Complex Numbers from A to ... Z

Andreescu, Titu; Andrica, Dorin

doi:10.1007/978-0-8176-8415-0

Cited by 22 publications

(23 citation statements)

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“…In anticipation of investigating this relationship between Pythagorean membership grades and complex numbered membership grades, we review some ideas about complex numbers …”

Section: Pythagorean Membership Grades and Complex Numbersmentioning

confidence: 99%

“…In anticipation of investigating this relationship between Pythagorean membership grades and complex numbered membership grades, we review some ideas about complex numbers. [11][12][13] A complex number Z is an ordered pair (x, y) interpreted as z = x + iy, where i = √ −1 is the so-called imaginary number. One can view a complex number as a point in a plane as shown in Figure 2.…”

Section: Pythagorean Membership Grades and Complex Numbersmentioning

confidence: 99%

“…Here we look in more detail at Pythagorean fuzzy sets (PFS). In addition to discussing the basic properties of these sets, we provide a formulation in terms of complex numbers . We consider the problem of multicriteria decision making when the degrees of satisfaction are expressed as Pythagorean membership grades.…”

Section: Introductionmentioning

confidence: 99%

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Pythagorean Membership Grades, Complex Numbers, and Decision Making

Yager

Abbasov²

2013

Int. J. Intell. Syst.

1,127

675

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We describe the idea of Pythagorean membership grades and the related idea of Pythagorean fuzzy subsets. We focus on the negation and its relationship to the Pythagorean theorem. We look at the basic set operations for the case of Pythagorean fuzzy subsets. A relationship is shown between Pythagorean membership grades and complex numbers. We specifically show that Pythagorean membership grades are a subclass of complex numbers called Π‐i numbers. We investigate operations that are closed under Π‐i numbers. We consider the problem of multicriteria decision making with satisfactions expressed as Pythagorean membership grades, Π‐i numbers. We look at the use of the geometric mean and ordered weighted geometric operator for aggregating criteria satisfaction.

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“…In anticipation of investigating this relationship between Pythagorean membership grades and complex numbered membership grades, we review some ideas about complex numbers …”

Section: Pythagorean Membership Grades and Complex Numbersmentioning

confidence: 99%

Section: Pythagorean Membership Grades and Complex Numbersmentioning

confidence: 99%

See 1 more Smart Citation

Pythagorean Membership Grades, Complex Numbers, and Decision Making

Yager

Abbasov²

2013

Int. J. Intell. Syst.

1,127

675

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“…There are many geometrical problems solvable with the aid of complex numbers. This is described in several books, one of the best being [1]. In this Note we shall study another such problem that we found in the century old paper [2], where it was solved using transformations.…”

Section: A Characterisation Of Midsquare Quadrilateralsmentioning

confidence: 97%

104.20 A characterisation of midsquare quadrilaterals

Josefsson¹

2020

Math. Gaz.

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“…If h = 0 we have α = π 4 and d = 3 . We see that a vertex of Γ can be constructed with ruler and compass if h ∈ 0, 1…”

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confidence: 99%

The Ptolemy–Alhazen Problem and Spherical Mirror Reflection

Fujimura

Hariri

Mocanu

et al. 2018

Comput. Methods Funct. Theory

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An ancient optics problem of Ptolemy, studied later by Alhazen, is discussed. This problem deals with reflection of light in spherical mirrors. Mathematically this reduces to the solution of a quartic equation, which we solve and analyze using a symbolic computation software. Similar problems have been recently studied in connection with ray-tracing, catadioptric optics, scattering of electromagnetic waves, and mathematical billiards, but we were led to this problem in our study of the so-called triangular ratio metric.2010 Mathematics Subject Classification. 30C20, 30C15, 51M99.

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Complex Numbers from A to ... Z

Cited by 22 publications

References 0 publications

Pythagorean Membership Grades, Complex Numbers, and Decision Making

Pythagorean Membership Grades, Complex Numbers, and Decision Making

104.20 A characterisation of midsquare quadrilaterals

The Ptolemy–Alhazen Problem and Spherical Mirror Reflection

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