Irrational Numbers, Square Roots, and Quadratic Equations

2015 ◽  
Vol 20 (8) ◽  
pp. 468-474
Author(s):  
Gorjana Popovic
Keyword(s):  
Irrational Numbers ◽  
Square Roots ◽  
Quadratic Equations

How do we introduce irrational numbers without simply telling students that they are irrationals?

The Great Art

2016 ◽  
Author(s):  
Joseph Mazur
Keyword(s):  
Eleventh Century ◽  
Negative Numbers ◽  
Square Roots ◽  
Quadratic Equations ◽  
Indian Mathematician ◽  
Northern India ◽  
First Millennium

This chapter discusses the origins of the art of algebra, beginning with the possibility that it may have come from the Greeks or from the Hindus. However, the Brahmins of northern India had some idea of algebra long before the Arabians learned it, contributed to it and brought that art to Spain in the late eleventh century. The Brahmasphutasiddhanta, written by the Indian mathematician Brahmagupta in 628, not only advanced the mathematical role of zero but also introduced rules for manipulating negative and positive numbers, methods for computing square roots, and systematic methods of solving linear and limited types of quadratic equations. The chapter also considers the contriburions of Abu Jafar Muhammad ibn Musa al-Khwārizmī and suggests that negative numbers originated in China, where they had been used since the beginning of the first millennium.

Les commentaires d'al-Māhānī et d'un anonyme du Livre X des Éléments d'Euclide

1999 ◽  
Vol 9 (1) ◽  
pp. 89-156 ◽  
Author(s):  
Marouane Ben Miled
Keyword(s):  
Negative Numbers ◽  
Irrational Numbers ◽  
Quadratic Equations ◽  
First Time

This paper presents the first edition, translation and analyse of al-Māhānī’s commentary of the Book X of Euclid’s Elements (9th century, the most ancient to have reached us) and of an anonymous’ one (prior to 968, among the first algebraic commentaries). For the first time, irrational numbers are defined and classified. The algebraisation of Elements’ X-91 to 102, on the basis of al-Khwārizmī’s Algebra, shows irrational numbers as solution to algebraic quadratic equations. The algebraic calculus makes here the first steps. On this occasion, negative numbers and their calculation rules appears. Simplifications imposed by the algebraic writings are sometimes in opposition with the conclusions of propositions conceived in a purely geometrical framework, revealing a contradiction between geometrical and algebraic goals. It will be resolved by the independant way algebra will take with mathematicians belonging to the tradition of al-Karajī and al-Samaw’al from the 11th-12th centuries on.

Teaching Irrational Numbers Through Trigonometry

Resonance ◽  
2021 ◽  
Vol 26 (6) ◽  
pp. 813-827
Author(s):  
Sameen Ahmed Khan
Keyword(s):  
Irrational Numbers

scholarly journals Algebraic singularities of scattering amplitudes from tropical geometry

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
James Drummond ◽  
Jack Foster ◽  
Ömer Gürdoğan ◽  
Chrysostomos Kalousios
Keyword(s):  
Scattering Amplitudes ◽  
Natural Language ◽  
Tropical Geometry ◽  
Cluster Algebras ◽  
Finite Alphabet ◽  
Square Root ◽  
Finite Sets ◽  
Yang Mills ◽  
Square Roots ◽  
Algebraic Singularities

Abstract We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

Why Square Roots are Irrational

1986 ◽  
Vol 93 (3) ◽  
pp. 213-214 ◽  
Author(s):  
William C. Waterhouse
Keyword(s):  
Square Roots
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