The End and Thank You! :)

Thank you for lending us your attention and have a nice day ahead! J

Please do visit our blog at archimedesmaths123.blogspot.sg to support our blog and find out more information about Archimedes and Pi! J

Our Project Objectives

Our Group's Project Objectives:

• To be able to find out more about the life of the famous mathematician, Archimedes, for us to be able to know more about him and be inspired by his mathematical contributions.

• In order for us to discover and learn more about Archimedes’ mathematical discoveries and contributions, mainly Pi and find out more on how he discovered these mathematical phenomena during his life in the past.

• We want to learn about some ways in which Archimedes’ mathematical discovery, Pi, can contribute and benefit us in what means of applications in our daily lives and give examples of such applications that we see or do all the time

Credits and Citations

Credits and Citations:

ü  Archimedes of Syracuse:

ü  Allen, James Dow. “Archimedes of Syracuse.” The Greatest Mathematician of All Time. 1998. 7 March, 2013. <http://fabpedigree.com/james/mathmen.htm#Archimedes>.

ü  “Archimedes of Syracuse.” The Archimedes Palimpsest. 7 March, 2013. < http://archimedespalimpsest.org/about/history/archimedes.php>.

ü  “Archimedes.” Wikipedia. 7 March, 2013. < http://en.wikipedia.org/wiki/Archimedes>.

ü  The Discovery of Pi:

ü  Allen, James Dow. “Archimedes of Syracuse.” The Greatest Mathematician of All Time. 1998. 7 March, 2013. <http://fabpedigree.com/james/mathmen.htm#Archimedes>.

ü  “A Brief History of Pi.” Tech Hive. 1998. 7 March, 2013. < http://www.techhive.com/article/191389/a_brief_history_of_pi.html>.

ü  Huberty, Michael D., Hayashi, Ko and Vang, Chia. Historical Overview of Pi. March, 1996. 7 March, 2013. < http://www.geom.uiuc.edu/~huberty/math5337/groupe/overview.html>.

ü  “Archimedes.” Wikipedia. 7 March, 2013. < http://en.wikipedia.org/wiki/Archimedes>.

ü  Applications of Pi:

ü  “Archimedes.” Wikipedia. 7 March, 2013. < http://en.wikipedia.org/wiki/Archimedes>.

Archimedes' Legacy

Archimedes' Legacy:

 

There is a crater on the Moon named Archimedes (29.7° N, 4.0° W) in his honor, as well as a lunar mountain range, the Montes Archimedes (25.3° N, 4.6° W).

The asteroid 3600 Archimedes is named after him.

The Fields Medal for outstanding achievement in mathematics carries a portrait of Archimedes, along with a carving illustrating his proof on the sphere and the cylinder. The inscription around the head of Archimedes is a quote attributed to him which reads in Latin: "Transire suum pectus mundoque potiri" (Rise above oneself and grasp the world).

Archimedes has appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963).

 

Our Individual Reflections

Our individual reflections:

    I have learnt that Pi is used many times in our daily lives. Even in simple activities such as simple calculations during Mathematics lessons, it is still used commonly in many other areas. In some occupations such are engineering, architectural designers and land developers require the use of Pi. I feel that we have good team spirit as we are all on task and we are able to finish our respective tasks on time and efficiently. ~~Sean

     I have discovered a lot about Pi. I used to think that Pi was just only used in equations to find areas and volumes of circles and spheres during Mathematics classes. I finally found out that Pi can also be applied in many aspects of our daily lives, such as in sculpturing and designing of buildings and structures, aircraft designing and as well as in GPS. I feel that the group was very efficient in performing tasks and keeping with deadlines. I feel that everyone has contributed to the group and no small effort from each member has contributed to the group’s project. ~~John

     I never knew that Pi could be used so much in our daily life such as architecture and designing that involve making use of Pi. It has deepened my interest in Mathematics and Physics as I never knew that Pi could also be used in other aspects other than equations. I feel that the group is very co-operative and work efficiently without meeting up with one another as we were able to communicate through the use of technology instead of meeting up in person. ~~Joseph

     I never knew that Archimedes was the inventor of Pi. I have discovered that Pi has changed the world as without it, we will not be able to design buildings and other applications of Pi would not be in existence now in the present. I think that the group’s teamwork is very efficient. Although we never met each other personally, we still managed to communicate with one another through other means and complete our respective tasks to contribute to the group. ~~Wei Feng

     I learned that Pi was invented by Archimedes. It is an interesting fact for me as I did not know about it before. I have also found out that without the invention Pi, it would have been impossible to calculate the area of a circle and the volume of a sphere, so architecture and designing of buildings would have not been able to advance as much. I find that the group is very fun and reliable. ~~Jasper (Yi Jie)

Mathematical Applications - Pi

Application of Pi:

     Pi is one of the most extremely useful and fundamental quantities we know of. It is defined as the ratio of a circle's circumference to its diameter (or it area to the square of its radius).

It can obviously be used to calculate:

- the circumference of a circle (2 X π X r)
- the area of a circle (π X r X r)

Let's look at some more interesting facts about pi.

Ø  It is an irrational number, which means that its value cannot be expressed exactly as a fraction, when the numerator and denominator are integers.

Ø  It has an infinite number of non-repeating digits.

Ø  It is a transcendental number. This means that no sequence of algebraic operations using integers such powers, roots or sums can be equal to its value.

     Throughout the history of mathematics, there has been much effort to determine π more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture.

Everyday Use:

     Pi appears in all sorts of calculations for physics, engineering, electrical systems. The double-helix is DNA revolves around pi. Pi is in the rainbow, the pupil of the eye, and when a raindrop falls into water pi emerges in the spreading rings. It appears in colors and in music. It is also used in probability and statistics.

     It also appears when calculating the number of deaths in a population. Interestingly, the ratio between the actual length of a river and its straight-line from source to mouth length tends to approach pi, Einstein being the one to show this. The current record for the decimal expansion of π, if verified, stands at 5 trillion digits.

     Thus, pi can be applied by many different types of people. Pi can turn out to be actually applied in many daily applications that many of us never even knew! To summarize some everyday uses, here are some of those uses:

1.      Electrical engineers used pi to solve problems for electrical application.

2.      Statisticians use pi to track population dynamics.

3.      Medicine benefits from pi when studying the structure of the eye.

4.      Biochemists see pi when trying to understand the structure/function of DNA.

5.      Physicists looking into the behavior of fluid ripples see pi and use it in their calculations.

6.      Clock designers use pi when designing pendulums for clocks.

7.      Aircraft designers use pi to calculate areas of the skin of aircrafts.

8.      Signal processing and spectrum analysis uses pi to find out what frequencies are in the wave that are used since the fundamental period of a sine wave is 2 X π.

9.      Pi can also be used for navigation, such as global positioning systems (GPS).

     Pi is commonly used for solving geometry, signalling, making probabilities and pi is also used for navigation.

     Let’s go into more details about pi being used for navigation. When planes fly great distances they are actually flying on an arc of a circle. The path must be calculated as such in order to accurately gauge fuel use. Additionally, when locating yourself on the globe, pi comes into the calculation in most methods.

     Pi can also be used in signalling during communication lines as well. Sine waves have a fundamental period of 2 X π, so pi becomes vital in signal processing, spectrum analysis such as when finding out what frequencies are in a wave that you have received or sent.                                                                                                                             

Mathematical Discovery - Archimedes' Principle

The discovery of the Archimedes’ Principle:

    Archimedes may have used his principle of buoyancy to determine whether the golden crown was less dense than solid gold.

 

     The most widely known anecdote about Archimedes tells of how he invented a method for determining the volume of an object with an irregular shape. According to Vitruvius, a votive crown for a temple had been made for King Hiero II, who had supplied the pure gold to be used, and Archimedes was asked to determine whether some silver had been substituted by the dishonest goldsmith. Archimedes had to solve the problem without damaging the crown, so he could not melt it down into a regularly shaped body in order to calculate its density. While taking a bath, he noticed that the level of the water in the tub rose as he got in, and realized that this effect could be used to determine the volume of the crown. For practical purposes water is incompressible, so the submerged crown would displace an amount of water equal to its own volume. By dividing the mass of the crown by the volume of water displaced, the density of the crown could be obtained. This density would be lower than that of gold if cheaper and less dense metals had been added. Archimedes then took to the streets naked, so excited by his discovery that he had forgotten to dress, crying "Eureka!" (Greek: "εὕρηκα!" meaning "I have found it!"). The test was conducted successfully, proving that silver had indeed been mixed in.

 

     The story of the golden crown does not appear in the known works of Archimedes. Moreover, the practicality of the method it describes has been called into question, due to the extreme accuracy with which one would have to measure the water displacement. Archimedes may have instead sought a solution that applied the principle known in hydrostatics as Archimedes' principle, which he describes in his treatise On Floating Bodies. This principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. Using this principle, it would have been possible to compare the density of the golden crown to that of solid gold by balancing the crown on a scale with a gold reference sample, then immersing the apparatus in water. The difference in density between the two samples would cause the scale to tip accordingly. Galileo considered it "probable that this method is the same that Archimedes followed, since, besides being very accurate, it is based on demonstrations found by Archimedes himself."

 

     So, with the clarifications that for a sunken object the volume of displaced fluid is the volume of the object. Thus, in short, buoyancy = weight of displaced fluid. Archimedes' principle is true of liquids and gases, both of which are fluids. If an immersed object displaces 1 kilogram of fluid, the buoyant force acting on it is equal to the weight of 1 kilogram (as a kilogram is unit of mass and not of force, the buoyant force is the weight of 1 kg, which is approximately 9.8 Newton.) It is important to note that the term immersed refers to an object that is either completely or partially submerged. If a sealed 1-liter container is immersed halfway into the water, it will displace a half-liter of water and be buoyed up by a force equal to the weight of a half-liter of water, no matter what is in the container.

 

     If such an object is completely submerged, it will be buoyed up by a force equivalent to the weight of a full liter of water (1 kilogram of force). If the container is completely submerged and does not compress, the buoyant force will equal the weight of 1 kilogram of water at any depth, since the volume of the container does not change, resulting in a constant displacement regardless of depth. The weight of the displaced water, and not the weight of the submerged object, is equal to the buoyant force.

 

     The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (if the surrounding fluid is of uniform density). In simple terms, the principle states that the buoyant force on an object is equal to the weight of the fluid displaced by the object, or the density of the fluid multiplied by the submerged volume times the gravitational constant, g. Thus, among completely submerged objects with equal masses, objects with greater volume have greater buoyancy.

 

     Suppose a rock's weight is measured as 10 Newton when suspended by a string in a vacuum with gravity acting upon it. Suppose that when the rock is lowered into water, it displaces water of weight 3 Newton. The force it then exerts on the string from which it hangs would be 10 Newton minus the 3 Newton of buoyant force: 10 − 3 = 7 Newton. Buoyancy reduces the apparent weight of objects that have sunk completely to the sea floor. It is generally easier to lift an object up through the water than it is to pull it out of the water.