Pythagorean Triple Essay
Introduction
A Pythagorean triple is an ordered triplet of positive integers (a,b,c) such that:
a^2+b^2=c^2.
It is evident that such integers correspond to the sides of a right triangle, a, b being the catheti (legs) and c being the hypotenuse of the triangle. The most well known Pythagorean triple is 3,4,5.
As early as in the 8th grade I started feeling that knowing at least a few commonly used Pythagorean triples allows solving various geometry problems with a bigger ease. For example, it helps you spot right triangles and solve for the third side in a triangle. This proved to be a valuable skill when dealing with comparatively primitive geometry problems in elementary school.
In this exploration I will investigate two ways how a Pythagorean triple can be generated. First, the Euclid’s theorem of generating these triplets will be explored and proved that the values generated, with the help of this formula, are in fact a Pythagorean triple corresponding to sides of a right triangle. Next, the Berggren’s Parent/Child relationships will be explored.
…show more content…
two methods of generating primitive Pythagorean triples were explored, such as, through the application of Euclid’s formula and through the application of Berggren’s Parent/Child Relationships. After having explored both methods, a way of generating not only primitive, but also non-primitive triples became evident. Furthermore, in this exploration I was able to prove that by using the Euclid’s formula it is possible to generate three values that correspond to the catheti and hypotenuse of a right triangle, hence all primitive Pythagorean triples. As a great advantage to this exploration I have gained a more thorough understanding of Pythagorean triples and how these can be generated, this topic can be easily linked to the 2D geometry course studied in
Nt1310 Unit 1 Programming Assignment
512 Words | 3 PagesGenerally this week programming assignment was not easy for me to do, there have been a lot of up and down and I find it a little challenging to finish my programming assignment. But unfortunately I was not able to submit my work because by the time I finished, it was too late for me to submit, but I learnt a lot on return values which only can you pass a parameter value into a function, a function can also produce a value. The built-in functions such as abs, pow, int, max, and range, have produced results. Calling each of these functions generates a value, which we usually assign to a variable or use as part of an expression. You have already seen this in some previous functions that you have used.
Comparison Of Herodotus And Cyrus Cylinder
616 Words | 3 PagesThe accounts of the defeat of Babylon from Herodotus and the Cyrus Cylinder are polar opposite, one defeat is through attack while the other is a peaceful take over. Herodotus’ version indicates that Cyrus was led by a need for power, Herodotus writes that after “having subdued the rest of the continent, [Cyrus] turned his attention to Assyria” (78). In Herodotus’ account, after Cyrus spent an entire summer “punishing” a river, he was met with Babylonians awaiting his arrival, and they “attacked him, but they were defeated and forced to retire inside their defences” (83). In order to concur Babylon, Cyrus manipulated Euphrates, allowing his troops to capture the outskirts “without the people in the centre knowing anything about it” (Herodotus 84). Herodotus’ version also states that the people of Babylon were happy before Cyrus invaded the city, they were celebrating during a festival and “continued to dance and enjoy themselves, until they learned the news the hard way” (84).
Herodotus And The Scythians
226 Words | 1 PagesHerodotus begins by describing the reasons for Darius invading Scythians. The Scythians, a century ago, invaded Media and defeated those who stood against them, thus starting the conflict with Persians. For the next twenty-eight years, they lorded the Upper Asia before returning home, only to discover a small army opposing their entrance (Herodotus, P.129) While the Scythians men were abroad, the women had intermarried with their slaves. Instead of fighting with the slaves, the military walked boldly into their territory because they thought that if they hold spears against their slaves, then the slaves might imagine themselves their equals.
Epicurean Conundrum Essay
2012 Words | 9 PagesThe Epicurean Conundrum in question four Everyday most people especially in the United States go through their average life without ever having to fear for where they are going to get their next meal or survive the night. Most of us don’t have to think about such idea’s that revolve around us suffering from dying slowly from preventable diseases or organ failure at an early age. Any other life is nearly unthinkable except when we watch the news and see people in other parts of the world who die from the lack of clean water, food or other curable diseases.
What Are The Achievements Of Islam Dbq
1003 Words | 5 PagesKhwarizmi and other mathematician’s established major advances in math and astronomy which proved to help generations to come after their deaths, the achievements made by these scholars proved to be huge and a history changing element used to this day in the modern era of math and astronomy. In conclusion, the huge impacts the achievements made in math and medicine helped shape the history of both subjects, providing a foundation to build upon through centuries later by rising scholars in the era’s to come. Muslim’s rebuilt the scholarship era, building the cultural center in Spain proved to be an incredible achievement itself as the center established a foundation for education and important figures in the subjects of math, astronomy, science, and religion. There is no doubt these 3 achievements produced by Muslims shaped the foundation of education and the subjects under schooling; the achievements impacted the world in a way which had not been seen yet and influenced other cultures to push for similar influences in their
Alcibiades Plato Analysis
1013 Words | 5 PagesSocrates in the dialogue Alcibiades written by Plato provides an argument as to why the self is the soul rather than the body. In this dialogue Alcibiades and Socrates get into a discussion on how to cultivate the self which they both mutually agree is the soul, and how to make the soul better by properly taking care of it. One way Socrates describes the relationship between the soul and the body is by analogy of user and instrument, the former being the entity which has the power to affect the latter. In this paper I will explain Socrates’ arguments on why the self is the soul and I will comment on what it means to cultivate it.
Odysseus As A Hero Essay
450 Words | 2 PagesThe odyssey, an epic told by Homer in ancient greece, has many major themes following odysseus’s adventures. While Odysseus is sentenced to never return home after the Trojan War. He is overcoming challenges to return home to his wife penelope and his son Telemachus. Throughout the story major themes of loyalty, hospitality and vengeance are hidden within the plot. The story continues to show his heroic side with three major traits.
Classical Athens Dbq Essay
714 Words | 3 PagesThe birthplace of democracy: A look into the life Aristotle once said, “The roots of education are bitter, but the fruit is sweet.” Athens was established in 3000 BC, while Sparta was respectively established in 431 BC. Athens was also referred to as the “birthplace of democracy,” a government system still in use today! Athens was a superior city state over Sparta. This is because of Athenian cultural achievements, government and social climate.
774 S. E. 2d 546
537 Words | 3 Pages1. Outlines principles of law in relation to variances of an indictment in general, in relation to the “manners and means” of committing a crime. • VARIANCE TO INDICTMENT occurs when facts proved at trial are different from those alleged or specified in the indictment. • MANNERS AND MEANS is the way the crime is done and the method of committing the crime. For example, Zizzi’s wife was beaten/ hit which is the manner and the means/method is either the golf club or the ornament from the bathroom.
Epictetus Analysis
726 Words | 3 PagesEpictetus’s way of philosophy is one that is purely Stoic, imploring that the solution to human finitude is one where humans can live life without showing feeling or complaining about pain and hardships towards unsavory situations. Each of his rules in his handbook offers advice in which the subject simply “deals” with disappointment, or rather, doesn’t expect something out of the scopes of reason and logic, so that, figuratively, when occurrences don’t go their way, they aren’t disappointed. This is because to Epictetus, all external events in life are pre-determined by fate, so it’s already out of our hands from the beginning. With a calm dispassion, or indifference, we approach our fate and accept it. This is shown in his rules in The Handbook,
Standardized Education In Charles Baxter's 'Gryphon'
1005 Words | 5 PagesThis quote proves the interest the children having in learning about these things. Rarely do fourth graders happily discuss arithmetic to any extent. Miss Ferenczi is a positive influence by teaching them to be excited about learning through the stories she tells them.
Essay On Odysseus Is An Epic Hero
726 Words | 3 PagesIf you were to think about a man with great abilities like being strong, courageous and intelligent, you could think of people you personally or like me, you could think of someone such as Odysseus. Odysseus fits the definition of an epic hero because he is quick thinking, strong and courageous. Odysseus is one quick thinking individual. He has gotten him and his crew out some some sticky situations with his clever plans. One of these plans were how to get away from Polyphemus, a cyclops who was the son of Poseidon.
13 Rules That Expire Summary
1584 Words | 7 PagesOften enough teachers come into the education field not knowing that what they teach will affect the students in the future. This article is about how these thirteen rules are taught as ‘tricks’ to make math easier for the students in elementary school. What teachers do not remember is these the ‘tricks’ will soon confuse the students as they expand their knowledge. These ‘tricks’ confuse the students because they expire without the students knowing. Not only does the article informs about the rules that expire, but also the mathematical language that soon expire.
Reflective Essay On Why I Want To Study Psychology
739 Words | 3 PagesBeing a health and fitness enthusiast I have always been interested in learning about how the mind works physically and emotionally, and in what ways the body reacts to various mental states is very intriguing. That is why I want to explore the mind and the essence of the human condition. We all question or try to make sense of our life; I believe that through its many perspectives, psychology has those answers. Psychology is accessible because not only is it a scientific study of thought and behaviour, it is also an art that can be applied by the individual to everyday life. I also believe we are all psychologists attempting to understand other people and ourselves.
Why Is Geometry Important In Education Essay
1613 Words | 7 PagesPart B Introduction The importance of Geometry Children need a wealth of practical and creative experiences in solving mathematical problems. Mathematics education is aimed at children being able to make connections between mathematics and daily activities; it is about acquiring basic skills, whilst forming an understanding of mathematical language and applying that language to practical situations. Mathematics also enables students to search for simple connections, patterns, structures and rules whilst describing and investigating strategies. Geometry is important as Booker, Bond, Sparrow and Swan (2010, p. 394) foresee as it allows children the prospect to engage in geometry through enquiring and investigation whilst enhancing mathematical thinking, this thinking encourages students to form connections with other key areas associated with mathematics and builds upon students abilities helping students reflect
More about Pythagorean Triple Essay
Nt1310 Unit 1 Programming Assignment
512 Words | 3 PagesComparison Of Herodotus And Cyrus Cylinder
616 Words | 3 PagesHerodotus And The Scythians
226 Words | 1 PagesEpicurean Conundrum Essay
2012 Words | 9 PagesWhat Are The Achievements Of Islam Dbq
1003 Words | 5 PagesAlcibiades Plato Analysis
1013 Words | 5 PagesOdysseus As A Hero Essay
450 Words | 2 PagesClassical Athens Dbq Essay
714 Words | 3 Pages774 S. E. 2d 546
537 Words | 3 PagesEpictetus Analysis
726 Words | 3 PagesStandardized Education In Charles Baxter's 'Gryphon'
1005 Words | 5 PagesEssay On Odysseus Is An Epic Hero
726 Words | 3 Pages13 Rules That Expire Summary
1584 Words | 7 PagesReflective Essay On Why I Want To Study Psychology
739 Words | 3 PagesWhy Is Geometry Important In Education Essay
1613 Words | 7 Pages