The first book I read this semester was The Math Book by Clifford A. Pickover. The second book that I read was Mathematician's Delight by W.W. Sawyer. I did not get to read the whole book, but I was able to skim through the entire book. On W.W. Sawyer's website the book is explained like this, "In this book Sawyer criticizes the teaching of math without context. The best way to learn geometry is to follow the road which the human race originally followed: Do things, makes things, notice things, arrange things, and only then reason about things." This book seemed interesting to me because it starts with simple arithmetic and algebra and proceeds by gradual steps through graphs, logarithms, and trigonometry to calculus and the world of numbers. This book is very thin and easy to read. It W.W. Sawyer knows how to teach and he shows that throughout this book. This book is a great place to start exploring mathematics! I would recommend this book to anyone that would like to explore mathematics, or would even like to know more about mathematics!
A magic square is an NxN matrix in which every row, column, and diagonal add up to the same number. Srinivasa Ramanujan was an Indian mathematician. He had almost no formal training in pure mathematics, but made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. When he was 12 years old, someone gave him a trigonometry book, and he taught himself. At 16 years old, a friend of his family gave him a math encyclopedia with over 6,000 theorems. That was all the math training he had. At age 23, he generated a formula that would calculate all primes up to 100,000,000. He then was invited to move to Cambridge and there he proved or conjectured over 3,000 results, including the best algorithms we have to this day for generating the digits of pi. He died at age 32. Ramanujan created a super magic square. The top row is his birthdate (December 22, 1887). This is a super magic square because not only do the rows, columns, and diagonals add up to the same number, but the four corners, the four middle squares (17, 9, 24, 89), the first and last rows two middle numbers (12, 18, 86, 23), and the first and last columns two middle numbers (88, 10, 25, 16) all add up to the sum of 139. Check out his magic square below. Seeing his magic square made me interested into making my own birthday magic square. My birthday is August 14, 1992, so that was my first row. Then I started looking at Ramanujan's magic square and subtracted or added the difference to each number in his magic square. I ended up with this... I realized this did not work because not all the rows, columns, and diagonals add up to the same number. Not only was this not a super magic square like Ramanujan's, but this was not a magic square at all. I then started playing around with the numbers row by row and I was then able to get every row and column to add to the same number (133), but not the diagonals. Here is what my new magic square looked like... Next, I messed around with these numbers to try and get the diagonals to add to 133, but I could not get it. Then I found this website that showed you how to do a Ramanujan's magic square for any special occasion date. I was finally able to solve my birthday special magic square! My number that everything needs to add to is 14+8+19+92=133. Here is my birthday magic square! I wanted to see if I could find another birthday using this same formula, so I decided to try my nieces birthday August 30, 2011. Her number that everything needs to add to is 30+8+20+11=69. Here is here birthday magic square! Having googled the answer to finally figure out the solution, I noticed a pattern within the way to solve a birthday magic square. I noticed two boxes together always equals a different two boxes added together. Then those four boxes are then apart of another equation to find out the next boxes. For example the second step is b+c=m+p. Well then the b, c, m, and p are all apart of a different set of equations. B is apart of b+n=g+k, and so on, but once a letter is used twice, it is not used again.The mathematics behind the birthday magic square is rich! It may look like simple addition, but the patterns within the addition is what makes it rich!
I think this is a fun activity to try. You can do this with any date! First, I would try it on your own to see if you can create your birthday magic square without the help of the website, just grab a pencil, paper, and calculator. I came close to figuring mine out without the help of the website! Then after spending some time and having different magic squares that do not work, go to the website here and try to see where you went wrong within your birthday magic square! In mathematics everyone talks about famous men mathematicians. I have been studying mathematics for four years and I have rarely heard of female mathematicians. I did some research and found 5 female mathematicians that everyone should know about! 1.) Hypatia (ca. 350-415) Hypatia was one of the earliest female mathematicians. She was the daughter of Theon. She followed in her father's footsteps in the study of astronomy and math and Hypatia was a philosopher. "It is thought that Book III of Theon’s version of Ptolemy’s Almagest—the treatise that established the Earth-centric model for the universe that wouldn’t be overturned until the time of Copernicus and Galileo—was actually the work of Hypatia." To learn more about Hypatia please click here. 2.) Sophie Germain (1776-1831) Sophie Germain studied mathematics and geometry. She taught herself Latin and Greek. Since she was female she was unable to study at École Polytechnique. Joseph Lagrange was a faculty member there and Sophie created a false name so she could attend. When Joseph found out Sophie was a woman, he became her mentor. She was the first woman to win a prize from the French Academy of Science. Her prize work was on a theory of elasticity. To learn more about Sophie Germain please click here. 3.) Ada Lovelace (1815-1852) Ada Lovelace never knew her father. Her mother was very protective of her. Her mother wanted her to study science and mathematics. Lovelace began to work with Charles Babbage, who was an inventor and mathematician. She was going to help translate an Italian mathematician's memoir. She went beyond completing the translation. She wrote her own set of notes about the machine and included a method for calculating a sequence of Bernoulli numbers. This is acknowledged as the world's first computer program. To learn more about Ada Lovelace please click here. 4.) Sofia Kovalevskaya (1850-1891) Sofia was Russian. She was not allowed to attend a university because she was a Russian woman. She married Vladimir Kovalevsky and moved to Germany. In Germany she was not attend university lectures, but she was privately tutored. After writing treatises on partial differential equations, Abelian integrals and Saturn's rings she received her doctorate. She eventually was able to teach lecture in math at the University of Stockholm. Then she became the first woman in that region of Europe to receive a full professorship. She won the Prix Bordin, the French Academy of Sciences in 1888, and a prize from the Swedish Academy of Sciences. Her prize work was focused on the rotation of a solid body. To learn more about Sofia Kovalevskaya please click here. 5.) Emmy Noether (1882-1935) Albert Einstein refereed to Emmy Noether as “the most significant creative mathematical genius thus far produced since the higher education of women began.” With rules against woman at universities in Germany, Emmy's mathematic education was delayed. She obtained her PhD and was still unable to obtain a position as a professor. She had the title of unofficial associate professor at the University of Göttingen. In 1933, she lost that position because she was Jewish. After losing her position at the University of Göttingen, she moved to America and did research and lectured at Bryn Mawr College and the Institute for Advanced Study in Princeton, New Jersey. In America she made significant advances in the field of algebra. To learn more about Emmy Noether please click here. There are many important woman in the field of mathematics. These are five that I enjoyed learning about. I learned about these five women through The Smithsonian Magazine here. The Smithsonian Magazine has wonderful articles about woman in mathematics. To search for articles on the Smithsonian Magazine click here. Many think that mathematics is a man field, but without some of these important woman, math may not have developed the same. Men and Women both have helped shape mathematics as we know it today! These women have influenced my mathematical career. Without them standing up for themselves, the number of women mathematicians would have decreased. In a report by Catherine Hobbs & Esmyr Koomen from the Department of Mathematical Sciences, School of Technology, Oxford Brookes University, Wheatley Campus, in 2006 they have shown the percentage of women mathematicians increasing. This graph below is one of many from their report. To read the full report please click here. These women helped me be able to study mathematics. Without them, mathematics may have been shaped differently and I may not have been able to be a mathematics major. One way things would have been different without these women is the first computer program was created by Ada Lovelace. I have used many different computer programs in the past years. If she did not invent the first computer program, we would not be able to know if someone else would have created one and life would be completely different. I use a computer for everything! These women changed not only mathematics, but they changed my life too!
I just finished reading The Math Book by Clifford A. Pickover. I would highly recommend this book to anyone! This book was very informative, and a variety of people could relate to the book. The layout of this book was very simple. It was in chronological order and provided the major milestones that have created mathematics. There were 250 milestones in the history of mathematics. These milestones include the greatest mathematicians. The author starts with a milestone in c. 150 Million B.C. and ends in the year 2007. Some of the milestones include history behind items that people do everyday, like play tic-tac-toe! The book was written in 2009. Each milestone is one page (left side) and then on the right side there is a picture to go along with the milestone. It is a very quick and easy read. I have picked a few examples from the book that were my favorite to share. The ones I have picked to share I think most people would enjoy learning about even if they do not like mathematics. · Ant Odometer (c. 150 Million B.C.) : This was the first milestone listed in this book. German and Swiss scientists discovered that ants “count” steps to judge distance. After ants had reached their destination, the scientists would add stilts or shorten their legs to see how the ants traveled back to their original destination. · Dice (c. 3000 B.C.) : Can you imagine a world without dice and random numbers? Dice were originally made from anklebones of hoofed animals. Dice are used to teach probability. Now dice are used for many things people do everyday. · Rubik’s Cube (1974) : Have you ever messed around with a rubik’s cube and got frustrated that you could not finish it? I have! After reading about the invention of the Rubik’s Cube I started playing around with one again because I was determined to figure it out. In this one page summary of the Rubik’s Cube, you find out that no configuration requires more than 20 moves to solve. That is just crazy to think about for me, since I have worked with Rubik’s Cubes for hours and could not figure them out! Those are just three of my favorites that I thought most people could relate to! This book is a quick, easy, and enjoyable read. It is just fascinating to read about the major milestones in mathematics that you would have never thought was a major milestone for mathematics. I would recommend everyone to read this book to see what you can relate to, and think about how your life would be different if these were never created within mathematics! In class as a group we had to create a bungee cord that would hold our action figure Thor. The object was to create the best bungee cord that made you come closest to the ground, but not to hit the ground. We were only allowed 10 rubber bands, and we had to base a prediction off of our 10 rubber bands. Then we were able to make our bungee cord with as many rubber bands as we predicted. Then as a class we let our characters bungee jump! Why did we do this and how is it related to Mathematics you ask? Well, it is related through Galileo. He was an Italian physicist, mathematician, engineer, astronomer, and philosopher. Galileo produced some mathematics. These were called Galileo's Paradox and it showed that "there are as many perfect squares as there are whole numbers, even though most numbers are not perfect squares." Galileo's Paradox Another thing Galileo explored was gravitational pull on an object. Galileo wanted to see what would happen if he had two of the same objects, but different weights, when released from his hands, what would reach the ground first. With how far science and math has come, we now know that the two objects would hit the ground at the same time because of gravitational forces, but Galileo did not know that then, so he collected data and made his assumption. You can explore Galileo's Experiments here. This is where the bungee jump came into play with our math class. Before our final predication of many things, we collected data. We tied our 10 rubber bands together and wrapped one rubber band completely around our figures legs, therefore leaving us with 9 rubber bands in our bungee cord. We then grabbed a meter stick and started measuring how far Thor fell with 2 rubber bands, 4 rubber bands, 6 rubber bands, 8 rubber bands, and 9 rubber bands. We saw an increase in length, but a decrease in length between each rubber bands. What this means is with 2 rubber bands we saw a fall of 85cm, 4 rubber bands we saw a fall of 105cm, and 6 rubber bands we saw a fall of 115cm. As you can see the total fall is increasing, but the distance between is decreasing. We could only experiment and get data with those 9 rubber bands, and then we had to make a prediction. We knew the ground was 505cm. We used a linear regression and graphed our data to make a prediction. The graph we used told us that we needed anywhere from 40 to 55 rubber bands, which we thought was high. We then used a linear regression and found we need 35 rubber bands. We wanted to be on the safe side so we used 34 rubber bands. This meant that we had 33 rubber bands in our bungee cord and one was completely tied around Thor's feet. You can watch how close it was below. As you can see our first attempt was very close. Thor did not hit the ground, but he was about 2-4cm from the ground. I would say that was a successful bungee jump! In the next video you can see another bungee jump (Hippo) and then Thor's bungee jump at a different angle, and in slow motion. I think this is a great lesson for students to try. Within this lesson there is history, science, mathematics, and it is hands on! Students get to make a hypothesis and try it to see how close they were. I think students would enjoy this lesson. I know I enjoyed it!
You should try it yourself. Find a ledge to do the bungee jump with an action figure or Barbie, grab some rubber bands, and TRY IT!! |