Pub Date : 2023-05-19DOI: 10.1080/0025570X.2023.2204794
H. Diamond
Summary Iterations of the form are used throughout mathematics, and convergence (or not) of an iteration to a fixed point is always a central question. We present an informal graphical technique for resolving whether a fixed point in one dimension is attracting or repelling. Specifically, given a real-valued function f, we are determining whether or at a point for which . The former condition on classifies the fixed point as attracting, and the iteration as locally convergent; the latter corresponds to a repulsive fixed point, in which the iterations get further away from the fixed point no matter how close you start them off. The technique requires only that we be able to compute points on the graph of and not necessarily on an explicit functional form or its derivative.
{"title":"Just Tilt Your Head: A Graphical Technique for Classifying Fixed Points","authors":"H. Diamond","doi":"10.1080/0025570X.2023.2204794","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2204794","url":null,"abstract":"Summary Iterations of the form are used throughout mathematics, and convergence (or not) of an iteration to a fixed point is always a central question. We present an informal graphical technique for resolving whether a fixed point in one dimension is attracting or repelling. Specifically, given a real-valued function f, we are determining whether or at a point for which . The former condition on classifies the fixed point as attracting, and the iteration as locally convergent; the latter corresponds to a repulsive fixed point, in which the iterations get further away from the fixed point no matter how close you start them off. The technique requires only that we be able to compute points on the graph of and not necessarily on an explicit functional form or its derivative.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42303235","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}
Pub Date : 2023-05-17DOI: 10.1080/0025570X.2023.2199695
Jake H. Lewis
Summary In classrooms where most students are simply told that , accept the fact, and move on, methods for finding lower or upper bound on are usually not taught. Here, I consider a University of Tokyo entrance exam problem: Prove that , I provide students with a simple, yet nontraditional, proof method. In particular, this method does not require a calculator (as in many exams), cumbersome circle geometry, direct use of calculus-based methods, or partial sums of any infinite series.
{"title":"A Simple Proof that π > 3.05","authors":"Jake H. Lewis","doi":"10.1080/0025570X.2023.2199695","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2199695","url":null,"abstract":"Summary In classrooms where most students are simply told that , accept the fact, and move on, methods for finding lower or upper bound on are usually not taught. Here, I consider a University of Tokyo entrance exam problem: Prove that , I provide students with a simple, yet nontraditional, proof method. In particular, this method does not require a calculator (as in many exams), cumbersome circle geometry, direct use of calculus-based methods, or partial sums of any infinite series.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42275893","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}
Pub Date : 2023-05-17DOI: 10.1080/0025570X.2023.2205820
S. Northshield
Summary We present some variations on the theme of “cubes”. We start with right and wrong ways to draw cubes and orthogonal axes. From there we classify cubes in 3-space with integral coordinates. Further, we find that four complex numbers are the vertices of a drawing of a regular tetrahedron if their average of squares equals the square of their average. Tangentially, we find short proofs of the Siebeck-Marden theorem and a method for solving a cubic equation. Finally, we consider sets of complex numbers whose average of squares equals the square of their average.
{"title":"Drawing Cubes, Finding Integral Cubes, and Solving Cubics","authors":"S. Northshield","doi":"10.1080/0025570X.2023.2205820","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2205820","url":null,"abstract":"Summary We present some variations on the theme of “cubes”. We start with right and wrong ways to draw cubes and orthogonal axes. From there we classify cubes in 3-space with integral coordinates. Further, we find that four complex numbers are the vertices of a drawing of a regular tetrahedron if their average of squares equals the square of their average. Tangentially, we find short proofs of the Siebeck-Marden theorem and a method for solving a cubic equation. Finally, we consider sets of complex numbers whose average of squares equals the square of their average.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45856645","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}
Pub Date : 2023-05-17DOI: 10.1080/0025570X.2023.2199678
Q. H. Tran
Summary We give another simple proof of Pascal’s theorem, based on the use of directed angles and directly similar quadrilaterals.
基于有向角和直接相似四边形的使用，我们给出了Pascal定理的另一个简单证明。
{"title":"Another Simple Proof of Pascal’s Theorem","authors":"Q. H. Tran","doi":"10.1080/0025570X.2023.2199678","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2199678","url":null,"abstract":"Summary We give another simple proof of Pascal’s theorem, based on the use of directed angles and directly similar quadrilaterals.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45067397","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}
Pub Date : 2023-05-17DOI: 10.1080/0025570X.2023.2206298
R. Viglione
Summary We offer a visual proof of Clough’s theorem on the semiperimeter of an equilateral triangle.
摘要我们提供了关于等边三角形半周长的Clough定理的直观证明。
{"title":"Clough’s Theorem","authors":"R. Viglione","doi":"10.1080/0025570X.2023.2206298","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2206298","url":null,"abstract":"Summary We offer a visual proof of Clough’s theorem on the semiperimeter of an equilateral triangle.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47700108","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}
Pub Date : 2023-05-15DOI: 10.1080/0025570X.2023.2203055
Matthew Just
Summary Let n be a positive integer, and k and integer with . Let g be the greatest common divisor of k and n. We use the cycle construction to give a combinatorial proof that the fraction n/g divides the binomial coefficient .
{"title":"Note on a Binomial Coefficient Divisor","authors":"Matthew Just","doi":"10.1080/0025570X.2023.2203055","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2203055","url":null,"abstract":"Summary Let n be a positive integer, and k and integer with . Let g be the greatest common divisor of k and n. We use the cycle construction to give a combinatorial proof that the fraction n/g divides the binomial coefficient .","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45960947","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}
Pub Date : 2023-05-11DOI: 10.1080/0025570X.2023.2199700
Karen S. Briggs, Caylee R. Spivey
Summary A professor and her students were working through a routine problem on isomorphisms that required finding the multiplicative inverse of 43 modulo 50. The professor, however, simply asked for “the inverse” of 43 modulo 50 and got a surprising response. One student quickly answered that the inverse of 43 modulo 50 was 7. Thinking that the student was responding with the additive inverse, the professor restated her question and asked for the multiplicative inverse. The student checked her work and said that 7 is the multiplicative inverse of 43 mod 50. The observation that the additive and multiplicative inverse of an element could be the same led to a research project into the questions of when, why, and how frequently this phenomenon occurs. Answering these questions led to tools such asthe Chinese remainder theorem, the fundamental theorem of cyclic groups, and quadratic residues.
摘要一位教授和她的学生正在解决一个关于同构的常规问题，该问题需要找到43模50的乘法逆。然而，这位教授只是简单地问了43模50的“逆”，得到了令人惊讶的回答。一个学生很快回答说43的倒数取50是7。考虑到学生的回答是加法逆，教授重述了她的问题，并要求使用乘法逆。这位学生检查了她的工作，说7是43 mod 50的乘法逆。一个元素的加法逆和乘法逆可能是相同的，这一观察结果导致了一个研究项目，研究这种现象何时、为什么以及发生频率。回答这些问题产生了诸如中国余数定理、循环群基本定理和二次余数等工具。
{"title":"When Additive and Multiplicative Inverses are the Same","authors":"Karen S. Briggs, Caylee R. Spivey","doi":"10.1080/0025570X.2023.2199700","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2199700","url":null,"abstract":"Summary A professor and her students were working through a routine problem on isomorphisms that required finding the multiplicative inverse of 43 modulo 50. The professor, however, simply asked for “the inverse” of 43 modulo 50 and got a surprising response. One student quickly answered that the inverse of 43 modulo 50 was 7. Thinking that the student was responding with the additive inverse, the professor restated her question and asked for the multiplicative inverse. The student checked her work and said that 7 is the multiplicative inverse of 43 mod 50. The observation that the additive and multiplicative inverse of an element could be the same led to a research project into the questions of when, why, and how frequently this phenomenon occurs. Answering these questions led to tools such asthe Chinese remainder theorem, the fundamental theorem of cyclic groups, and quadratic residues.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44205855","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}
Pub Date : 2023-05-05DOI: 10.1080/0025570X.2023.2231336
J. Cohen
Summary In 1845, Bertrand conjectured what became known as Bertrand’s postulate or the Bertrand-Chebyshev theorem: twice and prime strictly exceeds the next prime. Surprisingly, a stronger statement seems not to be well-known: the sum of any two consecutive primes strictly exceeds the next prime, except for the only equality . Our main theorem is a much more general result, perhaps not previously noticed, that compares sums of any number of primes. We prove this result using only the prime number theorem. We also give some numerical results and unanswered questions.
{"title":"Generalizations of Bertrand’s Postulate to Sums of Any Number of Primes","authors":"J. Cohen","doi":"10.1080/0025570X.2023.2231336","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2231336","url":null,"abstract":"Summary In 1845, Bertrand conjectured what became known as Bertrand’s postulate or the Bertrand-Chebyshev theorem: twice and prime strictly exceeds the next prime. Surprisingly, a stronger statement seems not to be well-known: the sum of any two consecutive primes strictly exceeds the next prime, except for the only equality . Our main theorem is a much more general result, perhaps not previously noticed, that compares sums of any number of primes. We prove this result using only the prime number theorem. We also give some numerical results and unanswered questions.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43761756","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}
Pub Date : 2023-05-02DOI: 10.1080/0025570X.2023.2199674
K. Williams
Summary In the 18th century, Euler evaluated the sum in terms of the Bernoulli number for any positive integer k. In this article we evaluate the sum where a and b are integers satisfying and .
在18世纪，欧拉用伯努利数计算了任意正整数k的和。在本文中，我们计算了a和b为满足和的整数的和。
{"title":"Beyond the Basel Problem: Part II","authors":"K. Williams","doi":"10.1080/0025570X.2023.2199674","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2199674","url":null,"abstract":"Summary In the 18th century, Euler evaluated the sum in terms of the Bernoulli number for any positive integer k. In this article we evaluate the sum where a and b are integers satisfying and .","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47332701","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}
Pub Date : 2023-04-28DOI: 10.1080/0025570X.2023.2199704
Rajib Mukherjee, Manishita Chakraborty
Summary In this article we are extending Galileo’s ratio to Hex numbers and beyond.
摘要在这篇文章中，我们将伽利略的比率扩展到十六进制数及以上。
{"title":"Extending Galileo’s Ratio to Hex Numbers and Beyond","authors":"Rajib Mukherjee, Manishita Chakraborty","doi":"10.1080/0025570X.2023.2199704","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2199704","url":null,"abstract":"Summary In this article we are extending Galileo’s ratio to Hex numbers and beyond.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42769000","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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