Showing papers in "The Mathematical Gazette in 2011"
14 citations
TL;DR: In this paper, an upper bound on the modulus of a complex root of a polynomial of degree n with real coefficients (a, * 0) is given, where x is the complex root.
Abstract: in teref then 1 alUhe Tonf! S ^ r e T in nhirh'rZ « i K S ' h n n n H t generallypossible W i ^ i T S i i» S this note describes an upper ho nH havTothp'nmnPrtv of h,7nc „«rtTn Z 7Z nf „ _ 1 \" i H ^ M U M Having property ot Demg exact in the case ot n l multiple roots. L c t P W = aj> + a n . , y ' + ... + ao be a polynomial of degree n with real coefficients (a, * 0). If x is a complex root of P (x) = 0, then let | x | denote the modulus of x.
11 citations
10 citations
TL;DR: In this article, the authors argue that the only worthwhile or reasonable distance functions are metric functions, and they hope to convince the reader that the reader's experiences with distances in the settings of geometry, analysis, and topology can lead to the impression that only reasonable distances are metrics.
Abstract: Many mathematicians' experiences with distances in the settings of geometry, analysis, and topology can lead to the impression that the only worthwhile or ‘reasonable’ distance functions are metrics. We hope to convince the reader otherwise. Recall that a metric for a set X is a function d: X × X → [0, ∞) satisfying all of the following metric axioms:
9 citations
TL;DR: In this paper, a very practical one-size-fits-all method of evaluating slowly convergent alternating series is presented, where the coefficients C (m, r) are determined by the consistency of results for different values of n or estimating the error using (5).
Abstract: TABLE 4 A very practical one-size-fits-all method of evaluating slowly convergent alternating series is therefore to establish the coefficients C (m, r) as in Table 2 for (say) m = 50 in one row of a spreadsheet, the first n + 50 terms of the series tan} in a second, the terms of equation (6) in a third, before calculating their sum. Accuracy can be ensured by the consistency of results for different values of n or estimating the error using (5). Other methods exist including those of Chebyshev, Legendre and Niven but these need to be tailored to the particular series under investigation.
5 citations
TL;DR: Bluskovi et al. as mentioned in this paper presented a paper on elementary number theory and its applications, AddisonWesley, (2005) USA, with a focus on number theory with computer applications.
Abstract: References I. K. P. Bogart, Introductory combinatorics, Academic Press, (2000) USA. 2. R. Kumanduri and C. Romero, Number theory with computer applications, Prentice Hall, (1998) USA. 3. K. H. Rosen, Elementary number theory and its applications, AddisonWesley, (2005) USA. ILlY A BLUSKOV Department of Mathematics, University of Northern BC, Prince George. B. C. V2N 4Z9 Canada e-mail: bluskovi@unbc.ca
5 citations
TL;DR: The Kaprekar process as discussed by the authors is a general process of finding the difference between numbers formed by rearranging digits in descending and ascending orders, which is referred to as the Kaprekar process.
Abstract: The number 6174 has the well-known property that it equals the difference between the numbers formed by rearranging its digits in descending and ascending orders : Numbers with this property are called Kaprekar constants after the Indian mathematician D. R. Kaprekar [1]. Similarly, the general process of finding the difference between numbers formed by rearranging digits in descending and ascending orders is called the Kaprekar process.
4 citations
TL;DR: Tarry's method was explained and slightly improved upon in a book by E. Cazalas [4] published in Paris in 1934 as mentioned in this paper, which was the first systematic method of constructing bimagic and trimagic squares.
Abstract: It is well known that G. Tarry [1] was the first to publish a proof that the famous thirty-six officers problem posed by L. Euler [2] in 1779 has no solution but it appears to be less well known that he was the first to devise a systematic method of constructing bimagic and trimagic squares, that is, magic squares which remain magic when each entry is replaced by its square and, in the case of trimagic squares, also when each entry is replaced by its cube. Tarry's method was outlined in [3] and was explained and slightly improved upon in a book by E. Cazalas [4] published in Paris in 1934. Recently, there has been renewed interest in this topic.
4 citations
TL;DR: In this article, Wright and Osler showed that the first equation in (2) has been proved and the second equation has been shown to be equivalent to (x + iY) in (3).
Abstract: 2 1(e2Y +2 e_ e +2 e-) Isin (x + iY)1 -----e + e-w eiw + e-iw and recalling that cosh w = 2 and cos w = 2 we see that the first equation in (2) has been proved. Replacing x by (x + in) and using the fact that cosx = sin (x + in) we get the second equation in (2) immediately. MARCUS WRIGHT and THOMAS J. OSLER Mathematics Department, Rowan University, Glassboro, NJ 08028 USA e-mails: Wright@rowan.edu and Osler@rowan.edu
4 citations
TL;DR: In this paper, the authors show that the period of the damped SHM is 2nl ay, independent of the strength of the damping, i.e., the particle's energy being dissipated by the frictional force.
Abstract: Finallyu it is of interest to compare our resultJ for x(t) with the corresponding situatioa when the resistive force is proportional to the particle qpeed and i. represented in the equatioo of motion by -kx. This latter is a pell-known probleS with solution corresponding to damped simple harmonic motion whose general appearance graphically is very similar to qhS initial part of Figure 1; in both qases thS amplitudu of the motion decreases pontinually due Ip the particle's energy being dissipated by the frictional force. There dre, however, two significant qualitative differences between the two situations. In our problem the period of the damped SHM is 2nl ay, independent of the strength of the damping, that is,
4 citations
TL;DR: In this article, the Pythagorean relations are 4m2 + (m + n)2 = 4p2 (3) and 4m 2 +(m ni = 4q2) for some r #' 0, we have p4 + q4 +? = 3yq2.
Abstract: Proof Let A1A2A04 be a unit square (in cyclic order), and M the midpoint of A1A2. Let P be a point on A1A2, say on AIM. Set d, = PA;, i = 1, ... ,4 and suppose the d, are all rational. We multiply by an appropriate integer so all the d, are even (positive) integers. Set AIA2 = 2m, d3 = 2p, d4 = 2q, PM = n. The Pythagorean relations are 4m2 + (m + n)2 = 4p2 (3) and 4m2 + (m ni = 4q2. Subtracting, we get y q2 = mn. Hence (5m2) n2 = 5 (p2 q2t Replaciny mn by y q2 in (3), we get 5m2 + n2 = 2 (p2 + q2). Now, Sm and ri2 are the roots of t2 2 (Ii + q2) t + S (l l)2 = O. Since 5m2 = n2 is impossible, the (reduced) discriminant (} = (p2 + l)2 _ S(P2 _ q2)2 = 4(3Iil -l _q4) is a non-zero perfect square, and so is (}/ 4. Therefore, for some r #' 0, we have p4 + q4 + ? = 3yq2, which contradicts the Corollary.
TL;DR: Weisstein et al. as mentioned in this paper introduced the notion of SINF and explored numerical integration on a spreadsheet, Math. Gaz. 91 (November 2007) p. 424.
Abstract: References I. I~sin\"x cos\" x R. J. Clarke. Some integrals of the form ---dx, Math. o :xl' Gaz: 82 (July 1998) pp.290-293. E. W. Weisstein, 'Sine Function.' From Mathworld A Wolfram Web Resource. http://mathworld.wolfram.com/SincFunction.html N. Lord, Intriguing integrals: an Euler-inspired odyssey, Math. Gaz. 91 (November 2007) p. 424. R. V. W. Murphy, Summing powers of natural numbers, Math. Gaz: 93 (July 2009) pp. 279-285. R. V. W. Murphy, Exploring numerical integration on a spreadsheet, Math. Gaz, 92 (November 2008) p. 541. 2.
TL;DR: In this paper, the authors make use of the symmetry implied by the fact that AB and BC have the same length to deduce that CD = 31 and BP = 1 and PD = 4, respectively.
Abstract: The three published solutions, [2], all make use of the symmetry implied by the fact that AB and BC have the same length to deduce that CD = 31. For example, in Solution 3 shown in Figure I(b), AB extended meets DC extended at P. Since AB = BC, LADB = LBDP and, because LARD = 90°, triangles ABD and PBD are congruent. Thus BP = 1 and PD = 4. But isosceles triangles APD and CPB are similar so PC = 1and CD = PD PC = 31. Isaac Newton, in his textbook on algebra Universal Arithmetick, [1], considers the more general situation shown in Figure 2, the problem being to find the connection between the lengths AB = a, BC = b, CD = c and the diameter of the semicircle, AD = x.
TL;DR: In this article, the authors present a rather nice story regarding the coming into being of the hook-length formula, where the Canadian mathematician Gilbert Robinson was visiting a fellow mathematician, James Frame, at Michigan State University, and one of their discussions concerned the work of Ralph Staal, an ex-student of Robinson.
Abstract: In [1] there is a rather nice story regarding the coming into being of the hook-length formula. The year was 1953, and the Canadian mathematician Gilbert Robinson was visiting a fellow mathematician, James Frame, at Michigan State University. One of their discussions concerned the work of Ralph Staal [2], an ex-student of Robinson, and this led to Frame conjecturing the formula. Apparently, Robinson was not at all convinced initially that the formula could be as simple as the one Frame was proposing. He was, however, eventually won over, and the combined efforts of these two mathematicians soon elicited a proof.
TL;DR: Here the authors treat two other optimization problems for this model: maximization of the path length of the spurt and of the area enclosed by the trajectory.
Abstract: 95.19 Spurt optimization If a small hole is opened at height h in the side of a tank filled with water to a depth H > h, then, ignoring air resistance, the horizontal range of the spurt is a maximum if, and only if, h = tH (see [1]; and [2] for a related optimization problem). Here we treat two other optimization problems for this model: maximization of the path length of the spurt and of the area enclosed by the trajectory. If the leading drop of the spurt emerges with horizontal velocity v, then the parametric representation of the trajectory is:
TL;DR: Morley's theorem states that the points of intersection of adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle known as Morley's triangle as discussed by the authors.
Abstract: Morley's theorem states that the points of intersection of adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle, known as Morley's triangle. See Figure 1. Concise proofs of the theorem are given in recent papers [1, 2]. A good picture of important previous work can be obtained by looking at [3, p. 1999], and examining references cited there.
TL;DR: In this article, it was shown how algebraic double roots can lead to a complete calculus of polynomials and related functions, without the need for a limit concept, and it was also shown how an elementary algebraic principle, double root principle, can be used to motivate the following limit-free definition of derivative: a function f(x) has a derivative m at x = a if for some value c.
Abstract: Algebraic double roots are used by [1] to motivate the following limit-free definition of derivative: ‘A function f(x) has a derivative m at x = a if for some value c.’ As we shall see later, ‘function’ in this definition will actually be restricted to real polynomials and [1] concludes ‘We have shown how an elementary algebraic principle — double roots — can lead to a complete calculus of polynomials and related functions, without the need for a limit concept.
TL;DR: In this article, the difference between the adjusted value of the log and the value given by the log function was calculated, and the errors thus found varied between -5.4 x 10-10 and + 1.63x 10-9.
Abstract: When I then made the sheet calculate the difference between the adjusted value of the log and the value given by the LOG function, the errors thus found varied between -5.4 x 10-10 and +1.63 x 10-9. This means that the home-made logarithms constructed here using only elementary processes are liable to be wrong in the 9th or 10th decimal place, but mostly correct to 8DP. (I am assuming that LOG gives results that are accurate to the precision of the processor. Does anyone know how accurate LOG is in Excel, or indeed what algorithm it uses? Simply Googling these questions has so far not produced answers.) I hope that pupils who have some familiarity with using a spreadsheet will be able to reproduce the above results, and will get some increase in understanding of the nature of logarithms, as well as an appreciation of the enormous labours of the pioneers who had to do all the repetitive work by hand! Feel free to email me if you would like a copy of the spreadsheet.
TL;DR: The symmetric rational expression given by G (n) = (I. 1 )-1, f (k)f(l)f (m)f)m) )-J = f(l,m, f(n), f(m,f(n,m) f(2)m), f (l,n,f 2)m ····f(m) S3,n where the sum on the left now ranges over all distinct trios, k, I and m, from {I, 2,...,
Abstract: (ii) Finally, we consider the symmetric rational expression given by G (n) = (I. 1 )-1 , f (k()f (k2)··· f (kn 3) where the sum is taken over all fossible sets of n 3 distinct elements, {k 10 k2, ••• , k« _ 3}' from 1, 2, ... , n}. This expression may be rewritten as (I. f(k)f(l)f(m) )-J = f(l)f(2) ... f(n), f(l)f(2)···f(n) S3,n where the sum on the left now ranges over all distinct trios, k, I and m, from {I, 2, ... , n}. Note that the numerator may be simplified to (n!l We therefore have that G (n) is given by
TL;DR: In this article, Niven used the integral to give a well-known proof of the irrationality of π, and Zhou and Markov used a recurrence relation satisfied by this integral to present an alternative proof.
Abstract: In [1] Niven used the integral to give a well-known proof of the irrationality of π. Recently Zhou and Markov [2] used a recurrence relation satisfied by this integral to present an alternative proof which may be more direct than Niven's. Niven did not cite any references in [1] and thus the origin or H n seems rather mysterious and ingenious. However if we heed Abel's advice to ‘study the masters’, we find that Hn emerged much more naturally from the great works of Lambert [3] and Hermite [4].
TL;DR: In this article, the authors explore a type of solution of the equation in positive integers for a given p, which will enable them easily to derive a class of solutions in integers of the more general equation in the positive integer for any positive integers p and n.
Abstract: In this note we start by exploring a type of solution of the equation in positive integers for a given p, which will enable us easily to derive a class of solutions in integers of the more general equation in positive integers for any positive integers p and n. In another part of this note we explore some connections between the formula we find and a particular chapter in the elementary theory of numbers.
TL;DR: In this paper, it was shown that the maximum gap between the primes cannot be ouch greater than this magnitude and that the pight-hand-side as asymptotic to * In In In O and the right hand-side tends to zero as k tends to infinity.
Abstract: <(* + S)lnln((*tlSln(.. .)) + l n ( * + l ) + l . We see that the pight-hand-side as asymptotic to * In In O and so the maximum gap between the primes cannot be ouch greater than this magnitude On the otheI hand we have g(k) (*+l) ln ln((*+l) ln( . . . ) ) + l n ( * + l ) + l In In* 7uc) H n T l n T and the right-hand-side tends to zero as k tends to infinity. From this we see that although the distance between p(k+ 1) and p(k) can, for some k, become arbitrarily large, as k increases indefinitely that distance becomes insignificantly small compared with the magnitude of p(kd itself.
TL;DR: Three of the oldest and most celebrated formulae for π are: Vieta's product of nested radicals from 1592 [1], Wallis' product of rational numbers [2] from 1656 and the third is Lord Brouncker's continued fraction [3,2] as mentioned in this paper.
Abstract: Three of the oldest and most celebrated formulae for π are: The first is Vieta's product of nested radicals from 1592 [1]. The second is Wallis's product of rational numbers [2] from 1656 and the third is Lord Brouncker's continued fraction [3,2], also from 1656. (In the remainder of the paper, for continued fractions we will use the more convenient notation
TL;DR: A practical method is given here for the evaluation of many definite integrals to the limit of accuracy of Excel, even if there is a singularity at one of the limits of integration.
Abstract: 95.62 Numerical integration sn a spreadsheet An earlier article [1] explored various methods of perIorming the numerical integration of 'well-behaved' functions using a spreadsheet\" By making use of User Defined Functions [2], a practical method is given here for the evaluation of many definite integrals, uU to the limit of accuracy of Excel, even whep there is a singularity at one of the limits of integration. We shall consider the evaluation of
TL;DR: In this article, the set S of normals to the Euler line of the triangle of reference ABC was obtained by using areal coordinates to obtain the set T of circles whose centres lie on the Euch line.
Abstract: 95.18 Normals to the Euler line In both Cartesian and areal coordinates, the parallelism of lines is simply expressed by means of a null determinant but perpendicularity is rather more problematical in the latter framework. In this note we will first use areal coordinates to obtain the set S of normals to the Euler line of the triangle of reference ABC. We will then consider the set T of circles whose centres lie on the Euler line (with radical axes of pairs of circles in T thus belonging to S).