When we worked on the Greek timeline in class, one of the developments in math that really caught my interest was that the Sumerians ‘invented’ zero as a placeholder around 300 B.C., but they did not consider it as a number (http://www.ccsdk12.org/mclemens/Projects/mathhist.htm). I found this fact very interesting because it made me wonder why the Sumerians did not regard zero as a number, and also what practical use for zero they had. Zero has always been such a mysterious number to me, so I decided to do a little digging to find the answers to some of my many questions about the number Zero. Here is what I found:
Zero has many functions in mathematics and number sense. To name a few, the digit “0” is used as a placeholder to distinguish between 1, 10, 100, etc., and it is also used to measure the length of a point and serves as a reference point for distance on a number line. Zero makes it possible for us to subtract a number from itself (ex: 4-4), is the additive identity, and also plays an important role in set theory as it is the cardinality of the empty set (http://www.ams.org/samplings/feature-column/fcarc-india-zero). Zero was not always considered to be a number as it is today. In the past, Zero was just seen as a placeholder and was not regarded as a true number. There is some debate in the math community about who created zero and when, and the digit appears in writings in many different cultures, namely in the fertile crescent, India, during the time of the ancient Mayans, and also during the time of the Sumerians. It is difficult to really tell when the first zero appeared, because there is not much of a surviving written record, and a lot of math was part of oral tradition (http://yaleglobal.yale.edu/about/zero.jsp). It wasn’t until around 1200 during the time of Fibonacci that Zero became the number we regard it as today. In class, when we debated what the definition of a number is, many of us considered a number to be a symbol that represents a finite distance from zero. I think that the reason that zero was not so widely accepted as a number in the past because it represented the absence of a number. Numbers were commonly used for measurement or for counting money or possessions (http://www.scientificamerican.com/article/history-of-zero/). How can we measure something or count something that is not there? Zero is the answer. We can count the number of coins we have in our pockets, but we can’t count them if we don’t have any. Since zero is not concrete like the other finite numbers, I think it is the number’s abstract quality that makes us question whether or not it is a true number. I also think that the definition of a number as a measure of distance also extends to zero because zero is also a measure of a distance of zero from itself, and it also describes the distance between every number and itself (ex: 0-0=1-1=2-2=3-3=4-4=…=0), therefore zero is a number.
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Part of being a Mathematician is being able to solve, represent, or understand a problem in different ways. After solving the Diophantus riddle algebraically in class, I wondered if there was a way that I could represent the problem visually so that I could better understand it. From a teacher viewpoint, I know that each student learns and understands differently, so some of my future students might better understand a problem like this if it is represented visually. In some of my teaching classes, when dealing with fractions, we have worked with visual models of fractions like arrays. I worked through the Diophantus riddle again and found a way to represent each part of the riddle visually in an array.Diophantus Riddle 'Here lies Diophantus,' the wonder behold. Through art algebraic, the stone tells how old: 'God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.' The first thing that I noticed when I worked through the riddle again was that I could establish a common denominator of 84 for the fractions 1/6, 1/7, and 1/12, since the least common multiple of 6, 7, and 12 is 84: Once I established a common denominator of 84, I knew that I could represent each of these fractions visually in a 7 x 12 array, which makes 84 equal-sized units. In the model below, I represented each value in the problem in a different color: Since Diophantus was 84 years old when he died, based on this model, we know that:
- His boyhood lasted 14 years (
*red*units) - He grew whiskers 7 years later when he was 21 (
*red*+*blue*units) - He was married 12 years later at the age of 33 (
*red*+*blue*+*green*units) - Five years later he had a son at the age of 38 (
*red*+*blue*+ green +*purple*units) - He was 42 when his son died (since 42 is half of 84); 42 is also the sum of the remaining
*white*units
algebraically in class, visually as an array, and also numerically when I found the sum of all the parts in the problem. In finding the sum, I was also checking my answer to see if it made sense, which attends to the SMP “Make Sense of Problems and Persevere in Solving Them.” When I was in Elementary, Middle, and High School, I thought that math was all about solving dozens of addition, subtraction, multiplication, long division, fraction, or trig calculations per night, using the same method my teacher showed me in class. I was not the best math student, and I remember feeling frustrated every night when I looked back at the notes I took in class and still couldn’t understand my homework. I thought that math was about finding an answer, and the answer could either be right or wrong, and there was no in-between.
When I came to Grand Valley, I chose to study math because all my life, it had been some big scary monster that I could never really understand. I wanted to overcome my fear and hatred of math and finally get some answers to those things about math that were always left unexplained. During my studies here at Grand Valley, my hatred and confusion of math turned into passion and curiosity. I realized that math is not all about numbers and finding the right answer-- for me, math is all about the process, not the answer. In the past, I believed that there was only one way to find the right answer to a problem, and that was the way that my teacher showed me how. All of my professors here at Grand Valley challenged me to find my own solutions to problems by approaching a problem in different ways, and they encouraged me to collaborate with my peers to find many different solutions to the same problem. Now, I think that math is more about solving problems creatively than about crunching numbers. Real math goes beyond pencil-and-paper and extends out into the real world. It is what we use to analyze and describe patterns and problems we observe around us. Math solves problems. Math challenges. Math creates. Math inspires. Although I’m a mathematician, I admit I don’t know much about math history. What I do know about the history of mathematics is that math has been around for a very, very long time, and it is responsible for some of humankind’s greatest accomplishments. Without math, the Egyptians would not have been able to design the pyramids, and ancient astronomers wouldn’t have learned very much about the Universe. Mathematics has always been a tool that people have used to help them make sense of the world around them, and it is a tool we all use in our daily lives today. I think that the most important development in the history of mathematics was the modern number system, because it made math universal. Some of the famous mathematicians whose work I am familiar with include Euler, Euclid, Hippocrates, Fibonacci, and Pythagoras. I am really looking forward to learning more about the history of mathematics in this course. |