applied to other values in the same type of problem. For example in the first problem, because the poor runner had a 100 pace head start, we knew our equation had to look like 60x+100=100x. To generalize this, we can represent our 100 as c, for constant. It can be replaced with any amount of head start and you could calculate how many paces it would take the good runner to catch the poor runner. Same follows by the second problem. The 3 and 4 in the problems could be replaced with any number and you could still take the same process to determine the total value and the number of people paying. Lastly the third problem follows suit. If you change the number of pieces of gold and silver to any other constants, you would use the same operations to determine the weight of silver and gold

Tuesday, May 10, 2016

What is Math?

I believe math is one of the hardest things to define. It involves so many different areas that each type of problem is unique. The best way for me to generalize what math is to say that math is a system to simplify problems into something more understandable. This can be taken in many different ways; however, what I get out of this definition is that we have a more complex form of some problem. For example, one of the most basic forms of mathematics is addition. If we are given 25+10, we can therefore simplify to 35 so that we better understand what 25+10 actually means. I don't believe there is one proper term to define mathematics but there are many ways to describe math. The simplest form is to use numbers, letters, and symbols to help understand the world around us.

As far as mathematical discoveries I cannot think of many. However, I tend to think about laws and other rules that we have came across our classes throughout our schooling.

I think one of the biggest discoveries in mathematics was the discovery of Euclidean Geometry. This was the turning point that made sense of every day life and objects that existed in the universe. It helped humans, and still helps humans understand the world. It is the most commonly used geometry by almost everyone and intrigues me how these rules and laws still are proven to be true.

In Discrete Mathematics, I think of Euler and the Eulerian Circuit. I like puzzles and other elements that involve problem solving and figuring things out, such as riddles. Therefore I think of the Eulerian Circuit as a fascinating puzzle that requires deep thinking and interests me.

The Fibonacci Sequence and variations of it are also interesting to me because I find it fun to discover the pattern and how to get to the next term. When first learning about the Fibonacci numbers I was well engaged and later after learning more complex sequences really enjoyed solving the patterns and finding the equations at which they changed by.

Lastly, I know most of these are kind of far fetched due to little knowledge I do have about famous discoveries, I really find the biggest discoveries are related to finding best ways to teach students. There are so many variations on how and why we teach the subjects the way we do and I think, with more studies and research, students should be encouraged to think differently. I think allowing students to find new patterns and ways to solve problems is the coolest thing about math. There isn't always one way to do something.