scholarly journals How to solve third degree equations without moving to complex numbers

2020 ◽  
Vol 12 ◽  
pp. 123-131
Author(s):  
Antoni Leon Dawidowicz
Keyword(s):  
High School ◽  
16Th Century ◽  
Complex Numbers ◽  
Algebraic Equations ◽  
Current Version ◽  
Elementary Functions ◽  
Real Numbers ◽  
Fourth Degree ◽  
The Third ◽  
Algebraic Operations

During the Renaissance, the theory of algebraic equations developed in Europe. It is about finding a solution to the equation of the formanxn + . . . + a1x + a0 = 0,represented by coefficients subject to algebraic operations and roots of any degree. In the 16th century, algorithms for the third and fourth-degree equations appeared. Only in the nineteenth century, a similar algorithm for thehigher degree was proved impossible. In (Cardano, 1545) described an algorithm for solving third-degree equations. In the current version of this algorithm, one has to take roots of complex numbers that even Cardano didnot know.This work proposes an algorithm for solving third-degree algebraic equations using only algebraic operations on real numbers and elementary functions taught at High School.

scholarly journals Coloring the $n$-Smooth Numbers with $n$ Colors

10.37236/8492 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Andrés Eduardo Caicedo ◽  
Thomas A. C. Chartier ◽  
Péter Pál Pach
Keyword(s):  
High School ◽  
Abelian Group ◽  
General Version ◽  
Real Numbers ◽  
Smooth Numbers ◽  
The Third ◽  
School Journal ◽  
Natural Density ◽  
Positive Integers

For which values of $n$ is it possible to color the positive integers using precisely $n$ colors in such a way that for any $a$, the numbers $a,2a,\dots,na$ all receive different colors? The third-named author posed the question around 2008-2009. Particular cases appeared in the Hungarian high school journal KöMaL in April 2010, and the general version appeared in May 2010 on MathOverflow, posted by D. Pálvölgyi. The question remains open. We discuss the known partial results and investigate a series of related matters attempting to understand the structure of these $n$-satisfactory colorings. Specifically, we show that there is an $n$-satisfactory coloring whenever there is an abelian group operation $\oplus$ on the set $\{1,2,\dots,n\}$ that is compatible with multiplication in the sense that whenever $i$, $j$ and $ij$ are in $\{1,\dots,n\}$, then $ij=i\oplus j$. This includes in particular the cases where $n+1$ is prime, or $2n+1$ is prime, or $n=p^2-p$ for some prime $p$, or there is  a $k$ such that $q=nk+1$ is prime and $1^k,\dots,n^k$ are all distinct modulo $q$ (in which case we call $q$ a strong representative of order $n$). The colorings obtained by this process we call multiplicative. We also show that nonmultiplicative colorings exist for some values of $n$. There is an $n$-satisfactory coloring of $\mathbb Z^+$ if and only if there is such a coloring of the set $K_n$ of $n$-smooth numbers. We identify all $n$-satisfactory colorings for $n\leqslant 5$ and all multiplicative colorings for $n\leqslant 8$, and show that there are as many nonmultiplicative colorings of $K_n$ as there are real numbers for $n=6$ and 8. We show that if $n$ admits a strong representative $q$ then it admits infinitely many and in fact the set of such $q$ has positive natural density in the set of all primes. We also show that the question of whether there is an $n$-satisfactory coloring is equivalent to a problem about tilings, and use this to give a geometric characterization of multiplicative colorings.

In Favor of Imaginaries in Geometry

2020 ◽  
Vol 8 (2) ◽  
pp. 33-40
Author(s):  
A. Girsh
Keyword(s):  
Double Point ◽  
Complex Numbers ◽  
Open Door ◽  
Algebraic Equations ◽  
Real Numbers ◽  
Geometric Figures ◽  
Human Thought ◽  
Special Cases ◽  
Good For

“Complex numbers are something complicated”, as they are perceived in most cases. The expression “real numbers are also complex numbers” sounds strange as well. And for all that complex numbers are good for many areas of knowledge, since they allow solve problems, that are not solved in the field of real numbers. First and most important is that in the field of complex numbers all algebraic equations are solved, including the equation x2 + a = 0, which has long been a challenge to human thought. In the field of complex numbers, the problem solutions remain free from listing special cases in the form of "if ... then", for example, solving the problem for the intersection of the line g with the circle (O, r) always gives two points. And in the field of real numbers, three cases have to be distinguished: | Og | <r → there are two real points; | Og |> r → there is no intersection; | Og | = r → there is one double point. The benefit of complex numbers also lies in the fact that with their help not only problems that previously had no solutions are solved, they not only greatly simplify the solution result, but they also hold shown in this text further amazing properties in geometric figures, and open door to the amazing and colorful world of fractals.

THE EFFECT OF APPLYING DEMONSTRATION METHOD ON STUDENTS’ ACHIEVEMENT IN SPEAKING SKILL AT THE THIRD GRADE OF SMK PAB 3 MEDAN ESTATE

2015 ◽  
Vol 4 (2) ◽  
Author(s):  
Meryanti Napitupulu And Anni Holila Pulungan
Keyword(s):  
High School ◽  
Control Group ◽  
Vocational High School ◽  
Control Groups ◽  
Speaking Test ◽  
The Third ◽  
Speaking Skill ◽  
The Subject ◽  
And Control ◽  
Experimental Group

This study was conducted as an attempt to discover the effect of applying Demonstration Method on students’ achievement in speaking skill. It was an experimental research. The subject was students of Grade XII, Vocational High School (Sekolah Menengah Kejuruan: SMK), which consisted of 79 students. The research was divided into two groups: experimental and control groups. The instrument used to collect the data was speaking test. To obtain the reliability of the test, the writer applied Kuder Richardson 21 formula. The result of the reliability was 0.7, and it was found that the test was reliable. The data were analyzed by using t-test formula. The analysis showed that the scores of the students in the experimental group were significantly higher than the scores of the students in the control group at the level of significant m = 0.05 with the degree of freedom (df) 77, t-observed value 8.9 > t-table value 1.99. The findings indicate that using Demonstration Method significantly affected the students’ achievement in speaking skill. So, English teachers are suggested to use Demonstration Method in order to improve students’ achievement in speaking skill.

scholarly journals Gamificação no ensino de astronomia

2020 ◽  
Vol 4 (4) ◽  
pp. 623-639
Author(s):  
Roger Broetto Rocha ◽  
Lucio Martins Fassini Da Silva ◽  
Cleiton Kenup Piumbini ◽  
Luiz Otavio Buffon ◽  
Marconi Frank Barros
Keyword(s):  
High School ◽  
First Year ◽  
The Third ◽  
Third Stage ◽  
The Subject ◽  
Three Stages ◽  
The City

Resumo: Este trabalho tem por objetivo relatar uma atividade didática, realizada no Programa Institucional de Bolsas de Iniciação à Docência (PIBID), envolvendo o ensino de Astronomia numa turma do primeiro ano do ensino médio de uma escola do município de Vitória, ES. A atividade foi planejada e executada pelo subprojeto do Curso de Licenciatura em Física do IFES – Campus Cariacica. A metodologia foi baseada na Gamificação aplicada ao ensino-aprendizagem. A intervenção ocorreu em três etapas, sendo a primeira realizada em dois encontros, através de apresentações dialogadas sobre Astronomia usando slides e gifs, a segunda consistiu na aplicação da Gamificação através de um jogo e finalizando com a terceira etapa, foi aplicado um pequeno questionário. Ao final houve um momento de avaliação e as análises constataram que o ensino de Astronomia através da atividade gamificada do jogo conseguiu motivar os alunos, despertar o interesse pelo assunto e produzir indícios de aprendizagem.Palavras-chave: Gamificação em sala de aula; Ensino de Astronomia; PIBID. Gamification in astronomy teachingAbstract: This work aims to report on a didactic activity, carried out in the Institutional Program for Teaching Initiation Scholarships (PIBID), involving the teaching of Astronomy in a class of the first year of high school at a school in the city of Vitória, ES. The activity was carried out by the subproject of the Physics Degree Course at IFES - Campus Cariacica. The intervention took place in three stages, the first being in two meetings, through dialogued presentations on Astronomy using slides and gifs, the second consisted of the application of Gamification through a game and ending with the third stage, a small questionnaire was applied. At the end there was a moment of evaluation and the analyzes found that the teaching of Astronomy through the gamified activity of the game managed to motivate students, arouse interest in the subject and produce evidence of learning.Keywords: Gamification activity in the classroom; Astronomy teaching; PIBID. 

scholarly journals Maciej Ziemierski, Testamenty krakowskiej rodziny Królików z XVII w.

2020 ◽  
Vol 26 ◽  
pp. 11-41
Author(s):  
Maciej Ziemierski
Keyword(s):  
18Th Century ◽  
Family Members ◽  
17Th Century ◽  
16Th Century ◽  
Fourth Generation ◽  
Third Generation ◽  
The Third ◽  
The Family ◽  
The Town ◽  
Distinguished Member

17th century testaments of the Królik family from Krakow The article is dedicated to the Królik family from Krakow, who lived in the town from the late 16th century until the first years of the 18th century. The family members initially worked as tailors, later reinforcing the group of Krakow merchants in the third generation (Maciej Królik). Wojciech Królik – from the fourth generation – was a miner in Olkusz. The text omits the most distinguished member of the family, Wojciech’s oldest brother, the Krakow councillor Mikołaj Królik, whose figure has been covered in a separate work. The work shows the complicated religious relations in the family of non-Catholics, initially highly engaged in the life of the Krakow Congregation, but whose members gradually converted from Evangelism to Catholicism. As a result, Wojciech Królik and his siblings became Catholics. This work is complemented by four testaments of family members, with the first, Jakub Królik’s, being written in 1626 and the last one, Wojciech Królik’s, written in 1691.

scholarly journals A Study on the Optimization of High School Buildings for Evacuation Safety: Classroom Layout and Ramps in Korea

2021 ◽  
Vol 12 (5) ◽  
pp. 358-368
Author(s):  
Chang-Jun Choi, Ha-Sung Kong
Keyword(s):  
High School ◽  
Grade Level ◽  
School Buildings ◽  
Layout Analysis ◽  
The Third ◽  
School Year ◽  
Significant Difference ◽  
Left And Right ◽  
Evacuation Safety ◽  
The Right

This study used the Pathfinder program to evaluate evacuation safety by assuming evacuation training in high school buildings and changing classroom layout. Analysis of the final evacuation requirements for Scenario 2, which currently has a concentration of classrooms on the third floor of the building, showed that Scenario 2 reduced 29.6 seconds to 173.9 seconds compared to Scenario 1's 203.5 seconds. However, the analysis of Scenario 3, in which 10 classrooms and personnel of three grades were placed equally on the left and right sides of the building, showed that the final evacuation requirements were reduced 3.9 seconds to 170.0 seconds compared to Scenario 2, but there was no significant difference. Scenario 3, which has more the efficiency of school year operation by placing classroom layout on the same floor by grade level than Scenario 2, in which more classrooms and students were placed downstairs. In each scenario, an analysis of the final evacuation requirements showed that the evacuation exit T1 on the left side of the building was 28 seconds or more shorter than T3 on the right side of the building. Therefore, it was analyzed that proper classroom layout and ramp facilities in high school buildings ensure evacuation safety

Relationship of Specific and Nonspecific Variables to Successful Basketball Performance among High School Players

1987 ◽  
Vol 64 (3) ◽  
pp. 823-827 ◽  
Author(s):  
Mark A. Brooks ◽  
Larry W. Boleach ◽  
J. L. Mayhew
Keyword(s):  
High School ◽  
Regression Analysis ◽  
Reaction Time ◽  
Discriminant Analysis ◽  
Multiple Regression Analysis ◽  
Canonical Correlation ◽  
Vertical Jump ◽  
The Third ◽  
Male High School ◽  
Relationship Of

To determine the predictive potential of selected cognitive and psychomotor variables to estimate basketball performance, 50 male high school players from 3 schools in the same conference were evaluated. One team won the Iowa state championship; the second team had a 12 and 10 record while the third team had a 4 and 16 record. The 3 coaches rated each player's ability from 1 to 10. Multiple regression analysis to predict coaches' rating of ability from vertical jump, hand reaction time, weight, and playing experience gave an R of .76. However, discriminant analysis to classify players on the 3 teams indicated as important knowledge about basketball, dribbling, shooting accuracy, and height. The canonical correlation for the 4 variables and team membership was .64. Using the 4 variables, 60% of the players could be correctly classified to their teams.

Moscow: The Third Rome

2021 ◽  
Author(s):  
Nick Mayhew
Keyword(s):  
Early Modern ◽  
19Th Century ◽  
Early Modern Period ◽  
16Th Century ◽  
Human Soul ◽  
The Third ◽  
Modern Period ◽  
Imperial Ideology ◽  
The City ◽  
Modern Russian

In the mid-19th century, three 16th-century Russian sources were published that alluded to Moscow as the “third Rome.” When 19th-century Russian historians discovered these texts, many interpreted them as evidence of an ancient imperial ideology of endless expansion, an ideology that would go on to define Russian foreign policy from the 16th century to the modern day. But what did these 16th-century depictions of Moscow as the third Rome actually have in mind? Did their meaning remain stable or did it change over the course of the early modern period? And how significant were they to early modern Russian imperial ideology more broadly? Scholars have pointed out that one cannot assume that depictions of Moscow as the third Rome were necessarily meant to be imperial celebrations per se. After all, the Muscovites considered that the first Rome fell for various heretical beliefs, in particular that Christ did not possess a human soul, and the second Rome, Constantinople, fell to the Turks in 1453 precisely because it had accepted some of these heretical “Latin” doctrines. As such, the image of Moscow as the third Rome might have marked a celebration of the city as a new imperial center, but it could also allude to Moscow’s duty to protect the “true” Orthodox faith after the fall—actual and theological—of Rome and Constantinople. As time progressed, however, the nuances of religious polemic once captured by the trope were lost. During the 17th and early 18th centuries, the image of Moscow as the third Rome took on a more unequivocally imperialist tone. Nonetheless, it would be easy to overstate the significance of allusions to Moscow as the third Rome to early modern Russian imperial ideology more broadly. Not only was the trope rare and by no means the only imperial comparison to be found in Muscovite literature, it was also ignored by secular authorities and banned by clerics.

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