Showing papers in "The Mathematical Gazette in 2015"

A simple proof of Stirling's formula for the gamma function

Abstract: where and the notation means that as . C = 2π f (n) ∼ g(n) f (n) / g(n) → 1 n → ∞ A great deal has been written about Stirling's formula. At this point I will just mention David Fowler's Gazette article [1], which contains an interesting historical survey. The continuous extension of factorials is, of course, the gamma function. The established notation, for better or worse, is such that equals rather than . Stirling's formula duly extends to the gamma function, in the form Γ (n) (n − 1)! n!

An audited elementary algebra

Abstract: Solving equations is one of the core reasons for elementary algebraic manipulation. As Chrystal wrote [1]: There are few parts of algebra more important than the logic of the derivation of equations, and few, unhappily, that are treated in more slovenly fashion in elementary teaching. No blind adherence to set rules will avail in this matter; while a little attention to a few simple principles will readily remove all difficulty.

99.06 A peculiar proof of an identity of Euler

Abstract: 99.06 A peculiar proof of an identity of Euler Introduction The history of complex analysis is a fascinating subject. Many mathematicians took part in the development of complex function theory. As one would expect, Euler was among them. At the age of 70, he became particularly interested in the use of complex variables in integration. According to Bottazzini and Gray [1, p. 93] ‘ In 1777 he presented a series of nine papers on this topic to the St. Petersburg Academy which, however, had little impact, if any, because they were published only posthumously between 1793 and 1805.’ These papers were devoted to the evaluation of certain definite integrals by making an appropriate change of variables. Euler believed that, in some cases at least, complex substitutions need not change the range of integration to a path in the complex plane. To a certain extent, he was correct. Allow us to elaborate more constructively. [0, ∞)

Artzt parabolas of a triangle

Abstract: The following article is about beauty in mathematics. It shows how simplicity can occur in apparently complex constructions. It explores the remarkable properties of parabolas which are tangent to two sides of a triangle where the tangent points are two vertices. The other side of the triangle is a chord of the corresponding parabola. These are known as the Artzt parabolas of the triangle since they were first described by Artzt in 1884 [1]. They are often described as being parabolas ‘inscribed in a triangle’ as in Bullard's papers [2, 3] and Eddy and Fritsch [4] (see Note 1). This article is also about how new tools for geometric drawing can enable the discovery of such properties in a way which is well within the grasp of students. Such explorations are a means to learn about geometry for students who are inspired to do so (see Note 2). The advent of dynamic geometry packages such as Geometer's Sketchpad, Cabri, Cinderella and GeoGebra allow easy drawing and manipulation of curves, such as conic sections, in a way mathematicians in the past could only have dreamed about. Results can be found without a mass of calculation. Moreover, the results are visual and so much more accessible. By moving key points in a drawing, observation shows whether or not a theorem is true by noticing how incidences remain (see Note 2). They also show that, in what is known as advanced euclidean geometry, while not being in the forefront of mathematical research, there are still many results to be found. Indeed, with the internet to communicate such geometrical ideas, this geometry is flourishing as strongly as its heyday at the end of the nineteenth century. This is evidenced by the Advanced Plane Geometry e-mail group [5] which can post ten or more results in a day; and by the electronic journal Forum Geometricorum [6]. The drawings shown here and discussion are based on the free program GeoGebra [7].

Evaluating Fresnel-type integrals

Abstract: Convergence of the integrals is not immediately apparent, since the integrands do not tend to zero as : in fact, for all . A formal proof of convergence will be given shortly, but informally it results from the increasingly rapid oscillation of and : as increases, these functions are alternately positive and negative on shorter and shorter intervals, which make contributions of decreasing magnitude to the integral. x → ∞ | eix2 | = 1 x

Minimal area of a bicentric quadrilateral

Abstract: A quadrilateral with an incircle is called a tangential quadrilateral and a quadrilateral with a circumcircle is called a cyclic quadrilateral. A bicentric quadrilateral is a convex quadrilateral that is both tangential and cyclic. In [1] we gave a short proof via an inequality that the bicentric quadrilateral with a given incircle and circumcircle that has the maximal area is a right kite. A year after the publication of that note, I received an e-mail from the German mathematician Heinz Schumann asking if I also had a simple proof of the minimal area case. At that time I had not considered that interesting question. Now I have and in this note such a proof is presented. Longer or more complicated proofs can be found in [2, pp. 6-8], [3, pp. 20-25], and [4, pp. 168-171], which uses derivatives.

The ‘hitchhiker triangle’ and the problem of perimeter = area

Abstract: In a recent Student's problem in the Gazette, a triangle is formed from three rectangular strips of paper lying on a flat surface. Their widths were given and the challenge was to find the sides of the triangle so formed. These turned out to be in the proportions 7: 15: 20 [1]. The triangle with sides , , is a curious one. It has both perimeter and area ( ) equal to 42. In his Hitchhiker's Guide to the Galaxy Douglas Adams thought ‘42’ the ‘answer to the ultimate question of life, the universe, and everything’ and it has achieved cult status. For this reason, we may term this particular triangle the ‘hitchhiker triangle’. a = 7 b = 15 c = 20

Elegant special cases of Van Aubel’s theorem

Abstract: Van Aubel's theorem for quadrilaterals concerns any quadrilateral on whose sides squares are constructed*. The theorem states that the segments connecting the centres of the squares lying on opposing sides are equal and perpendicular. In other words, is equal and perpendicular to , as shown in Figure 1. The theorem also holds for non-convex quadrilaterals, as shown in Figure 2, and for a self-intersecting quadrilateral, as shown in Figure 3. The squares may be constructed outwards (Figure 1) or inwards (Figure 4) on the quadrilateral. The theorem holds even when the quadrilateral degenerates into a triangle, a straight segment, and even two connected segments. PR SQ

The Heron parameters of a triangle

Abstract: 1. The problem If a triangle has sides of integer lengths, and an inscribed circle of unit radius, then it is a right-angled triangle with sides of lengths 3, 4 and 5. To see that this is the only right-angled triangle with these properties draw a right-angled triangle with sides of integer lengths , and (the hypotenuse), and inscribed circle of unit radius. Obviously and , so we can write and for positive integers and . As the two tangents from a point outside a circle to the circle are of equal length, we see that . Thus a b c a > 1 b > 1 a = 1 + a1 b = 1 + b1 a1 b1 c = a1 + b1 (a1 + b1) = c = a + b = (1 + a1) + (1 + b1) , or ; hence as required. However, how do we show that this is the only (not necessarily right-angled) triangle with these properties? (a1 − 1) (b1 − 1) = 2 {a1, b1} = {2,3}

99.33 An integral inequality with applications to probability theory

Abstract: 3.3! + π 5.5! − π 7.7! + ... ) ≈ 1.17898 × π4 , which corrected value indicates an ‘overshoot’ of roughly 18% at each jump discontinuity. Other stationary values of invoke the corresponding multiple of for the upper limit of integration and a graph of the integrand (or , say) shows these to be insignificant. f n (x) π sin x / x f 10 (x) A conventional Fourier treatment of the Gibbs phenomenon employing a different waveform is given in reference [3].

99.29 On the sum of the reciprocals of the primorials

Abstract: f (m) = (p1p2... pm) for , and . Now let , for , denote an infinite, strictly-increasing sequence of positive integers such that and, for each , . The sequence , for , is defined by setting and, for each , letting for . m > 0 f (0) = 1 (g (m)) m ≥ 0 g (0) = 1 m ∈ pg(m) > f (m) + 1 (an) n ≥ 0 a0 = 0 m ∈ an = f (m − 1) g (m − 1) ≤ n ≤ g (m) − 1 Similarly, we define the sequence , for , by (bn) n ≥ 1

99.37 Anthropologic

Abstract: 99.37 Anthropologic It seems that your work as an anthropologist is getting ever harder. Once you visited ocean islands with just two tribes, Truth Tellers (T) and Liars (L), but now might also be faced with Random Responders (R). On your first visit to such an island, you are certain to meet a group of three with one person of each propensity. Can you sort them out in just three yes/no questions? Each question can be addressed to any one of the three welcomers, who, by the way, speak perfect English, are experts in logic and know each others' tribes. It seems at first that this is an impossible challenge, because if the first question happens to be addressed to R, you will gain no information and the remaining 2 questions can only have answers, not enough to distinguish permutations. Indeed an eminent mathematician recently produced this argument and was so convinced by it that he refused to listen to our answer. 2 = 4 3! = 6

Textbooks from Shanghai and Singapore A National Debate

Abstract: The Minister for Schools, Nick Gibb, is insistent that what England requires to improve primary maths (and, apparently, other subjects as well) are textbooks − preferably the same textbook − in front of all children in all classes. I would like to see all schools, both primary and secondary, using high quality textbooks in all subjects, bringing us closer to the norm in high performing countries In this country, textbooks simply do not match up to the best in the world, resulting in poorly designed resources, damaging and undermining good teaching ...

On the Routh-Steiner theorem and some generalisations

Abstract: He emphasised that this result was discovered by Steiner, but simultaneously cited two references: the first was Steiner's work [2, pp. 163168] and the second was Routh's work [3, p. 82]. Later in his book he referred to the result as ‘Routh's theorem’ [1, p. 219], admitting to the contribution of both scientists in revealing the theorem. Coxeter gave a general proof of this result usingbarycentric coordinates attributed to Möbius. These arehomogeneous coordinates , where are masses at the vertices of a triangle of reference . In particular (1,0,0) is , (0,1,0) is , (0,0,1) is and corresponds to a point such that the areas of the triangles , , are proportional to the barycentric coordinates of , respectively (see Figure 1). If then the normalised barycentric coordinates are calledareal coordinates. In this case the areas of the triangles , , are times the area of the whole triangle , respectively. (t1, t2, t3) t1, t2, t3 A1A2A3 A1 A2 A3 (t1, t2, t3) P PA2A3 PA3A1 PA1A2 t1, t2, t3 P t1 + t2 + t3 = 1 (t1, t2, t3) PA2A3 PA3A1 PA1A2 t1, t2, t3 A1A2A3

99.35 On the asymptotes of the Kiepert hyperbola

Abstract: 99.35 On the asymptotes of the Kiepert hyperbola The asymptotes of the Kiepert rectangular hyperbola may be employed as Cartesian axes for the general scalene triangle (see [1]). ABC The origin of coordinates is the Kiepert centre and the vertices , , , centroid and orthocentre lie on the curve . Thus , with for rectangularity. Note also that and two of the parameters may be taken as positive. K (0, 0) A (a, a) B (b, b) C (c, c) G (g, g) H (h, h) xy = 1 3g = a + b + c g = a + b + c abch = −1 g, h a, b, c

A search for sums of powers

Abstract: and using standard formulae for sums of terms and squares of terms we have the quadratic equation in given by which has the solution , and since is a sequence composed of every second triangular number (which are 1, 3, 6, ). So equation (1) is a formula for our sequence of sums of squares when p is every second triangular number and . p p2 + 2n2p − n2 (2n + 1) = 0 p = n (2n + 1) n (2n + 1) 10, ...

99.03 Quick proofs that certain sums of fractions are not integers

Abstract: A generalisation Finally, let the integers and satisfy and , respectively. We generalise here to , the generating function for the increasing and infinite sequence of all positive integers such that when each of these numbers is written in decimal notation, there is an in the column, noting that we adopt here the convention that the right-most column is termed the 0th column (or units column), the next column is termed the 1st column (the tens column), and so on. It is relatively straightforward to show that this generating function is given by m n 0 ≤ m ≤ 9 n ≥ 1 F (x) Fm,n (x)

On some elementary functions

Abstract: We gather here some material, ‘well-known to the experts’ — see the short historical discussion at the end — but not often found in the massmarket textbooks for high schoolers and undergraduates (who would benefit from it the most), that can be used to discuss , and rigorously in terms of simple inequalities and high school algebra of numbers, polynomials, and rational functions along with the completeness axiom of the real line: ex x log x

The Siddhantasundara of Jnanaraja, an English translation by Toke Lindegaard Knudsen, pp. 351, £58.00, ISBN 013 978-1-4214-1442-3, John Hopkins University Press (2014).

Abstract: could have got by with a rudimentary understanding of how the decimal system used on the eastern trade routes worked, without needing either to forgo his own system of calculating or to be au fait with how calculating in the ‘new‘ system worked. As Orstein Ore put it in the memorable quotation cited on page 77, ‘Not until the sixteenth century had the new numerals won a complete victory in schools and trade. Even as late as Nikolaus Copernicus' famous work De revolutionibus, published in 1543, the year of his death, one finds a strange mixture of Roman and Indian numerals and even numbers written out fully in words.’ In Part 2, Mazur examines the development of algebra from its rhetorical origins to its eventual symbolic form. He traces the story from Diophantus, via Indian and Arabic mathematicians to Renaissance Europe, up to the introduction of rival calculus notations by Newton and Leibniz. He makes some very pertinent points about the ‘survival of the fittest’ in terms of competing notations. For example, in Pacioli'sSumma of 1494, is denoted by and by while is denoted by . Such notation is adequate for (say) notating a quadratic equation, but there is little hope of recognising the laws for exponents, and . In the latter form, the symbols develop a life of their own rendering natural such nonobvious statements as , and . But − and this is the cautionary jolt that a historical narrative such as Mazur's delivers − there is an apparently inexorable golden rule that ideas precede their optimal expression in symbolic form. For example, Napier invented and used logarithms without our exponential notation or a formal definition of inverse functions. x RX3 x RX4 2 RX2 xx = x + n (xm) = x