Wednesday, 28 February 2018

On A Mathsy St David's Day

Q: What links Bertrand Russell, the equals sign, the word in the English language with the most Zs, the use of π as the ratio of a circle's circumference to its diameter, the Online Encyclopedia of Integer Sequences, and the number 36?

A: Wales.

By way of explanation, and in mathsy celebration of St David's Day, I offer below a handful of mathematical somethings with a Welsh bent, peppered with one or two suggested explorations or diversions that teachers may wish to share and use with students on the day or, indeed, any other.

Dydd Gŵyl Dewi Hapus.

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The Equals Sign
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Welsh Doctor and mathematician Robert Recorde, born in 1510 in Tenby, Pembrokeshire, South Wales, was a popular author of a number of mathematical books — which he wrote, unusually for the time, in the English vernacular, thus making his writing more accessible than most scholarly books of the age, which were usually written in Latin.

In his 1557 book, The Whetstone of Witte, whiche is the seconde parte of Arithmetike: containyng thextraction of Rootes: The Cossike practise, with the rule of Equation: and the woorkes of Surde Nombers [1] (which you can peruse electronically here), Recorde 'invented' the '=' symbol, 'to avoide the tediouse repetition of these woordes : is equalle to :', which he had already used some 200 times in the book [2].

During Recorde's time, much of the mathematical notation we take for granted today was not yet in use.  In designing the symbol '=' — 'a paire of paralleles, or Gemowe [twin] lines of one length, thus: =====, bicause noe 2 thynges can be moare equalle' — Recorde's initial motivation to abbreviate was quickly overtaken by something more profound, more enduring.  As Joseph Mazur (2014) eloquently puts it, 'the concise character of the symbol came with an unintended benefit: it enabled an unadorned picture in the brain that could facilitate comprehension'.

Recorde was also, for example, and in the same book, the first to use the plus and minus signs in English: 'There be other 2 signes in often use of which the first is made thus + and betokeneth more: the other is thus made – and betokeneth lesse'.  And neither was there an easy way in the 16th century of denoting the powers of numbers, so Recorde coined the now unsurprisingly obsolete term ‘zenzizenzizenzic' [3] to ‘doeth represent the square of squares squaredly’, or in other words to denote the square of the square of a number's square:

\[{\left( {{{\left( {{n^2}} \right)}^2}} \right)^2} = {n^8}\]
Recorde also used another word (which didn't quite catch on), the 'sursolid', meaning to be raised to a prime number greater than three.  So a power of five would be the first sursolid, a power of seven the second sursolid, a power of eleven the third, and so on.

Suggested explorations/diversions with/for students:
  • Find the zenzizenzizenzic of n for 0 < n < 10.
  • Devise questions in Recordian notation and answer them, for example: 'What is the fourth sursolid of two divided by the zenzizenzizenic of two?  Give your answer in modern and Recordian form'.
  • Consider and explore the difference between:

\[{\left( {{{\left( {{n^2}} \right)}^2}} \right)^2}\;{\rm{and}}\;{n^{{2^{{2^2}}}}}\]

The first use of π to denote C/d
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Before being denoted π, the ratio of the circumference of a circle to its diameter was referred to typically in the Latin 'quantitas in quam cum multiflicetur diameter, proveniet circumferencia ('the quantity which, when the diameter is multiplied by it, yields the circumference)' (Rothman, 2009).

Welsh mathematician William Jones, born in 1675 in Llanfihangel Tre'r Beirdd, on the Isle of Anglesey, North Wales, was the first person to use π to denote the ratio of a circle's circumference to its diameter, doing so in his 1706 book Synopsis palmariorum matheseos: or, A new introduction to mathematics: containing the principles of arithmetic & geometry demonstrated, in a short and easie method; with their application to the most useful parts thereof ... Design'd for the benefit, and adapted to the capacities of beginners [4].  You can peruse the book electronically here [5].

This first ever appearance of π denoting the ratio of a circle's circumference to its diameter can be seen on p243, then more explicitly on p263, as excerpted below.  It can also be seen that Jones gave π correct to 100 decimal places, 'as Computed by the Accurate and Ready Pen of the Truly Ingenious Mr. John Machin', using an infinite series whose sum converged to π (see this on 'Machin's Formula' from Peter Rowlett in The Aperiodical).

Using π in this way was a significant philosophical step forwards; Jones was more than merely abbreviating.  Although unable to prove it, Jones recognised that the ratio of a circle's circumference to its diameter could not be expressed as a rational number — or in other words, that π was an irrational number — as can be seen in the p243 excerpt above: 'For as the exact Proportion between the Diameter and the Circumference can never be expres'sd in Numbers' [6].  Jones recognised, as such, that 'to represent an ideal that can be approached but never reached.... only a pure platonic symbol would suffice' (Rothman, 2009).

Jones' use of π as C/d was popularised when the great Swiss mathematician Leonhard Euler adopted it in his Introductio in analysin infinitorum (Introduction to the Analysis of the Infinite) in 1748 [7].

Suggested explorations/diversions with/for students:
  • Share the excerpts above with students and try to make sense of them together.
  • Have students work out their π-related birthdays, past or future, discussing precision (see this post).
  • Note, for example, that at some point on St Davids' Day 2018:
    • 3 year-olds born on 8 January 2015, will be π years old.
    • 9 year-olds born on 17 April 2008, will be ππ years old.
    • 36 year-olds born on 8 January 2015, will be ππ years old.

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Mathematician Neil Sloane, born on 10 October 1939 in Beaumaris, North Wales — described by Erica Klarreich as 'The Connoisseur of Number Sequences' in this article in Quanta magazine, and as 'the Guy who Sorts All the World's Numbers in his Attic' in this reprint of the article in Wired — is considered by some to be one of the most influential mathematicians of our time, because in 1964 Sloane founded The Online Encyclopedia of Integer Sequences (OEIS).

The OEIS, as the name suggests — or Sloane, as it is often referred to by its users — is an online database of at the current count, over one quarter of a million integer sequences.  It is designed to be used by researchers in mathematics, but as John Conway and Tim Hsu put it in 2006, 'most Nerds should be able to get some enjoyment out of it'.

Enjoy this short selection of some gems that I first discovered through the OEIS:

And consider this sequence (sequence A168087): a(n) = the smallest number whose Welsh name (masculine or feminine versions) in the modern Decimal System contains n letters of the alphabet.  (For example, 224, dau gant dau ddeg pedwar, is the 20th number in the sequence, and is the smallest number with 20 letters.)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 2 3 7 4 11 12 15 16 14 27 24 47 44 127 124 147 144 244

Suggested explorations/diversions with/for students:
  • Try and ascertain the rule that describes each of my selection of 'gems' above.
  • For sequence A168087
    • Continue the sequence to n = 40.
    • Find a(100).
  • Do the same for sequence A168085 (using the traditional Vigesimal System).  
  • Generate the same sequence for numbers in English, and other languages.
  • Describe this sequence 4, 2, 3, 3, 6, 4, 6, 5, 4, 3, 3, 8, 5... (sequence A140396), entered into the OEIS by Sloane himself in 2008, perhaps as a nod to his Welsh heritage.
  • Generate a sequence of numbers that have the same amount of letters in Welsh as in English.  For example, a(1) = 2, because 'two' in English and 'dau' in Welsh has 3 letters.  
    • Maybe submit the sequence to the OEIS (it's not there; I've checked) on behalf of a student (with parental consent of course) who generates it and defines it best, according to the OEIS' format.  
  • Try this puzzle set by Sloane in Quanta Magazine:
    • Can you figure out the 'simple' rule that describes this sequence 13, 26, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, 24, 16, … (click here for the solution when you're ready).
  • The image below shows the 'zigzag triangle', via JohnConway and Tim Hsu (2006).  On the LHS of the triangle are the Zig (or secant or Euler) numbers, and on the RHS are the Zag (or tangent) numbers.  
    • Find the next Zig and Zag numbers.
    • How far can you keep going?


The Number 36
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Charles Yang, associate professor in the University of Pennsylvania Department of Linguistics, and not Welsh, has shown (2005, cf. 2015) that if there is a linguistic rule, a generalisation in other words that can be applied to a set of N words, but within this set of N words there is a subset of words, e, that do not follow this rule and that must therefore be memorised, then

\[e < {\theta _N}\;where\;{\theta _N}: = \frac{N}{{\ln \left( N \right)}}\]
This model, dubbed the 'Tolerance Principle', can be applied to how we, as children, learn to count.  In short (see this post, 'On 73'for more detail), by using the Tolerance Principle, we can find the least amount of words that we need to learn ‘to overcome the exceptions we have to memorise’.  In Welsh, in the modern Decimal System of counting, the numbers 1 to 10 are the only 'exceptions': Un (one), Dau (two), Tri (three), Pedwar (four), Pump (five), Chwech (six), Saith (seven), Wyth (eight), Naw (nine), and Deg (ten).  All numbers beyond this are generalised from them, for example eleven is un deg un (one ten one), twelve is un deg dau (one ten two), seventy three is saith deg tri (seven ten three), etc.  Thus, the smallest value of N in Welsh such that θN = 10 is 36:

\[\begin{array}{l}\begin{align}N &= 36\;\\\because10 &= e < {\theta _N} = \frac{{36}}{{\ln \left( {36} \right)}}\\\;where\;{\rm{ }}\frac{{36}}{{\ln \left( {36} \right)}} &= 10.045991...\end{align}\end{array}\]
Or in other words, once a child has learned to count to 36 in Welsh, they have learned the rules of the game sufficiently to overcome the cognitive need for memorisation, and thus to keep going.  

Bertrand Russell
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Bertrand Russell, the mathematician, logician and humanist, was born on 18 May 1872, in Trellech, Monmouthshire.  Russell lived for most of his later years at Plas Penrhyn in Penrhyndeudraeth, Merioneth, North Wales, with a view south to Cardigan Bay and north to the mountains of Eryri (Snowdownia).  He died at Plas Penrhyn on February 2, 1970 (read his obituary in the New York Times here), was cremated at Colwyn Bay and had his ashes scattered over the Welsh hills.

Russell won a scholarship to read mathematics at Trinity College, Cambridge University, and with Alfred North Whitehead wrote his monumental three-volume work, Principia Mathematica, between 1910 and 1913.  ('Logicomix: An Epic Search for Truth', a wondrous graphic novel 'inspired by the epic story of the quest for the Foundations of Mathematics', described Russell and Whitehead's Principia as 'a heroic intellectual adventure.')   In 1950, Russell was made a Nobel Laureate in Literature “in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought”.

Read a short biography of Russell here, and watch his 1959 BBC 'Face to Face' interview with John Freeman here, and/or read the transcript here.

Notes, References & Links:

[1] Ian Stewart (2009, p20) explains that 'Cossike practise; refers to algebra: the Reanaisssance Italian algebraists refered to the unknown, which we now call x, as cosa, Italian for 'thing'.  As in cosa nostra, 'this thing of ours', referring to the Mafia.  'Surde nombers' are things like square roots, and the word 'surd' still exists in English, though it is seldom used nowadays'.

[2] For a little more, see this from Mark Dominus and this from Samuel Arbesman in Wired.

[3] Zenzizenzizenzic is also has more Zs than any other word in the English language (According to the Oxford English Dictionary).

[4] Phew!

[5] Jones was also noted for his 1711 work, with assistance from Newton, Analysis per quantitatum series, fluxiones ac differentias, available to view digitally here, in which he introduced the dot notation for differentiation in calculus.

[6] π was proven to be irrational by Johann Heinrich Lambert in 1761.  See this from The World of Pi website.

[7] For more on William Jones and the origins of π, in addition to Rothman's excellent 2009 History Today article, see this as equally excellent Guardian article from Gareth Ffowc Roberts.

Arbesman, S.  (2014, 30 October).  'The Invention of the Equals Sign'.  Wired.  Retrieved February 2018 from

BBC Wales Arts.  (2008, 15 December).  Bertrand Russell.  Retrieved February 2018 from

BHA Website.  (n.d.).  Plas Penrhyn, Wales.  In Humanist Heritage.  Retrieved February 2018 from

Cavanaugh, R.  (2016, March 4).  'Robert Recorde, Inventor of the equals sign', Mental Floss.  Retrieved February 2018 from

Conway, J.H., and Hsu, T.  (2006).  Some Very Interesting Sequences.  Retrieved January 2018 from
Dominus, M.  (2006, April 7).  Robert Recorde Invents the Equals Sign [Blog post].  Retrieved February 2018 from

Doxiadis, A. and Papadimitriou, C.  (2009).  Logicomix: An Epic Search for Truth.  London: Bloomsbury.

Ffowc Roberts, G.  (2015).  'Pi Day 2015: meet the man who invented π', The Guardian.  Retrieved February 2018 from

Gourévitch, B.  (2013, 13 April).  Johann Heinrich Lambert (1728 - 1777) [Blog post].  Retrieved February 2018 from

Irvine, A.D.  (1996, December).  'Bertrand Arthur William Russell', School of Mathematics and Statistics, University of St Andrews.  Retrieved February 2018 from

Jones, W.  (1706).  Synopsis Palmariorum MatheseosOr, A New Introduction to the Mathematics: Containing the Principles of Arithmetic and Geometry.  London.  Retrieved February 2018 from

Johann Heinrich Lambert.  (n.d.).  In Wikipedia.  Retrieved February 2018 from

Jones, W.  (1711).  Analysis per quantitatum series, fluxiones ac differentias.  London.  Retrieved February 2018 from

Klarreich, E.  (2015).  'The Connoisseur of Number Sequences', Quanta Magazine.  Retrieved February 2018 from

Mazur, J.  (2014).  Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers.  Princeton: Princeton University Press.

O'Connor, J.J. and Robertson, E.F.  (2003, April).  'John Machin', School of Mathematics and Statistics, University of St Andrews.  Retrieved February 2018 from

O'Connor, J.J. and Robertson, E.F.  (1998, September).  'Leonhard Euler', School of Mathematics and Statistics, University of St Andrews.  Retrieved February 2018 from

O'Connor, J.J. and Robertson, E.F.  (2002, April).  'Robert Recorde', School of Mathematics and Statistics, University of St Andrews.  Retrieved February 2018 from

O'Connor, J.J. and Robertson, E.F.  (2005, February).  'William Jones', School of Mathematics and Statistics, University of St Andrews.  Retrieved February 2018 from

Plus and minus signs.  (n.d.).  In Wikipedia.  Retrieved February 2018 from

ReasonPublic (2012, March 13).  Bertrand Russell - Face to Face Interview (BBC, 1959) [Video File].  Retrieved February 2018 from

Recorde, R.  (1557).  The whetstone of witte, whiche is the seconde parte of Arithmetike: containyng thextraction of Rootes: The Cossike practise, with the rule of Equation: and the woorkes of Surde Nombers.  London.  Retrieved February 2018 from

Rothman, P.  (2009).  'William Jones and his Circle: The Man who Invented Pi', History Today, Vol. 59, Issue 7.  Retrieved February 2018 from

Rowlett, P.  (2015, March 13).  'Pi Approximation: Machin's Formula'.  The Aperiodical.  Retrieved February 2018 from

Springer, M.  (2012, 31 August).  Face to Face with Bertrand Russell: ‘Love is Wise, Hatred is Foolish’.  In Open Culture.  Retrieved February 2018 from

Stewart, I.  (2009).  Professor Stewart's Hoard of Mathematical Treasures.  London: Profile Books.

The Whetstone of Witte.  (n.d.).  In Wikipedia.  Retrieved February 2018 from

Whitman, A.  (1970, 3 February).  Bertrand Russell is Dead; British Philosopher, 97.  Retireved February 2018 from

Yang, C.  (2015).  'Negative Knowledge from Positive Evidence', Language, Vol. 91, No.4, pp938-953.  Retrieved from:

Yang, C.  (2005).  'On Productivity', in Linguistic Variation Yearbook 5, John Benjamins Publishing Company.  Retrieved from:

Zenzizenzizenzic.  (n.d.).  In Wikipedia.  Retrieved February 2018 from

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