The Somewhat Secret Life of Pi

It became a secret by default but is the overtly pragmatic outlook of contemporary mathematics educators causing undue mysticism of the “life” of pi? 

By: Ringo Bones 

Most contemporary mathematics educators discuss pi only as far as the ratio of the circumference of a circle to its diameter. Though it may allow students to solve the area of a given circle and other related engineering problems, but should today’s mathematics educators be “more adventurous” to have an extensive discussion on the somewhat secret historic life of pi? 

The symbol of pi that used the Greek letter was first employed in its present meaning by William Jones in 1707 and its adoption by Leonhard Euler in 1737 brought it into general use. In far earlier times, the determination of the value of pi and the calculation of the area of a circle of a given diameter were recognized early as closely related problems. 

The mathematicians of antiquity had learned how to calculate precisely the areas of plane figures bounded by straight lines, such as the rectangle, parallelogram and triangles. The circle, however, eluded the mathematicians of antiquity’s methods, although many approximations for its area were developed. It remained for the mathematicians of the last 150 years to disclose the real nature of the problem. 

The ancient Chinese and the ancient Hebrews seem to have taken pi as equal to 3. In the Ahmes Papyrus of ancient Egypt back in 1650 BC, the area of a circle is found by subtracting from the diameter 1/9 of its length and squaring the remainder. This is equivalent to taking pi as 256/81 or 3.1605. 

In ancient Greece, the problem took the form of requiring the construction of a square, by straight edge and compass alone, whose area equals that of a given circle – i.e. squaring the circle. Back around 225 BC, Archimedes approximated the area of a circle by comparing the areas of regular inscribed and circumscribed polygons of a large number of sides. Using polygons of 96 sides, Archimedes found that the ratio of the circumference of a circle to its diameter lies between 3 1/7 and 3 10/71. After the time of Archimedes, 3 1/7 and the square root of 10 were taken as a satisfactory approximation of pi. An approximation equivalent to the now familiar decimal 3.1416 dates from before 200 AD. 

After mathematicians invented decimals and promoted its widespread use, the calculation of pi was carried out to 35 decimal places by Ludolph van Ceulen (1540 – 1610). In Germany, pi is sometimes called the Lundolphian Number in his honor. 

Toward the end of the 17^th Century, new methods of analysis were developed that made it possible to express pi as the sum or product of an infinite number of terms. One of the earliest of such expressions was formulated by Franciscus Vieta (1540 – 1603). In 1655, John Wallis (1616 – 1703) and in 1658 William Brouncker and Leibniz in 1673 published their various expressions of series to compute the value of pi.

By the use of such series, the value of pi may be computed to any desired number of decimal places. By 1873, William Shanks had carried the computation of the value of pi to 707 decimal places; in 1946, however, it was discovered that Shanks’ vas incorrect after the 527^th decimal place. With the advent of mechanical – and later electronic – computing machines, more extensive calculations of the value of pi became feasible. Even as far back as 1958, pi was already accurately calculated to 10,000 decimal places using the most advanced electronic computers of the time. 

In our more pragmatic contemporary engineering practices computing the value of pi beyond 10 decimal places offer little practical value. To ten places, pi is given as 3.1415926536 – this is of sufficient accuracy to permit computing the circumference of a sphere the size of the planet Earth with an error less than 1/8 of an inch. Out to about 40 decimal places, pi would give the circumference of the entire visible universe with an error imperceptible even with an electron microscope. Thus these extensive calculations have no practical value and serve chiefly to show the power of modern methods of computation. 

Though in the field of number theory, the value of pi manifest itself in an exiting way that can sometimes only appreciated by pure mathematics enthusiasts. In 1761, Johann Lambert showed that pi is irrational – i.e. cannot be expressed as the ratio of two integers. While in 1794 Andre Marie Legendre made the proof a rigorous one, thus finally establishing that the value of pi is not a repeating decimal. 

Joseph Liouville in 1884 proved the existence of transcendental numbers – i.e. numbers that cannot be expressed as roots of algebraic equations with rational coefficients. Charles Hermite in 1873 proved that e – the base of natural logarithms – was such a number. Using this information and Euler’s theorem that e raised to the power of pi times the square root of negative one is equal to zero, Ferdinand Lindermann proved in 1882 that pi is transcendental thus assuring the impossibility of squaring the circle. 

The number pi enters into the measurement of many more curved plane figures other than the circle. Thus the area of an ellipse is given in terms of half the length of its axes as A = pi (a) (b). It also appears in many of the relations of mathematics, physics and engineering other than those directly concerned with areas or lengths of arcs. 

Was The 2012 US Presidential Election Results Mathematically Predicted?

Did a 34 year old statistician and “poker number expert” used tried and true mathematics to predict the outcome of the 2012 US Presidential Elections with better than 90 percent accuracy?

By: Ringo Bones

Despite attracting widespread ridicule from traditional political pundits in what was regarded as the closest US Presidential Race in years, a 34 year old statistician and poker numbers expert named Nate Silver gave incumbent US President Barack Obama a 90.9 % chance of success by using tried and true mathematical techniques way before the final voting results came through. Nate Silver used an elaborate series of tried and true calculations to correctly call the outcome in all 50 states by running thousands of computer calculations based on innumerable factors such as polling results and voting outcomes of previous elections. His achievement opens the door for a more statistics-based approach to polling says experts. Given Silver’s mathematically predictive success, will the outcome of the next US Presidential Election – in 2016 – be predicted with better than 90 percent accuracy using his method?

The highly mathematical approach used by Nate Silver – which has echoes of the book and a Hollywood film that stars Brad Pitt about baseball statistics titled “Moneyball” – involves running hundreds of mathematical calculations for the polling data of each state based on a myriad of factors, including election results from the past and more recent polling data. Even though Silver is an avowed Obama supporter, his “magic mathematical formula” calculated that there was a 90.0 % likelihood of an Obama win and the President would win 332 Electoral College seats compared to 206 for Republican challenger Mitt Romney – correct if the Democrat wins Florida which, at the time, was yet to declare when Silver made his prediction.

Years before gaining fame for his mathematical predictions for the 2012 US Presidential Race, Nate Silver’s mathematical skills were already well known in the American journalistic world because back in 2010 his existing blog at FiveThirtyEight.com was bought by the New York Times for a princely sum. And then Silver revealed to them his own proprietary mathematical models for predicting election results based on a combination of electoral history, demographics and polling results. I wonder if Silver’s mathematical method works in the actuary world of determining cost competitive premium rates for insurance policies with “sketchy” statistical data. 

Does Srinivasa Ramanujan’s Mathematical Musings Have Divine Origins?

First dismissed as a minor under-educated bureaucrat from Madras, India before his mathematical musings got the scrutiny eminent Cambridge don named G.H. Hardy, does Srinivasa Ramanujan’s number theories have “divine” origins?

By: Ringo Bones

Before the advent of String Theory when many theoretical physicists still clung on to the simplicity of the Standard Model, the numerical insights and brilliant conjectures of Srinivasa Ramanujan was perceived by the global mathematical community as nothing more than a mere exercise in “number theory”. But in recent years, Ramanujan’s brilliant mathematical insights have become de rigueur in explaining the possibility of the existence of wormholes and quantum mechanical phenomena beyond the Standard Model. Ramanujan’s high-ordered number theories has been recently seen as a mathematical model explaining how artificial wormholes / stargates and other faster-than-light interstellar travel technologies works as they are often used in science fiction stories.

According to Srinivasa Ramanujan, his unique and brilliant number theories were apparently inspired by the Hindu goddess named Namagiri. Namagiri is worshiped especially in the Namakkal district of Tamil Nadu state in Southern India where Ramanujan lives. Namagiri’s devotees worship her as a consort of Narasimha – an avatar / incarnation of the deity Vishnu. Namagiri was the mathematician Srinivasa Ramanujan’s family’s deity. But according to ancient astronaut theorists, it is much more than that.

Ancient astronaut theorists believes that Hindu deities were nothing more than extraterrestrial biological beings much more advanced than us that had been guiding the development of the indigenous inhabitants of the Hindus River Valley develop their own civilization over 5,000 years ago. And many of them see Srinivasa Ramanujan’s mathematical musings predictive modeling String Theory, wormholes and other theoretical science behind faster-than-light interstellar travel nothing more than an advanced civilization’s technology bequeathed to humanity.

The Slide Rule: Mathematical Relic?

Supposedly superseded by the electronic pocket calculator back in 1974, are the days of the slide rule as a viable mathematical computational aid finally over?


By: Ringo Bones


While the electronic calculator and every similarly battery-operated model that superseded it since it was introduced back in 1974 had been decried by “old-school” math educators as the primary cause of the current “dumbing-down” of prospective math students, I sometimes wonder if those under-40-old-school-math-educators had ever experienced the joy of handling a slide rule. But is the slide rule still relevant in today’s world? Especially when prospective math students can easily install – if they chose to - scientific calculator apps in their PDAs and other hand-held computers?

A slide rule is a mechanical analog computer, a mechanical device consisting of moveable scales, arranged to slide opposite each other, by means of which certain mathematical operations may be carried out quickly. Also known colloquially as a “slip-stick”, it is used primarily for multiplication and division and also for “scientific” functions such as roots, logarithms and trigonometry. More often than not, slide rules do not generally perform addition or subtraction.

The working principle behind the slide rule was made possible when John Napier invented the concept of logarithm, his treatise of the subject being published in 1614. In 1632, William Oughtred, an English clergyman and teacher, arranged two logarithmic scales – which he invented back in 1622 – to form the first slide rule. The slide rule in its modern form – before being superseded by the modern pocket calculator – was developed by Amédée Mannheim, a French artillery officer, in 1850.

Amédée Mannheim’s version of the slide rule consists of a 10-inch rule, with three parts: the stock, or two fixed parallel rules, each with a scale on its inner edge; the slide, a single rule, moving between them, having a scale on each outer edge corresponding to the fixed scale to which it is adjacent; and the cursor, a square glass with a hair line which may be moved the length of he rule as an aid to reading it.

Before the advent of the pocket calculator, the slide rule was the most commonly used calculation tool in science and engineering. The use of slide rules continued to grow through the 1950s and the 1960s even as digital computing devices were being gradually introduced. Famed aerospace engineer Clarence “Kelly” Johnson was purported to have used the slide rule in designing the U2 and SR-71 Blackbird spy planes. High fidelity audio equipment engineered using slide rules during the Golden Age of Stereo – i.e. 1950s to the 1960s – were famed for their warm and smooth natural sound. By around 1974, the pocket-sized electronic scientific calculator introduced by Texas Instruments and other competing consumer electronics manufacturers, largely made the slide rule obsolete and most suppliers exited the business around 1976.

The only saving grace of the slide rule – if you’re fortunate enough to find one in a garage sale or antiques shop being offered at a keen price - over competing electronic calculating devices is that it doesn’t require batteries. And also, more “mature” old-school math teachers praise the slide rule for keeping math students on their toes in its use thus making their users smarter. To me, the slide rule or slip-stick’s primary advantage over the electronic pocket calculator is that the slide rule is 100% water proof and dust proof – as in desert conditions dust proof.

Given that the “modern” slide rule was developed by a French artillery officer named Amédée Mannheim, nobody has yet pitted the slide rule side-by-side against those dedicated palm-held ballistic computer. Like Knights Armament Company’s KAC Bullet Flight 2.0.0 described as a palm-held ballistic computer designed to provide quick firing solutions in the field. I wonder how this palm-held push-button firing solution fares well against an early 1960s era slide rule when providing firing solutions for a 123-grain Lapua Scenar round with a ballistic coefficient of 0.547 and a muzzle velocity of 2,600 feet-per-second.

Seki Kówa: Japan’s Forgotten Mathematical Genius?

Credited for independently discovering his version of integral calculus and determinants in 17th Century isolationist era Japan; Was Seki Kówa Japan’s forgotten mathematical genius?


By: Ringo Bones


Fortunately for us in the West, Japanese intellectuals was still “literate” enough to jot down their musings for posterity even in the harshly despotic and isolationist regimes of the Hideoshi and Tokugawa shogunates of 16th and 17th Century era Japan. It probably wasn’t until Commodore Matthew Perry imposed his “Gunboat Diplomacy” back in 1853 and 1854 that finally allowed the curious West to scrutinize what Japanese intellectuals managed to discover during their country’s isolationist period. And with the Meiji Restoration period of 1868 to 1912, the Western world finally uncovered that the Japanese development in mathematics was as advanced that of in Europe.

Enter Seki Kówa, traditionally credited in Japan for discovering his version of the integral calculus during the 17th Century. His yenri or circle principle, which was documented back in 1670, uses a series of triangles to measure the area of a circle. Like that used by 17th Century contemporary calculus discoverers in Europe, Isaac Newton and Gottfried Wilhelm von Leibniz.

Seki Kówa also independently discovered the mathematical principle of determinants back in 1683. While in Europe, calculus pioneer Gottfried Wilhelm von Leibniz also independently discovered the principle of determinants 10 years later – in 1693. While integral calculus in our day and age is indispensable in determining the volume of all manner of irregular shapes, such as airplane fuselages and oil storage tanks and the areas of curved surfaces to find the exact amount of sheet metal to use in a car body or the lifting surfaces of a modern jet. We – in the present - are still thankful for a now largely forgotten 17th Century Japanese mathematical genius named Seki Kówa for being curious enough for contributing his own ideas on integral calculus and other then still obscure mathematical concepts in the 1600s.

Alfred North Whitehead: Mathematical Unifier?

Inspired by the underlying commonality of the existing mathematics of his day, would Alfred North Whitehead be as successful as Rene Descartes in establishing a new branch of mathematics?


By: Ringo Bones


Ever since Rene Descartes made possible the happy marriage of curves and quantities – i.e. by merging all the arithmetic, algebra and geometry of ages past into a single technique – to produce analytic geometry, many a wannabe great mathematician had tried to create their own branch of mathematics by combining existing ones. During the latter half of the Victorian Era, none got closer than Alfred North Whitehead. But today’s kids would certainly ask who the heck is he?

Alfred North Whitehead (1861 – 1947) English mathematician and philosopher, was born at Ramsgate, the Isle of Thanet, Kent, on February 15, 1861, of a family of teachers and ministers. His father was an Honorary Canon of Canterbury. Upon entering Trinity College, Cambridge, Alfred North Whitehead devoted himself to mathematics. But by being a member of the Apostles Club – philosophy, literature, history, politics and religion were all subjects of intense discussion.

In 1885, Whitehead became a Fellow of Trinity and began teaching mathematics. His Treatise on Universal Algebra led to his election to the Royal Society in 1903. A decade of collaboration with the most brilliant of his former pupils – Bertrand Russell – resulted in the publication of the monumental Principia Mathematica consisting of three volumes between 1910 and 1913.

Alfred North Whitehead’s mathematical work that gained him fame was the 1898 publication of his pioneering work called A Treatise on Universal Algebra. Now called Abstract Algebra, it was his unfinished attempt to unify “the various systems of Symbolic Reasoning allied to ordinary algebra.” The first and only volume that was published is a detailed investigation of H. G. Grassmann’s Calculus of Extension – Ausdehnungslehre in German – which was first published in 1844 but insufficiently appreciated and George Boole’s Algebra of Logic. These mathematical works had attracted Whitehead’s attention by their bold extension of algebraic methods beyond the traditional realm of the quantitative.

Whitehead restated Grassmann’s calculus and employed it to unify a variety of geometries; thus the theorems of projective geometry were exhibited as consequences of the definitions of the calculus. Some, but comparatively few, additions to the superstructure of mathematics were included – resulting in a main achievement that was more a novel unification. But this was along relatively unorthodox lines, thus the work had little influence among mathematicians.

Returning to the great Principia Mathematica, the first portion of which is a deductive elaboration of formal logic from a few axioms; the remainder is a detailed deduction, from this alone, of the basic concepts and principles, first called cardinal arithmetic and then of the other recognized mathematical sciences; and many new sciences suggested by the broad definitions being laid down.

The whole is written in exact and elaborate symbolism, taken partly from Giuseppe Peano. The work is basic for students of the foundations of mathematics, in spite of the fact that the first portion, for which Russell was mainly responsible, was involved in difficulties which have challenged experts ever since, and that the fourth volume, which was to deal with geometry alone, and was to be written by Whitehead alone, never appeared. Thus his attempts to unify various geometries were seen nothing more by mathematicians – then and now – as nothing more that a novel unification.

Is Prison A Good Place to Hone One's Math Skills?

From Jean Victor Poncelet to the imprisoned Tiananmen Square dissidents of 1989, is prison really a good place hone one’s mathematics skills?


By: Ringo Bones


Most prisoner of war inmates opt intricate scrimshaw carving or feeling sorry for themselves, but there are an exceptional few who used their time spent in captivity to develop and improve their existing math skills. Some have even managed to contribute indispensable facts to the still growing collective mathematical knowledge.

Captured during Napoleon’s disastrous Russian campaign of 1812, French mathematician Jean Victor Poncelet conquered the boredom of his prison camp by aligning a lot of disorganized non-Euclidian insights into a new branch of mathematics now known as projective geometry. Its aim is to study the properties of geometric shapes that stay unchanged when seen from a distance. An example of which is when Poncelet created a set-up during his captivity – a pyramid that contains a seemingly chaotic arrangement of colored cards. When the eye looks at this particular pyramid’s base, it sees an orderly pattern due to the angle at which “chaos” - or the chaotically arranged colored cards, is projected through space to the viewer’s eye. While some of his comrades are probably busied themselves carving intricate pieces of scrimshaw, Jean Victor Poncelet managed to create a new branch of mathematics now called projective geometry.

Professor Jackow Trachtenberg, a brilliant engineer who managed to invent a very handy set of mathematical shortcuts now known as the Trachtenberg Speed System of Mathematics during the years that he spent in captivity at Hitler’s concentration camps as a political prisoner. There’s even a Trachtenberg Mathematical Institute in Zurich, Switzerland established in honor of Professor Jackow Trachtenberg.

There also had been anecdotes during 1990 that some Chinese students who became political prisoners after their participation of the pro-democracy protests in Tiananmen Square back in 1989 have been making good use of their time spent in captivity as a political prisoner. Some have even tried to continue the unfinished work of Albert Einstein that he started since the1950s of formulating an equation that could unite the two very disparate systems of Quantum Mechanics and General Relativity. But is prison really an ideal place to hone one’s mathematics skills? Only time will tell.