Beyond Pythagoras - Mathematical Investigation
1) Do both 5, 12, 13 and 7, 24, 25 satisfy a similar condition of :
(Smallest number)² + (Middle Number)² = (Largest Number) ² ?
5, 12, 13
Smallest number 5² = 5 x 5 = 25
Middle Number 12² = 12 x 12 = 144+
169
Largest Number 13² = 13 x 13 = 169
7, 24, 25
Smallest number 7² = 7 x 7 = 49
Middle Number 24² = 24 x 24 = 576+
625
Largest Number 25² = 25 x 25 = 625
Yes, each set of numbers does satisfy the condition.
They are both Pythagorean triples.
Area = 12 x 5
2
Area = 6 x 5
Area = 30
Perimeter = 5
12
13+
30
13
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5
12
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Area = 24 x 7
2
Area = 12 x 7
Area = 84
Perimeter = 7
24
25+
56
7
24
Area = 40 x 9
2
Area = 20 x 9
Area = 180
Perimeter = 9
40
41+
90
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41
40
9
b)
3)
From my data I can construct a table to make the identification of
patterns easier.
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So far I have observed the following patterns:
· The smallest side length advances by two each time.
· All the middle numbers are even
· The middle length is always a multiple of four.
· The longest length is always one unit more than the middle length.
· Both the shortest and longest side lengths are always odd.
· Both the area and perimeter are always even numbers.
I can immediately see the formula to derive the shortest side length
12. If d = 3 + e, and e = 4, what is the value of (20 - d) + e
Geometry is the branch of mathematics that deals with the properties of space. Geometry is classified between two separate branches, Euclidean and Non-Euclidean Geometry. Being based off different postulates, theorems, and proofs, Euclidean Geometry deals mostly with two-dimensional figures, while Demonstrative, Analytic, Descriptive, Conic, Spherical, Hyperbolic, are Non-Euclidean, dealing with figures containing more than two-dimensions. The main difference between Euclidean, and Non-Euclidean Geometry is the assumption of how many lines are parallel to another. In Euclidean Geometry it is stated that there is one unique parallel line to a point not on that line.
in 212 B.C. at the age of 75 in Syracuse. It is said that he was killed
Born between 530-569 B.C. Pythagoras of Samos is described as the first "pure mathematician." Pythagoras' father was Mnesarchus of Tyre and Pythais of Samos. Mnesarchus was a merchant who was granted citizenship after he brought corn to Samos during a famine. The citizenship was an act of gratitude. There are accounts that Pythagoras traveled widely with his father, even back to his father's home, Tyre and Italy. During these travels Pythagoras was educated by Chaldaeans and learned scholars in Syria.
Archimedes, a name commonly associated with the beginning of science, was an engineer and one of the greatest mathematicians in history. His impact on modern science rests on his use of experiment and invention to test ideas and his use mathematics to describe the basic principles of physical phenomena.
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.
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.
Archimedes (287BC-212BC) was truly one of the greatest mathematical minds of all time. The discoveries and inventions of Archimedes formed the basis of many of the fundamental concepts of modern physics and mathematics.
Livio, Mario. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway, 2002. Print.
are how to follow lines of reasoning, how to say precisely what is intended, and
Mathematicians, engineers and scientists encounter numerous functions in their work: polynomials, trigonometric and hyperbolic functions amongst them. However, throughout the history of science one group of functions, the conics, arise time and time again not only in the development of mathematical theory but also in practical applications. The conics were first studied by the Greek mathematician Apollonius more than 200 years BC. Essentially, the conics form that class of curves which are obtained when a double cone is intersected by a plane. There are three main types: the ellipse , the parabola and the hyperbola . From the ellipse we obtain the circle as a special case, and from the hyperbola we obtain the rectangular hyperbola as a special case. These curves are illustrated in the following figures. cone-axis
Born the son of an astronomer, Phidias, in 287 B.C., Archimedes' education began as a young man in Syracuse. He furthered his education in Alexandria, where he studied with fellow scholar Conon, an Egyptian mathematician.
The Nature of Mathematics Mathematics relies on both logic and creativity, and it is pursued both for a variety of practical purposes and for its basic interest. The essence of mathematics lies in its beauty and its intellectual challenge. This essay is divided into three sections, which are patterns and relationships, mathematics, science and technology and mathematical inquiry. Firstly, Mathematics is the science of patterns and relationships. As a theoretical order, mathematics explores the possible relationships among abstractions without concern for whether those abstractions have counterparts in the real world.