Mathematics & Technology
August 9, 2015
Mathematics & Technology
Mathematics is often described as the science of pattern. Mathematicians are looking for patterns and ways of expressing them. Algebra is the study of pattern in number, calculus is the study of pattern in variation, geometry is the study of pattern in space, and probability is the study of pattern in random events (Knight, 2013). One thing about technology is it cannot do proof, as in mathematics (not yet anyway). A computer program cannot reason to create a mathematical proof and mathematical proof requires a human brain. A mathematical proof means that we have demonstrated rigorously the veracity of our conjecture in all cases beyond any doubt. There are many techniques of proof in mathematics but what they all share is that they are a formal series of statements showing that if one thing is true something else necessarily follows from it leading to a proof of our conjecture (Knight, 2013).
The Fibonacci sequence of numbers occurs in many places including Blaise Pascal’s (1623-1662) triangle, the binomial formula, probability, the golden ratio, the golden rectangle, plants and nature, and on a piano keyboard, where one octave contains two black keys in one group, three black keys in another, five black keys all together, eight white keys, and 13 keys in total (Lewinter & Widulski, 2002). Leonardo Fibonacci (1170-died after 1240) listed the following sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 …where the …… means keep going, the sequence continues forever. The pattern is simple. Each term is the sum of the two preceding terms. Thus, 13 appears where it does because it is the sum of the two preceding terms (Lewinter & Widulski, 2002). An example of this would be the number of petals on a flower (8), or the spirals on the cactus plant above (144), these are Fibonacci numbers.
In 1202, Fibonacci changed everything with his book Liber Abbaci, Latin for Book of Calculation, the first arithmetic textbook in the west. His book revolutionized everything from education to economics by making arithmetic available to the masses (Popova, 2011). Liber Abacci described the Hindu-Arabic numerals which were new to the west. The Hindu-Arabic numeral system is a positional decimal numeral system. Today, it is the most common symbolic representation of numbers in the world and was invented between the first and fourth centuries by Indian mathematicians. The book addressed the applications of both commercial tradesmen and mathematicians, it helped to convince the public of the superiority of the new numerals. Until that time the Roman numeral system was in place. One intention of Liber Abacci was to describe methods of doing mathematical calculations without the need of an abacus. The abacus had been around since about 2500 BC Mesopotamia, and is still widely used by merchant’s traders, and clerks in Asia and Africa.
Since the dawn of mankind and the Stone Age, we can see signs of the earliest mathematical forms in the artifacts collected over the years. We have come from etchings in prehistoric bones, to sending men to the moon and into orbit above the earth. As mathematics has evolved over the centuries and millenia, technology has never been far behind. Mathematics and technology is a very broad subject that can hardly be covered in this short paper. There is some controversy over when humans first started sailing. Archaeologists tell us the first sign of sailing ships appeared in Egypt or Mesopotamia around 3500 BC. They also tell us that the Egyptians used “sailing ships” to transport people and goods on the Nile. That being the case, the ancient Egyptians must have had a good grasp of mathematics in order to build sailing ships so long ago. Fast forward to the 21st century and we have middle school children building boats, or skiffs for class projects.
“For their first-ever boat trip, Destinee Rogers and Angel Scott, both 13, took a ride Friday on a wooden skiff that they and their classmates helped to build. They and other members of the school’s Math Club studied geometry and fractions this year by building two boats in a mentoring project sponsored by the 100 Black Men of Mobile Alabama. On Friday afternoons during the school year, Math Club members have gathered in a workshop in a vacant wing at Scarborough Middle School. Led by Wooden Boat Ministry founder Jonathan Stebbins, they measured, cut, nailed, glued and drilled pieces of Okoume wood, painstakingly turning the parts into two 12 1/2-foot skiffs.
“We need to install the deck beam,” Stebbins, 37, told the class one Friday in April. “Here’s our task: You need a helper and a measuring tape.”
Stebbins, an ordained minister, showed several students how to measure 16 inches from the center point of the bow. He swung the measuring tool side to side. “Where you find where it hits the plywood, mark that with a pencil,” he said. “Congratulations! You’ve just made an isosceles triangle”(McPhail, 2015).
The point of this story is, what it must have took the ancient Egyptians to do in 3500 BC, middle school children are doing today, only on a smaller scale. The math is the same. There have been times when technology has advanced by leaps and bounds. Times of war often see new technology arrive, all for the purpose of destroying an enemy. “There is probably no better example of this than the Manhattan Project which developed the first nuclear reactors and atomic bombs. The project also established high expectations for the effectiveness of mathematical modeling and computer simulations that continue to the present day. The Manhattan Project is remarkable in the history of invention and discovery in the number of scientific and technological advances made in a short period of time (1939-1945) and in that the first full system test, the detonation of the first implosion plutonium bomb at the Trinity Test Site on July 16, 1945, was a success. It is a case in which theoretical calculations and simple numerical simulations using electromechanical devices such as Marchant Calculators and IBM punched-card machines with much less computing power than a 1980’s Apple II computer were seemingly able to correctly design extremely destructive weapons that worked right the first time (McGowan, 2013). The remarkable, perhaps unprecedented, success of the Manhattan Project established high expectations for so-called “Big Science” and for the effectiveness of theoretical calculations, mathematical models, and computer simulations. The Manhattan Project had and continues to have a powerful influence on expectations for science in general and specifically for the effectiveness of theoretical calculations and today computer simulations (McGowan, 2013).
Knight, T. (2013, July 2). Mathematics and Technology. Retrieved from news.pamojaeducation.com/: http://www.news.pamojaeducation.com/mathematics-and-technology/
Lewinter, M., & Widulski, W. (2002). The Saga of Mathematics: A Brief History. Prentics-Hall.
McGowan, J. F. (2013, July 8). The Mathematics of the Manhattan Project. Retrieved from MATHblog: http://math-blog.com/2013/07/08/the-mathematics-of-the-manhattan-project/
McPhail, C. (2015, May 17). When boat building is about something more–in this case, middle school math. Retrieved from AL.com: http://www.al.com/living/index.ssf/2015/05/when_boat_building_is_about_so.html
Popova, M. (2011, July 21). The Man of Numbers. Retrieved from The Man of Numbers: How Fibonacci Changed the World: http://www.brainpickings.org/2011/07/21/the-man-of-numbers-keith-devlin-fibonacci/