A method is presented for creating a problem, solving it, and confirming that the solution is correct. The problem is to analyze a second-degree polynomial equation in two variables, in order to identify and draw the defined conic section, with its axes, without changing coordinates. One creates the problemby choosing conjugate diameters that are not axes.
{"title":"Conics in Place","authors":"David Pierce","doi":"10.24917/20809751.13.2","DOIUrl":"https://doi.org/10.24917/20809751.13.2","url":null,"abstract":"A method is presented for creating a problem, solving it, and confirming that the solution is correct. The problem is to analyze a second-degree polynomial equation in two variables, in order to identify and draw the defined conic section, with its axes, without changing coordinates. One creates the problemby choosing conjugate diameters that are not axes.","PeriodicalId":33912,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis Studia Naturae","volume":"278 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86568408","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a proof of the Hurwitz Theorem about construction and properties of real numbers, complex numbers, quaternions and octonions. In the proof we use the Dickson double for Cayley-Dickson algebras.
{"title":"A Proof of the Hurwit'z Theorem About Composition Algebras","authors":"Aleksandra Galecka","doi":"10.24917/20809751.13.4","DOIUrl":"https://doi.org/10.24917/20809751.13.4","url":null,"abstract":"We present a proof of the Hurwitz Theorem about construction and properties of real numbers, complex numbers, quaternions and octonions. In the proof we use the Dickson double for Cayley-Dickson algebras.","PeriodicalId":33912,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis Studia Naturae","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82243141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we refer to different way of measuring distances than commonly used Euclidean metric. It is shown how some objects of two-dimensional geometry look like in the jungle river metric. Some examples of the objects are presented in pictures made in the free available Geogebra program, the conditions of occurrence of this objects are exactly described. Reading the text, one may learn that a sphere need not be round and a parabola may not be a bounded curve. Some task for the reader are included.
{"title":"Conic curves in jungle river metric","authors":"Rafał Stypka","doi":"10.24917/20809751.13.3","DOIUrl":"https://doi.org/10.24917/20809751.13.3","url":null,"abstract":" In this article, we refer to different way of measuring distances than commonly used Euclidean metric. It is shown how some objects of two-dimensional geometry look like in the jungle river metric. Some examples of the objects are presented in pictures made in the free available Geogebra program, the conditions of occurrence of this objects are exactly described. Reading the text, one may learn that a sphere need not be round and a parabola may not be a bounded curve. Some task for the reader are included.","PeriodicalId":33912,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis Studia Naturae","volume":"325 ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72506125","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
(Hartshorne, 2000) interprets Euclid’s Elements provides an interpretation of Euclid’s Elements in the Hilbert system of axioms, specifically propositions I.1-I.27, covering the so-called absolute geometry. We develop an alternative interpretation that explores Euclid’s practice concerning the relation greater-than. Discussing the Postulate 5, we present a model of nonEuclidean plane in which angles in a triangle sum up to π. It is a subspace of the Cartesian plane over the ordered field of hyperreal numbers R*.
{"title":"Commentary to Book I of the Elements. Hartshorne and beyond","authors":"Piotr Błaszczyk, Anna Petiurenko","doi":"10.24917/20809751.13.7","DOIUrl":"https://doi.org/10.24917/20809751.13.7","url":null,"abstract":"(Hartshorne, 2000) interprets Euclid’s Elements provides an interpretation of Euclid’s Elements in the Hilbert system of axioms, specifically propositions I.1-I.27, covering the so-called absolute geometry. We develop an alternative interpretation that explores Euclid’s practice concerning the relation greater-than. Discussing the Postulate 5, we present a model of nonEuclidean plane in which angles in a triangle sum up to π. It is a subspace of the Cartesian plane over the ordered field of hyperreal numbers R*.","PeriodicalId":33912,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis Studia Naturae","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73401520","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recenzja z M. Pater, R. Sochacki, Wokółgeometrii trójkąta, Wydawnictwo UniwersytetuOpolskiego, Opole 2020
{"title":"Recenzja z M. Pater, R. Sochacki, Wokół geometrii trójkąta, Wydawnictwo Uniwersytetu Opolskiego, Opole 2020","authors":"D. Wójcik","doi":"10.24917/20809751.13.10","DOIUrl":"https://doi.org/10.24917/20809751.13.10","url":null,"abstract":"Recenzja z M. Pater, R. Sochacki, Wokółgeometrii trójkąta, Wydawnictwo UniwersytetuOpolskiego, Opole 2020","PeriodicalId":33912,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis Studia Naturae","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75333818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Amine Amimour, Lynda Benbessai, Kada Becharef, Hafida Mokhenfer
Teaching mathematics using programming has cognitive benefits, improves practical skills, and provides powerful independent thinking and problem-solving strategies. R is a free software environment and one of the most popular programming tools, which allows for solving various mathematical problems through coding. In this work, we will use the R language to solve some problems posed in complex numbers and cover some exercises from the third-grade mathematics textbook in Algeria.
{"title":"Modernize the teaching of mathematics at the secondary level through R programming","authors":"Amine Amimour, Lynda Benbessai, Kada Becharef, Hafida Mokhenfer","doi":"10.24917/20809751.13.8","DOIUrl":"https://doi.org/10.24917/20809751.13.8","url":null,"abstract":"Teaching mathematics using programming has cognitive benefits, improves practical skills, and provides powerful independent thinking and problem-solving strategies. R is a free software environment and one of the most popular programming tools, which allows for solving various mathematical problems through coding. In this work, we will use the R language to solve some problems posed in complex numbers and cover some exercises from the third-grade mathematics textbook in Algeria.","PeriodicalId":33912,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis Studia Naturae","volume":"11 3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77585867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main aim of the article is to present and discuss the evolution of L2 education and research expressed in two major paradigm shifts that could be observed over the last century: from the modernist approach, through postmodernism and postmethod, to transmodernity. The article also offers an overview of new approaches and trends in L2 education and research that emerged as a result of those watershed changes, in particular: complexity theory, the ecological approach, transnational identity and translanguaging. The paper ends with a brief discussion on how these changes affected L2 researchers and educators and what implications can be grasped.
{"title":"The evolution of L2 education and research: An overview of paradigm shifts in applied linguistics","authors":"Agata Wolanin","doi":"10.24917/20831765.16.18","DOIUrl":"https://doi.org/10.24917/20831765.16.18","url":null,"abstract":"The main aim of the article is to present and discuss the evolution of L2 education and research expressed in two major paradigm shifts that could be observed over the last century: from the modernist approach, through postmodernism and postmethod, to transmodernity. The article also offers an overview of new approaches and trends in L2 education and research that emerged as a result of those watershed changes, in particular: complexity theory, the ecological approach, transnational identity and translanguaging. The paper ends with a brief discussion on how these changes affected L2 researchers and educators and what implications can be grasped.","PeriodicalId":33912,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis Studia Naturae","volume":"109 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77454317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An inalienable property of the linguistic reality is the multi-language nature of the world and the multi-variation character of the ethnic-national communication space. As regards Polish pace, one can distinguish a series of axes around which the processes of long lasting concentrate. The author discusses the following five axes: (1) idiolectal (individual languages), (2) one determined by the dichotomy: spoken language – written language (contemporarily it assumes the form: spoken language – media language – written language), (3) one connected with the opposition: dialects – literary language (today in the form: dialects – mixed languages – general language), (4) functional variations (their ‘canonical’ series: colloquial language – scientific language – journalistic language – language of the administration – religious language – artistic language), and (5) one of three styles (high – medium – low).
{"title":"Polska sieć komunikacyjna i jej główne osie","authors":"S. Gajda","doi":"10.24917/20831765.16.4","DOIUrl":"https://doi.org/10.24917/20831765.16.4","url":null,"abstract":"An inalienable property of the linguistic reality is the multi-language nature of the world and the multi-variation character of the ethnic-national communication space. As regards Polish pace, one can distinguish a series of axes around which the processes of long lasting concentrate. The author discusses the following five axes: (1) idiolectal (individual languages), (2) one determined by the dichotomy: spoken language – written language (contemporarily it assumes the form: spoken language – media language – written language), (3) one connected with the opposition: dialects – literary language (today in the form: dialects – mixed languages – general language), (4) functional variations (their ‘canonical’ series: colloquial language – scientific language – journalistic language – language of the administration – religious language – artistic language), and (5) one of three styles (high – medium – low).","PeriodicalId":33912,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis Studia Naturae","volume":"135 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89801145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
There are same chronological diversified lexis layers In the Bible edited by Jan Leopolita and published in Cracow in 1561. The analysis of Polish equivalents for the Greek lexeme γάμος (or its Latin equivalents in the Vulgate – nuptiae, nuptialis) ‘a marriage, wedding, weddingceremony; plur: a wedding-feast’, i. e. swadziebny, gody, wesele, referring to empirical system and text data, acquired from lexica of the historic Polish and from texts of other Renaissance Polish renderings of the New Testament, proved chronological diversity of the equivalents. The lexeme swadziebny belongs to an older lexis layer, probably representing the lexicon of the former translation, perhaps medieval. The lexeme wesele represents a new lexis layer which may have been introduced into the text by Jan Leopolita, as the effect of his editorial efforts. Chronological status of the lexeme wesele in the analyzed Biblical rendering lexicon is ambiguous. It should be considered as an evidence of an traditional lexis layer. It may have been introduced into the text as a substitute of the lexeme swadziebny.
{"title":"Polskie ekwiwalenty greckiego leksemu γάμος w Nowym Testamencie Biblii Leopolity (1561) – swadziebny, gody, wesele – a problem stratyfikacji chronologicznej leksyki tego przekładu","authors":"T. Lisowski","doi":"10.24917/20831765.16.9","DOIUrl":"https://doi.org/10.24917/20831765.16.9","url":null,"abstract":"There are same chronological diversified lexis layers In the Bible edited by Jan Leopolita and published in Cracow in 1561. The analysis of Polish equivalents for the Greek lexeme γάμος (or its Latin equivalents in the Vulgate – nuptiae, nuptialis) ‘a marriage, wedding, weddingceremony; plur: a wedding-feast’, i. e. swadziebny, gody, wesele, referring to empirical system and text data, acquired from lexica of the historic Polish and from texts of other Renaissance Polish renderings of the New Testament, proved chronological diversity of the equivalents. The lexeme swadziebny belongs to an older lexis layer, probably representing the lexicon of the former translation, perhaps medieval. The lexeme wesele represents a new lexis layer which may have been introduced into the text by Jan Leopolita, as the effect of his editorial efforts. Chronological status of the lexeme wesele in the analyzed Biblical rendering lexicon is ambiguous. It should be considered as an evidence of an traditional lexis layer. It may have been introduced into the text as a substitute of the lexeme swadziebny.","PeriodicalId":33912,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis Studia Naturae","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89814301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The article is a research attempt to show the participation of a group of negative adjectives and adjectival participles formed using prefixes nie- and bez- in 8 romantic poems representing the textual corps of Polish romantic poetry. The negatives with prefix nie- are nine times more numerous than negatives formed using prefix bez-. Almost each negative is in romantic poems less numerous than his not negative equivalent. The participation of negatives reach 1,45% of the vocabulary and 0,34% of the text of analyzed corps. From among 7 romantic poets C. Norwid uses negatives most willingly. Most frequently represented are negatives: niemy, nieruchomy, niewinny, nieznajomy and niezwykły. Negatives in romantic poems are used for semantic and stylistic purposes. Most often they serve the poets to specify the description of characters and things.
{"title":"Zaprzeczone przymiotniki i imiesłowy przymiotnikowe w poematach romantycznych","authors":"Edward Stachurski","doi":"10.24917/20831765.16.17","DOIUrl":"https://doi.org/10.24917/20831765.16.17","url":null,"abstract":"The article is a research attempt to show the participation of a group of negative adjectives and adjectival participles formed using prefixes nie- and bez- in 8 romantic poems representing the textual corps of Polish romantic poetry. The negatives with prefix nie- are nine times more numerous than negatives formed using prefix bez-. Almost each negative is in romantic poems less numerous than his not negative equivalent. The participation of negatives reach 1,45% of the vocabulary and 0,34% of the text of analyzed corps. From among 7 romantic poets C. Norwid uses negatives most willingly. Most frequently represented are negatives: niemy, nieruchomy, niewinny, nieznajomy and niezwykły. Negatives in romantic poems are used for semantic and stylistic purposes. Most often they serve the poets to specify the description of characters and things.","PeriodicalId":33912,"journal":{"name":"Annales Universitatis Paedagogicae Cracoviensis Studia Naturae","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74499294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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