Beyond Pythagoras
What this coursework has asked me to do is to investigate and find a
generalisation, for a family of Pythagorean triples. This will include
odd numbers and even numbers.
I am going to investigate a family of right-angled triangles for which
all the lengths are positive integers and the shortest is an odd
number.
I am going to check that the Pythagorean triples (5,12,13) and
(7,24,25) cases work; and then spot a connection between the middle
and longest sides.
The first case of a Pythagorean triple I will look at is:
[IMAGE]
[IMAGE][IMAGE][IMAGE]The numbers 5, 12 and 13 satisfy the connection.
5² + 12² = 13²
25 + 144 = 169
169 = 13
The second case of a Pythagorean triple I will look at is:
[IMAGE][IMAGE]The numbers 7, 24 and 25 satisfy the connection.
7² + 24² = 25²
[IMAGE][IMAGE][IMAGE]49 + 576 = 625
625 = 25
There is a connection between the middle and longest side. This is
that there is a one number difference.
So if M= middle and L= longest
L = M + 1
I am going to use the triples, (3,4,5), (5,12,13) and (7,24,25) to
find other triples. Then I will put my results in a table and look for
a pattern that will occur. I will then try and predict the next
results in the table and prove it.
[IMAGE]
[IMAGE]
[IMAGE]
n
Smallest
O
Middle
O
Longest
O
1
3
4
5
2
5
12
13
3
7
24
25
There is a clear pattern between the middle and longest side.
There is also a sequence forming.
n = 1 S = 3 M = 4 L = 5
n = 2 S = 5 M = 12 L = 13
8. Repeat from step 4 twice more so you end up with three results for
You solve this problem by plugging in 2, 3, 4, 5, and 6 for x.
Empedocles was born in Acragas, Sicily about 492 BCE to a distinguished and aristocratic family. His father, Meto, is believed to have been involved in overthrowing Thrasydaeus who was the tyrant of Agrigentum in the year 470 BCE. Empedocles is said to have been somewhat wealthy and was a popular politician and a champion of democracy and equality.
Archimedes was born in 287 BC in Syracuse, a Greek seaport colony in Sicily. Archimedes’ father was Phidias. He was an astronomer; this is all we know about his father and we learn this from Archimedes’ work, The Sandreckoner. Archimedes was educated in Alexandria, Egypt. Archimedes’ friend, Heracleides, wrote a biography about him, but this work was lost. Some authors report that he visited Egypt and there invented a tool known as Archimedes' screw. This is a pump, still used today in parts of the world. It is likely that, when he was a young man, Archimedes studied with the followers of Euclid. Many of his ideas seem to correspond with the mathematics developed there. This speculation is much more certain because he sent his results to Alexandria with personal messages. He considered Conon of Samos, one of the greatest achieving mathematicians at Alexandria, both for his abilities as a mathematician and he also respected him as a close friend.
Pythagoras is considered, not only as one of the greatest mathematicians in history, but also for his works concerning music, philosophy, astrology, and many others for all the discoveries made by him. One of the greatest discoveries attributed to Pythagoras is the discovery of the musical scale used nowadays. This scale was based on the principle in which all Pythagoreans base their thought: the existence of numbers in every single aspect in existence. A philosophical belief of universal creation based upon the perfect harmony between numbers and nature. From this argument, he built a whole theory about the harmonies (referring to musical harmonies) which exists in our solar system, which was later developed by several philosophers, physicists, musicians, and so on. He stated that the distances between the different planets had a direct relation with those discovered by him in the musical scale, and that each planet would make a special sound that combined with those of the other planets would create a perfect this harmony that is known as the “Music of the Spheres” (also called “Harmony of the Spheres” or “Universal Harmony”). Based upon his geocentric theory of the solar system, his theory about the celestial harmony created by the spheres, stated that those bodies, with smaller distances to the center of the solar system, or those bodies that orbit closer to the Earth, would make a lower notes that would stay constant and would produce and sound without an end.
in 212 B.C. at the age of 75 in Syracuse. It is said that he was killed
Euclid Of Alexandria may be the best-known mathematician of the world, he is best known for his work on mathematics The Elements. The fact that his work has survived so long, 2000 years in fact, is a tribute to his mathematical genius, however very little of him is known. Three theories abound as to the true nature of this historical figure. Not all historians agree that Euclid was in fact a historical figure, some argue that the school in Alexandria took up the name Euclid to publish their works. But the more accepted theories are that Euclid was in fact a real historical figure who may have been the leader of a team of mathematicians.
The first degree of belief are physical objects, as the second degree of belief are shadows and images of the physical objects. In the last book, Plato criticizes poetry and the fine arts. Plato feels that art is merely the imitation of the imitation of reality, and that poetry corrupts the soul. Socrates says that artists merely create things. As an example, if a painter draws a couch on his canvas, he is creating a couch. But the couch he creates is not the real couch, it is nothing but a copy of an ordinary, physical couch which was created by a craftsman. But the ordinary, physical couch is nothing more than an imperfect copy, or image of the Form of Couch. So, the couch on the canvas is nothing but a copy of a copy of the real couch and is therefore three times removed from reality. Socrates then goes on to explain that an artist's knowledge is also third-rate. If an artist is painting a picture of a table, for example, he is copying a table that has been manufactured by a furniture-maker, and this furniture-maker has more knowledge of the table than the painter does. But there is someone who has ever more knowledge about the table, the person who wants to have the table made. He is the one who gives the furniture-maker instructions to follow when making the table, according to its purpose for the buyer. So, the buyer of the table knows more about the table than the furniture-maker, and the furniture-maker knows more about the table than the painter.
The focus of Socrates at this time in Plato’s Republic is of the ideal city and how it can be traced to the human soul. Socrates believes that the city he has proposed to the other men is perfect in itself. He says that this city possesses four virtues which are the base for the city being perfect. These are the virtues of wisdom, courage, moderation and lastly but most importantly is the virtue of justice. He breaks down the city into classes and he says how each man within the city is responsible for what his life work is. He says that the people of the city whom the mass will see as most educated will be most fit for rule. “You remember the original principle which we were always laying down at the foundation of the State, that one man should practise one thing only, the thing to which his nature was best adapted; now justice is this principle or a part of it.”(433a) It is here that each man concentrates on his own possessions and his own business where we find a just city. He explains that being able to compare social classes within the city is very important because it has produced the important virtue of justice. With Socrates being able to do this he now has to establish a proper dialogue for explaining justice and the soul of an individual.
What is trigonometry? Well trigonometry, according to the Oxford Dictionary ‘the branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles.’ Here is a simplified definition of my own: Trigonometry is a division of mathematics involving the study of the relativity of angles and sides of triangles. The word trigonometry originated from the Latin word: trigonometria.
For the Greeks philosophy wasn’t restricted to the abstract it was also their natural science. In this way their philosophers were also their scientist. Questions such as what is the nature of reality and how do we know what is real are two of the fundamental questions they sought to answer. Pythagoras and Plato were two of the natural philosophers who sought to explain these universal principles. Pythagoras felt that all things could be explained and represented by mathematical formulae. Plato, Socrate’s most important disciple, believed that the world was divided into two realms, the visible and the intelligible. Part of the world, the visible, we could grasp with the five senses, but the intelligible we could only grasp with our minds. In their own way they both sought to explain the nature of reality and how we could know what is real.
(XXY) – three unknowns, test is which is odd one out (Y = 1, Y = 2, or Y = 3).
Euclid of Alexandria was born in about 325 BC. He is the most prominent mathematician of antiquity best known for his dissertation on mathematics. He was able to create “The Elements” which included the composition of many other famous mathematicians together. He began exploring math because he felt that he needed to compile certain things and fix certain postulates and theorems. His book included, many of Eudoxus’ theorems, he perfected many of Theaetetus's theorems also. Much of Euclid’s background is very vague and unknown. It is unreliable to say whether some things about him are true, there are two types of extra information stated that scientists do not know whether they are true or not. The first one is that given by Arabian authors who state that Euclid was the son of Naucrates and that he was born in Tyre. This is believed by historians of mathematics that this is entirely fictitious and was merely invented by the authors. The next type of information is that Euclid was born at Megara. But this is not the same Euclid that authors thought. In fact, there was a Euclid of Megara, who was a philosopher who lived approximately 100 years before Euclid of Alexandria.
Trigonometry is one of the branches of mathematical and geometrical reasoning that studies the triangles, particularly right triangles The scientific applications of the concepts are trigonometry in the subject math we study the surface of little daily life application. The trigonometry will relate to daily life activities. Let’s explore areas this science finds use in our daily activities and how we use to resolve the problem.