Most mathematicians will be familiar with the above picture. This diagram, credited to the Ancient Chinese mathematical text Zhou Bi Suan Jing, is a charmingly simple visual proof of the Pythagorean Theorem, one of mathematics’ most fundamental results. It would be hard to argue that this proof is not convincing. In fact, most standard proofs of the Pythagorean Theorem still use this picture, o...

We show that Möbius transformations preserve the rotationminimizing frames which are associated with space curves. In addition, these transformations are known to preserve the class of rational Pythagorean-hodograph curves and also rational frames. Based on these observations we derive an algorithm for G Hermite interpolation by rational Pythagorean-hodograph curves with rational rotation-minim...

Multi-criteria group decision-making is a process in which decision makers assess the performance of alternatives on the basis of conflicting criteria to opt the most worthy alternative as solution. TOPSIS and ELECTRE are effective and commonly used methods to solve multiple criteria decision-making problems. The aim of this study is to propose two new models, namely, complex Pythagorean fuzzy ...

Recently, a new model based on Pythagorean fuzzy set (PFS) has been presented to manage the uncertainty in real-world decision-making problems. PFS has much stronger ability than intuitionistic fuzzy set to model such uncertainty. In this paper, we define some novel operational laws of PFSs and discuss their desirable properties. For the multicriteria decision-making problems with PFSs, we prop...

This document gives a formal proof of the cases n = 3 and n = 4 (and all their multiples) of Fermat’s Last Theorem: if n > 2 then for all integers x, y, z: x + y = z =⇒ xyz = 0. Both proofs only use facts about the integers and are developed along the lines of the standard proofs (see, for example, sections 1 and 2 of the book by Edwards [Edw77]). First, the framework of ‘infinite descent’ is b...

The Pythagorean triples of integers satisfying x + y = z have been studied and enumerated since Babylonian times. Since Diophantus, it has been known that this set of triples is related to the standard rational parameterization of the unit circle, ( t 2−1 t2+1 , 2t t2+1 ). The Pythagorean triple solutions, which are relatively prime, have the elementary and beautiful characterization as integer...

This study aims to determine the difference in learning achievement of class VIII students SMP Negeri 4 Mataram through online using WhatsApp Group and Google Classroom applications for 2020/2021 academic year. research is an experimental with a posttest-only non-equivalent group design used . Determination sample probability sampling technique being 2 receiving application as 1 3 2. The instru...

In this paper we present two new types aggregation operators such as, induced Pythagorean fuzzy ordered weighted averaging aggregation operator and induced Pythagorean fuzzy hybrid averaging aggregation operator. We also discuss of important properties of these proposed operators and construct some examples to develop these operators.

A Clifford algebra has three major multiplications: inner product, outer product and geometric product. Accordingly, the same Clifford algebra has three versions: Clifford vector algebra, which features on inner products and outer products of multivectors; Clifford bracket algebra, which features on pseudoscalars and inner products of vectors; Clifford geometric algebra, which features on geome...

In 1998, in the winter issue of Mathematics and Computer Education ([1]) Monte Zerger posed the following problem. He had noticed or discovered the Pythagorean triple (216, 630, 666); (216) + (630) = (666). Note that 216 = 6 and 666 is the hypotenuse length of this Pythagorean triangle. The question was, then whether there existed a digit d (in the decimal system) and a positive integer k (othe...