Print Сite this

Generating Pythagorean Triples

Introduction

Pythagorean triple, originated from the terminology referred to as Pythagorean Theorem, which states that each right-angled triangle has its sides that satisfy the formula x2+y2=z2 and thereby, the 3 sides of a right-angled triangle are actually described by a Pythagorean triple” (Stillwell, 2003, p.5). Fundamentally, three numerals that are positive are comprised of in a Pythagorean triple. For instance, if e, f, and g are positive numerals, then e2+f2=g2. In numerical stipulations, a Pythagorean triple is simply a set of 3 numerals x, y, z that “make-up the sides/lengths of a right-angled triangle” (Sierpinski, 2003, p. 7).

Our experts can deliver a customized essay
tailored to your instructions
for only $13.00 $11.05/page
308 qualified specialists online
Learn more

Idyllically, 3, 4, 5 are the smallest set of positive numerals in a Pythagorean triple. In this paper, someone has to complete reading Chapter ten from the textbook that is entitled, ‘Mathematics in Our World’, and which was written by Bluman G. Allan. In doing so, he/she will choose not less than five Pythagorean triples. Indeed, he/she ought to be concise on his/her reasoning. Thus, this paper aims at illustrating why the selected five groups of Pythagorean Triples are capable of working, specifically in the formula Pythagorean Theorem.

Generating Pythagorean Triples

There exists a formula that is uncomplicated, and which is able to engender Pythagorean triples. Ideally, the numerals 5, 4, and 3 are referred to as Pythagorean triples because 52=42+32. Similarly, the numerals 13, 12 and 5 are Pythagorean triples. This is in view of the fact that 132=122+52. Suppose j and k are two numerals that are positive, such that j is less than k, then k2-j2, 2kj, and k2+m2 form a Pythagorean triple (Bluman, 2005).

Algebraically, it is easier to notice that the sum-of-squares of the first-two is similar to square of the last-one,” (Alperin, 2005, p. 809). In consequence, this technique is capable of engendering each triple. This simply means that a Pythagorean triple is a set of (x, y, z) that is capable of working in the equation x2+y2=z2. By employing this formula, we are capable of finding any numeral of this kind, that is, “the square of x plus the square of y is equal to the square of z” (Alperin, 2005, p. 811).

By employing this formula, we can observe that Pythagorean triples that can be engendered, and which actually works encompass:

[3, 4, 5] i.e. 32+42=52;

[8, 6, 10] i.e. 82+62=102;

On-Time Delivery! Get your 100% customized paper
done in
as little as 3 hours
Let`s start

[15, 8, 17] i.e. 152+82=172;

[27, 36, 45] i.e. 272+362=452;

[33, 56, 65] i.e. 332+562=652;

[45, 26, 53] i.e. 452+262=532;

[13, 84, 85] i.e. 132+842=852;

[20, 21, 29] i.e. 202+212=292; and

[45, 108, 117] i.e. 452+1082=1172

We’ll deliver a custom paper tailored to your requirements.
Cut 15% off your first order
Use discount

Conclusion

In a Pythagorean triple that is primitive, x and y are co-prime, that is, they do not share any numerals or prime factors. In this kinds of Pythagorean triples, either x or y can be an odd numeral, or an even numeral; basing on this, it can be seen that z will be odd (Eckert, 1992). Nonetheless, the notion of Pythagorean triples can be generalized by manifold ways, such as Pythagorean quadruple, Pythagorean n-tuple, Fermat’s Last-Theorem; and Heronian triangle-triples.

References

Alperin, C.R. (2005). Pythagoras Model Tree. American Mathematical Journal, 112 (8), 806-815.

Bluman, A. G. (2005). Mathematics In Our World. College of Allegheny: McGraw-Hill Publishers.

Eckert, E. (1992). Pythagorean Triples that are Primitive. College Mathematics Journal, 23 (4), 412-418.

Sierpinski, W. (2003). Pythagorean Triangles. London: Dover Publications.

Stillwell, J. (2003). Fundamentals of Number Theory. New York: Springer Publishers.

Cite this paper

Select style

Reference

StudyCorgi. (2022, August 26). Generating Pythagorean Triples. Retrieved from https://studycorgi.com/generating-pythagorean-triples/

Reference

StudyCorgi. (2022, August 26). Generating Pythagorean Triples. https://studycorgi.com/generating-pythagorean-triples/

Work Cited

"Generating Pythagorean Triples." StudyCorgi, 26 Aug. 2022, studycorgi.com/generating-pythagorean-triples/.

* Hyperlink the URL after pasting it to your document

1. StudyCorgi. "Generating Pythagorean Triples." August 26, 2022. https://studycorgi.com/generating-pythagorean-triples/.


Bibliography


StudyCorgi. "Generating Pythagorean Triples." August 26, 2022. https://studycorgi.com/generating-pythagorean-triples/.

References

StudyCorgi. 2022. "Generating Pythagorean Triples." August 26, 2022. https://studycorgi.com/generating-pythagorean-triples/.

References

StudyCorgi. (2022) 'Generating Pythagorean Triples'. 26 August.

This paper was written and submitted to our database by a student to assist your with your own studies. You are free to use it to write your own assignment, however you must reference it properly.

If you are the original creator of this paper and no longer wish to have it published on StudyCorgi, request the removal.