Free

Pythagorean triples - Essay Example

Comments (0) Cite this document
Summary
It is about a property of all right triangles showing a relationship between their sides given by the equation c2 = a2 + b2; that is, the square of the longest side is equal to the…
Download full paperFile format: .doc, available for editing
GRAB THE BEST PAPER99% of users find it useful
Pythagorean triples
Read TextPreview

Extract of sample "Pythagorean triples"

PYTHAGOREAN TRIPLES The Pythagorean Theorem is probably one of the first principles learned in a Trigonometry It is about a property of all right triangles showing a relationship between their sides given by the equation c2 = a2 + b2; that is, the square of the longest side is equal to the sum of the squares of the remaining two sides (Knott 2009). Hence, given any two sides of a right triangle (triangle with a 90-degree angle) the unknown side can always be solved by algebraically manipulating the equation presented above.
A special case of such triangles is when all the three sides, a, b and c, are integers, thus generating a Pythagorean Triple (Bogomolny 2009). The most popular example of which is the 3-4-5 triangle, the triple which, according to Knott (2009), was known to the Babylonians since way back 5,000 years and was possibly used as a basis in making true right angles in ancient building construction.
Then again, the 3-4-5 triangle is just one of the infinitely many Pythagorean Triples, and mind you, there are various ways of generating such triples. One is, given two integers n and m, where n > m, then sides a, b and c are define as n2 - m2, 2nm and n2 + m2, respectively, following a simple proof (Bogomolny 2009):
Taking for instance the triple 8-15-17, which is generated by taking n = 4 and m = 1, then a = n2 - m2 = 4­­2 - 12 = 16 - 1 = 15; b = 2mn = 2(1)(4) = 8, and; c = n2 + m2 = 4­­2 + 12 = 16 + 1 = 17. Another example is 7-24-25, which can be verified using n = 4 and m = 3. Such triples are examples of Primitive Pythagorean Triples, or those triples that are not multiples of another and are found using the n-m formula (Knott 2009). Other Pythagorean Triples can be found using a variety of methods as presented by Bogomolny (2009) and Knott (2009), some of which are:
a) by generating multiples of Primitive Pythagorean Triples, ka, kb, kc: 2 * (3,4,5) = 6, 8, 10;
b) by taking a series of multiples 1, 11, 111, …: 3-4-5, 33-44-55, 333-444-555, …;
c) by Two-fractions method—choose any two fractions whose product is 2, add 2 to each fraction, then cross multiply, getting the two shorter sides of the triple: 4/2, 2/2 → 8/2, 6/2 → 16, 12 → 162 + 122 = 202, and;
d) by taking (m+1) for n, with m as powers of 10, hence simplifying the triples into 2m+1, 2m(m+1), 2m2+2m+1: 2(10) + 1 = 21, 2(10)(10+1) = 220, 2(10)2 + 2(10) + 1 = 221.
To sum it up, there are infinitely many Pythagorean Triples existing. But one thing is for sure, a variety of techniques are available that will serve useful in generating patterns among such triples. Hence, if you cannot list them all, be familiar of their patterns at least.
References:
Bogomolny, A. (2009). Pythagorean Triples. Retrieved November 20, 2009, from Interactive Mathematics Miscellany and Puzzles Web site: http://www.cut-the-knot.org/pythagoras/pythTriple.shtml
Knott, R. (2009). Pythagorean Triangles and Triples. Retrieved November 20, 2009, from The University of Surrey, Mathematics Web site: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html Read More
Cite this document
  • APA
  • MLA
  • CHICAGO
(“Pythagorean triples Essay Example | Topics and Well Written Essays - 500 words”, n.d.)
Retrieved from https://studentshare.org/miscellaneous/1560180-pythagorean-triples
(Pythagorean Triples Essay Example | Topics and Well Written Essays - 500 Words)
https://studentshare.org/miscellaneous/1560180-pythagorean-triples.
“Pythagorean Triples Essay Example | Topics and Well Written Essays - 500 Words”, n.d. https://studentshare.org/miscellaneous/1560180-pythagorean-triples.
  • Cited: 0 times
Comments (0)
Click to create a comment or rate a document

CHECK THESE SAMPLES OF Pythagorean triples

Nursing - Use of Herbals

6 Pages(1500 words)Research Paper

Ionian and Pythagorean schools

...Ionian and Pythagorean views The first Greek philosophers who laid the foundation of the recorded and well-known philosophical works of Plato and Aristotle were the Ionians and the Pythagorean. They also influence the works of succeeding revolutionary scientists such as Galileo, Copernicus, Newton and even Einstein in the modern period. Their scientific philosophies also underpin the theoretical and basic methodological approaches of current scientific practice. It can be said that the Ionian School with its naturalist approach in observing the world and phenomena provided the empirical or inductive method that is seen as one way of arriving at scientific knowledge. The Pythagoreans on the other hand, with their basic conception... of...
4 Pages(1000 words)Case Study

Pythagorean triplets

...Pythagorean Triplets Pythagorean triplets are integer solutions to the Pythagorean Theorem. Pythagoras lived around 500 BC, but Pythagorean triplets were first recorded as far back as 1700-1800 BC in Babylonia, inscribed on a clay tablet, known as Plimpton 322. A large number of integer pairs (a,c) for which there is an integer b satisfying Pythagoras' equation were systematically listed on it. Let a, b, and c be positive real numbers. Then c is the length of the hypotenuse of a right triangle with side lengths a, b, and c if and only if; a2 + b2 = c2 The Pythagorean triplet is an ordered triplet (a, b, c) of three positive integers such that a2 + b2...
2 Pages(500 words)Essay

MATH 1)NUMBER SYSTEMS 2)PYTHAGOREAN BROTHERHOOD 3)BOOK 'HOW TO SOLVE IT' 4) JOHN NAPIER

...the alteration of the scale of logarithms from the hyperbolic 1 / e from which John Napier had given to that which unity is assumed as the logarithm of the ratio of 10 to 1 [2]. During their conversation, the alteration proposed by Briggs was agreed upon by Napier and published the first chiliad of his logarithms during his second visit to Edinburgh in 1617. In 1624 Henry Briggs gave a numerical approximation to the base 10 logarithm. Pythagorean Brotherhood The Pythagorean Brotherhood is one of Greece's ancient societies. Pythagoreans, as the members were called combined philosophy and politics. This so-called brotherhood was established by Pythagoras, a Greek philosopher...
5 Pages(1250 words)Essay

The Pythagorean Theorem

...THE PYTHAGOREAN THEOREM [Source: http www.arcytech.org/java/pythagoras/preface.html] April 30, 2008 The Pythagorean Theorem In mathematics, the Pythagorean Theorem is a relation in Euclidean geometry among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras who lived in the 6th century B.C. The theorem is as follows: "In any right angle triangle, the area of the square of the side opposite the right angle i.e. whose side is the hypotenuse is equal to the sum of the areas of the squares of the two sides that meet at a right angle i.e. whose sides are the two legs" In other words The square on the hypotenuse is equal to the sum of the squares on the other two sides Geometric... ...
14 Pages(3500 words)Essay

Pythagorean Thagorean equations

...Pythagorean Triples A Pythagorean triple refers to the three positive integers, a,b, and c, in the Pythagorean equation a2 + b2 = c2. These three positive integers represent the three sides of a right triangle. Suppose we have m and n which are both positive integers at which mPythagorean triple which represents the values of a,b, and c. In this assignment, we are required to generate five more Pythagorean Triples. 1. [8,6,10] 2. [12,16,20] 3. [9,40,41] 4. [91,60,109] 5. [28,96,100] After identifying five triples, we should verify each of them using the Pythagorean Theorem equation. 1. [8,6,10] let a = 8, b = 6, and c = 10 a2 + b2 = c2 82 + 62 = 64 + 36 = 100 102  = 100 2. [12,16,20] let a = 12... , b = 16, and c = 20 a2 + b2 = c2 122 + 162 = 144 + 256 = 400 202  = 400 3. [9,40,41] let a = 9, b = 40, and c = 41 a2 + b2 = c2 92 + 402 = 81 + 1600 = 1681 412  = 1681 4. [91,60,109] let a = 91, b = 60, and c = 109 a2 + b2 = c2 912 + 602 = 8281 + 3600 = 11881 1092  = 11881 5. [28,96,100] let a = 28, b = 96, and c = 100 a2 + b2 = c2 282 + 962 = 784 + 9216 = 10000 1002  = 10000 In all the triples identified, it was verified that a, b, and c are in...
1 Pages(250 words)Essay

Pythagorean Theorem

...Pythagorean Theorem (Add (Add (Add Pythagorean Theorem Introduction Evidences show that Pythagorean Theorem was popular even among ancient civilizations. This famous theorem was developed by the Greek mathematician and philosopher Pythagoras. Historical writings argue that though Babylonian mathematicians had knowledge in the theorem, they could not develop it into a mathematical framework. Unlike any other mathematical theorem, the Pythagorean Theorem is supported by both geometric and algebraic...
3 Pages(750 words)Essay

Math Project

...by the transcendence done by Lindemann which stated that the circle could not be squared. Lastly, the Twentieth Century idiosyncrasies exposed the division between analytical and computational studies. Ramanujans Modular Equations and Approximations exhibits a remarkable series for 1/ π. 4. The number 3,4 and 5 are called Pythagorean triples since 32+42=52. The numbers 5,12,and 13 are also Pythagorean triples since 52+122=132 .More Pythagorean triples include; 8,15 and 17. 9,20 and 41(92+402=412). 12,13 and 37(122+132=37 2). 13,84 and 85(132+842=852). 15,112and 113( 152+ 1122+1132) . 16,63 and 65(162+632=652)....
2 Pages(500 words)Speech or Presentation

Pythagorean Triples

...Pythagorean Triples A Pythagorean triple refer to a set of three positive integers a, b, c such that the sum of the squares of two smaller integers of them equal to the square of the third largest integer i.e. if c > a, b then a2 + b2 = c2, c2 – a2= b2, or, c2 – b2 = a2. Pythagorean triples are part of Pythagorean Theorem which says that in a right triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides 1. Each triple set has its many multiples which will be regarded as the same triple set i.e. the set (a, b, c) and its any multiple k, ka, kb, kc,...
1 Pages(250 words)Speech or Presentation

Pythagorean Quadratic

...Pythagorean Theorem and Quadratic Equation Introduction Discovered and developed by scientist and mathematician, Pythagoras (c. 570 BC – c. 495 BC),‘Pythagorean Theorem’ is a widely applied mathematical statement which illustrates the relation of the three sides of a right triangle that consists of two legs and a longest side, known as the hypotenuse. In equation, it is given by -- c2 = a2 + b2 where the variables ‘a’ and ‘b’ refer to the length measures of the right triangle’s legs while the variable ‘c’ pertains to the hypotenuse. Applications of Pythagorean Theorem are recognized in various fields of maths such as algebra, trigonometry, and calculus. Problem 98: ...
1 Pages(250 words)Speech or Presentation
sponsored ads
We use cookies to create the best experience for you. Keep on browsing if you are OK with that, or find out how to manage cookies.

Let us find you another Essay on topic Pythagorean triples for FREE!

Contact Us