Document details

Tag’s

advanced mathematical skills
approach to homosexuality
behavioral modification approach
carl rogers approach
childlike approach
cognitice approaches
communications theoretical approach
comparing mathematics and english
complex numbers
comprehensive
comprehensive marketing audit
comprehensive personality questionnaire
comprehensive view of the world
copyright approach
denominators
equator
euclids
father of modern mathematics
generativity
generators
getting on
gore
helping students understand mathematics
holy vs. humanistic approach
human capital approach
humanistic centered approach
idealistic approaches to dreams
improving the teaching of mathematics
influence of pythagorean thinking
international approach methods
islamic mathematics
let it be
long-term approaches
major theoretical approaches
managing values a systematic approach to business ethics
mannerisms
math
mathematical branch
mathematical discipline
mathematical tablet
mathematics
minskyâ€™s balance-sheet approach
multiplicity
non-muscle sparing approaches
noting
oac
ore
organizational change approach
originator
pedagogical approaches to religious education
pedagogical approaches to student involvement
person-centered approach to therapy
possible triples
practical approaches to dreams
pro-active approach to crime
probabilities
pythagorean
pythagorean literature
pythagorean triple
pythagorean triples
pythagoreanism
pythagoreans
questioners
rational number
rationing
right realist approach
supranational approach
the bank street approach
the comprehensive everglades restoration plan
the encoding-decoding approach
the golden rule approach
the human approach to world peace
the lewinian approach
the posterior approach
the pythagorean formula
the pythagoreans
three integers
trip
triple-therapy drug regimen
underage drinking approaches
use of geometry
why students fail mathematics
willingness-to-pay approach

Document details

Pythagorean TriplesThree integers a, b, and c that satisfy a2 b2 c2 are called Pythagorean Triples. There are infinitely many such numbers and there also exists a way to generate all the triples. Let n and m be integers, nm. Then define() a n2 - m2, b 2nm, c n2 m2. The three number a, b, and c always form a Pythagorean triple. The proof is simple (n2 - m2)2 (2mn)2 n4 - 2n2m2 m4 4n2m2 n4 2n2m2 m4 (n2 m2)2. The formulas were known to Euclid and used by Diophantus to obtain Pythagorean triples with special properties. However, he never raised the question whether in this way one can obtain all possible triples.The fact is that for m and n coprime of different parities, () yields coprime numbers a, b, and c. Conversely, all coprime triples can indeed be obtained in this manner. All others are multiples of coprime triples ka, kb, kc.As an aside, those who mastered the arithmetic of complex numbers might have noticed that (m in)2 (n2 - m2) i2mn. Which probably indicates that () has a source in trigonometry. But the proof below only uses simple geometry and algebra.First of all, note that if a2 b2 c2, then (ac)2 (bc)2 1. With x ac and y bc we get x2 y2 1. This is the well known equation of the unit circle with center at the origin. Finding Pythagorean triples is therefore equivalent to locating rational points (i.e., points (x,y) for which both x and y are rational) on the unit circle. For if (pq)2 (rs)2 1, multiplication by a common denominator leads to an identity between integers.Rational numbers approximate irrational to any degree of accuracy. Therefore, the set of rational pairs is dense in the whole plane. So, perhaps, one might...

similar examples

An Introduction to the Pythagorean Theorem Used in Maths and Physics

Category:
Mathematics

1088 words

2 pages

The Pythagorean Theorem is a geometrical expression used often in math and physics. It used to2 2 2find the unknown side of a right triangle. The exponential form of this theorem a b c . That is the equation you use when you are looking for the unknown side of ...

The Life of Leonardo Pisano and His Role in Shaping the Field of Mathematics

Category:
Mathematics

841 words

1 pages

Leonardo Pisano was the first great mathematician of medievalChristian Europe. He played an important role in revivingancient mathematics and made great contributions of his own. After his death in 1240, Leonardo Pisano became known as LeonardoFibonacci. Leonardo Fibonacci was born in Pisa in about 1180, the son of a member ...

An Analysis of the Pythagorean Theorem in Geometry, Mathematics, and Physics

Category:
Mathematics

1088 words

2 pages

The Pythagorean Theorem is a geometrical expression used often in math and physics. It used to 2 2 2find the unknown side of a right triangle. The exponential form of this theorem a b c . That is the equation you use when you are looking for the unknown side ...

An Introduction to the History and Analysis of Pythagorean Triples

Category:
Mathematics

646 words

4 pages

Pythagorean TriplesThree integers a, b, and c that satisfy a2 b2 c2 are called Pythagorean Triples. There are infinitely many such numbers and there also exists a way to generate all the triples. Let n and m be integers, nm. Then define()a n2 - m2, b 2nm, c n2 m2. ...

An Analysis of Pythagorean Theorem by the Greek Mathematician Pythagoras

Category:
Mathematics

478 words

2 pages

According to Laubenbache and Pengelley (1999)The Pythagorean Theorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy. He is credited with many ...

An Essay on Pythagorean Triples

Category:
Mathematics

649 words

4 pages

Pythagorean TriplesPythagorean TriplesThree integers a, b, and c that satisfy a2 b2 c2 are called Pythagorean Triples. There are infinitely many such numbers and there also exists a way to generate all the triples. Let n and m be integers, nm. Then define() a n2 - m2, b 2nm, c ...

Services

We accept:

eliteessaywriters.com © 2018