Tuesday, 1 May 2018

On Mountain Numbers

Cairn Toul, a beautifully wild and remote mountain in the Cairngorms massif of the eastern Scottish highlands, is notable not only for being the fourth highest mountain in the UK, but also for being the only mountain in the UK whose elevation in metres — 1291m above sea level — is also a 'mountain'.  Großglockner, by way of a further example, the highest mountain in Austria, is a mountain whose elevation in feet — 12461ft above sea level — is also a 'mountain'.

To elaborate: A number is a 'mountain' if its decimal digits start with 1, i.e. 'base camp', ascend continuously to a unique summit, i.e. to one largest digit, then descend continuously back to 1, i.e. back to base camp.  Such mountain numbers are recorded as sequence A134941 (itself a mountain) in the On-line Encyclopedia of Integer Sequences (OEIS), from which you can also find this table of all possible 21,846 mountains (including 1 itself).

Mountain primes, as the name suggests, are prime mountain numbers, prime numbers in other words whose decimal digits start with 1, ascend continuously to the summit of a single largest digit, then descend continuously back to 1.  (The area chart image above shows a mountain range of the first 17 such numbers.)  Mountain primes are recorded in the OEIS as sequence A134951 (yes, also a mountain prime), from which you can find this table of all 2620 such primes.

When climbing a mountain, it is typical to descend back to base camp, i.e. back to where you started your ascent, but it is not always the case.  Mountain numbers that ascend from one location (as an elevation) but descend to another — e.g. 3,598,432 — are considered to be generalised mountain numbers, recorded as sequence A134853 in the OEIS (there are 173,247 such mountains).

Suggested explorations, diversions & links:
  • Explain why there must be a finite number of mountain numbers.
  • Construct a mountain range diagram for mountains of your choice.
  • What is the Everest of mountain numbers?
    • How will you define 'Everest'; how will you define 'elevation'?  (In the image above, the 'height' of mountain 1291, for example, is greater than the 'height' of mountain 1571.)  
  • Explore the distribution of mountain number digits, i.e. how many mountains have 1 digit, 2 digits, 3 digits, etc. (OEIS sequence A135417).
  • If we regard the number of digits in a mountain number as the horizontal distance travelled when climbing such a mountain:
    • What mountain(s) has (have) the shallowest ascent?
    • What mountain has the steepest ascent and descent?
    • What about generalised mountains?
  • Many people find the most beautiful mountains to be those most pyramidal in shape, such as for example the Matterhorn in the Alps and Machapuchare in the Himalaya.
    • Define and find your most beautiful mountain numbers: 
      • Palindromic mountain numbers perhaps (OEIS sequence A173070),
      • Or palindromic mountain prime numbers (the largest is 123467898764321).
    • Find all possible — i.e. 45 — Giza numbers.
      • Giza numbers are so-called because they represent the pyramids of Giza in the sense that their first digits increase in consecutive order to a largest central digit, and their last digits decrease in the same consecutive order as they increased.  
      • The largest Giza number is 12,345,678,987,654,321 (OEIS sequence A134810).
  • Of two mountain numbers chosen at random, what is the probability that their numerical heights — i.e. the magnitude of the number — will be in alignment with the elevation of actual mountains?  
    • For example, mountain number 1291 is 'higher' than mountain number 1571, but as actual elevations, 1291m < 1591m, so these two mountains are not in 'alignment'. 
  • Find as many actual mountains whose heights are mountain numbers, or mountain primes.
    • What is the highest mountain in the world whose height is also a mountain (use metres and/or feet for the height)?

Friday, 27 April 2018

On Overheard Conversations About Maths #1

The floor.  Just before bedtime.
Three siblings are islands in a sea of Lego.
The eldest is trying to make a Lego Penrose triangle.


(Silently counting the studs on a red 2 × 4 Lego brick,
mouthing each number until, to no-one in particular...)


(He opens up his thumb and fingers successively
from the fist of one hand as he counts, this time out loud.) 
One, two, three, four, five...

(Now to the other hand.)

...six, seven, eight.

(Some time passes as he collects all the 2 × 4 bricks
he can find, piles them together, and pretends to count them all.)

A hundred!  A hundred is bigger than four.  Or five.  Or eight.


It’s not as big as a million though.


(With a nod to his Numberphile education...)

Or Graham.


Or ten.  Ten's bigger.


No, a million's bigger than ten.  A hundred is bigger than ten.  A million is really big.


Who's Graham?


Infinity is bigger than Graham.  




No, infinity.  In-fin-ity.


(Straining to speak; arms outstretched as wide as he can make them go.)  

Is Nifity big like this?


That’s not even a number.




Infinity.  In-fi-nity.  In-finity.  No, it’s not a number it just means the biggest thing.  People say it's a number but it's not because you could just say infinity plus one but that's still just infinity because...


Infinity plus a hundred then.



Infinity plus infinity.


Two times infinity.


Duh, that's the same.  And it's still just infinity.


(Quietly, in the ear of his 6 year old sibling.)

Is Graham friendly?


Infinity times infinity then.


Do you even know what that means?  OK then, infinity times infinity infinity times.  It's still just infinity!  It's still...

(Establishing the correct perspective
from which to see the Lego Penrose triangle
doesn't seem to matter quite as much right now.)

Isn't it...?  Erm...




Suggested explorations, diversions and links:

What is infinity?
  • Share this typically accessible introduction to the idea of infinity from MathsIsFun.com.  (Play, for example, with the provocation 'An infinite series of A's followed by a B will NEVER have a B'.)  The page gently introduces infinity as 'the idea that something has no end', that doesn't grow, and subsequently moves into arithmetic properties and the exciting idea that there are different sizes of infinity.  

    Why does \(\infty  + 1 = \infty  + 100 = \infty  + \infty  = \infty  \times \infty  = \infty \)?

    We can (sometimes) add an infinity of numbers.

    Play around with infinite sums, starting perhaps with this article by Luciano Rila from Plus Magazine.   And consider, for example:
    \[\underbrace {1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...}_\infty \]Or, to put it another way:
    \[\begin{array}{c}\begin{align}\sum\limits_{k = 0}^\infty  {\left( {\frac{1}{{{2^k}}}} \right)}  &= \frac{1}{{{2^0}}} + \frac{1}{{{2^1}}} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ...\\ &= \frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...\end{align}\end{array}\]
    If we add up the first few terms (i.e. find the first few partial sums):
    \[\begin{array}{c}\begin{align}\frac{1}{1} + \frac{1}{2} &= 1\frac{1}{2}\\\frac{1}{1} + \frac{1}{2} + \frac{1}{4} &= 1\frac{3}{4}\\\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} &= 1\frac{7}{8}\\\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} &= 1\frac{{15}}{{16}}\\etc.\end{align}\end{array}\]
    We can see that each iteration produces a number that gets closer and closer to 2.  We produce another infinite sequence in others words with a limit of 2, thus:
    \[\left\{ {1,\;1\frac{1}{2},\;1\frac{3}{4},\;1\frac{7}{8},\;1\frac{5}{{16}},\;...} \right\}\]
    And we can therefore show that the sum of infinite series is 2.  Say:
    \[\sum\limits_{k = 0}^\infty  {\left( {\frac{1}{{{2^k}}}} \right)}  = s = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...\]
    \[\begin{array}{c}\begin{align}\ 2s &= 2\left( 1 \right) + 2\left( {\frac{1}{2}} \right) + 2\left( {\frac{1}{4}} \right) + 2\left( {\frac{1}{8}} \right) + ...\\2s &= 2 + 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...\end{align}\end{array}\]
    And so:
    \[\begin{array}{c}\begin{align}\ 2s - s &= \left( {2 + 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...} \right) - \left( {1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...} \right)\\s &= 2\end{align}\end{array}\]
    \[\sum\limits_{k = 0}^\infty  {\left( {\frac{1}{{{2^k}}}} \right)}  = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... = 2\]

    Friday, 9 March 2018

    On π Day

    March 14th every year is \(\pi \) day[1], because it is the 3rd month of the year, and 14th day of that month, and in the mm/dd format this is written as 3/14, or 3.14, which is \(\pi \) correct to two decimal places (or three significant figures) — meaning, of course, that the day should be more accurately referred to as Pi-To-Two-Decimal-Places day, or Pi-To-Three-Significant-Figures day.  It is, nevertheless, a day for schools to unashamedly revel in a pure celebration of \(\pi \) and, by implication and association, mathematics.

    I offer in this post therefore a few little somethings that teachers may wish to share with their students, a few little somethings moreover that teachers may want to delve deeper into and thus take further with their students, marking \(\pi \) day in the spirit of celebration that it promotes.  In addition, and in complement, schools and teachers may wish to carry out some Random Acts of Maths (download them here) over the course of the day, and/or hold a Favourite Number Election (find suggestions for use and download a ballot paper here).  You might also find some inspiration from the #PiDay2018 Twitter highlights here.

    Click to jump to:

    Archimedes' Constant
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    Archimedes (c. 287-212 BC) was the first to theoretically calculate \(\pi \) — i.e. the ratio of a circle's circumference to its diameter (he never used the symbol) — rather than estimate it.  In proposition three of his ‘Measurement of the Circle’, Archimedes used a geometrical approach in which he repeatedly enclosed a circle by circumscribing it (i.e. constructing a regular polygon outside the circle) and by inscribing it (i.e. constructing a regular polygon inside the circle).

    Starting with a regular hexagon, Archimedes progressively doubled the number of sides of the polygon in order to approximate a circle with increasing accuracy (from 6 to 96 sides).  He was thus able to prove that:

    \[\frac{{223}}{{71}} < \pi  < \frac{{22}}{7}\]
    Or, in mixed numbers:
    \[3\frac{{10}}{{71}} < \pi  < 3\frac{1}{7}\]
    Or, to six decimal places:
    \[3.140845 < \pi  < 3.\dot 14285\dot 7\]
    This 'method of exhaustion' continued to be used for many years, until the invention of calculus and the use of arctangents allowed us to find \(\pi \) to ever increasing numbers of decimal places (\(\pi \) was found to 100 decimal places in such a way by John Machin, as published in William Jones' 1706 book Synopsis palmariorum matheseos).  The Archimedean 'method of exhaustion' was used most notably by German mathematician Ludoph van Ceulen in the 16th Century, who reached a regular polygon with 4,611,686,018,427,387,904 sides to arrive at a value of \(\pi \) correct to 35 digits, calculated, of course, by hand.

    Following Archimedes' approach with younger secondary students can influence their perception of mathematics, of what mathematics is, profoundly.  For more on how the Archimedes' upper and lower bounds for \(\pi \) were found, see this video (17min).  With this interactive demonstration from the Wolfram Demonstrations Project, students can explore how the upper and lower bounds for \(\pi \) change as the number of sides to the polygons change.

    The origin of the symbol for \(\pi \) as \(C/d\)
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    \(\pi \) was first used to denote the ratio of a circle's circumference to its diameter by Welsh mathematician William Jones in his 1706 book Synopsis palmariorum matheseos.  In the same book, as mentioned above, \(\pi \) was shown to 100 decimal places, as 'Computed', in Jones' words, 'by the Accurate and Ready Pen of the Truly Ingenious Mr. John Machin', using what has since become known as 'Machin's formula'.  (See this this post for a little more on Jones' notation.)

    You can peruse the book electronically here, finding π (as C/d) for the first recorded time in history on p243, as extracted in the first image below, then more explicitly on p263, as extracted in the second image below.

    \(\pi \) formulae
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    Many awe-inspiring formulae have been found for \(\pi \), in exact or approximate, iterative forms.  Below is a small selection of some classics that may be interesting to explore with students, possibly comparing the relative accuracy of respective formulae and their relative rates of convergence.

    Viète's formula.  François Viète published in 1593 what is commonly accepted to be the first instance of an infinite product known in mathematics.  It is derived by comparing the areas of regular polygons with 2n and 2n + 1 sides inscribed in a circle, and converges to \(\pi \) fairly rapidly.  (Play with Viète's formula with this interactive Wolfram demonstration.)

    \[\pi  = 2 \times \frac{2}{{\sqrt 2 }} \times \frac{2}{{\sqrt {2 + \sqrt 2 } }} \times \frac{2}{{\sqrt {2 + \sqrt {2 + \sqrt 2 } } }} \times \frac{2}{{\sqrt {2 + \sqrt {2 + \sqrt {2 + \sqrt 2 } } } }} \times ...\]

    Wallis' product.  John Wallis published this formula in his 1656 Arithmetica Infinitorum.  Recently, Wallis' product was 'discovered,' quite incredibly, 'hidden in [the] hydrogen atom', as shared with the world through this paper — 'Quantum mechanical derivation of the Wallis formula for pi' — from the Journal of mathematical Physics, revealing a hitherto unknown and beautiful connection between mathematics and Physics.  (Visualise Wallis' sieve approximation for \(\pi \) with this interactive Wolfram demonstration and watch this typically wonderful video from Grant Sanderson providing a 'new proof of the Wallis formula'.)

    \[\frac{\pi }{2} = \prod\limits_{n = 1}^\infty  {\left( {\frac{{2n}}{{2n - 1}} \times \frac{{2n}}{{2n + 1}}} \right)} \]Giving
    \[\frac{\pi }{2} = \frac{2}{1} \times \frac{2}{3} \times \frac{4}{3} \times \frac{4}{5} \times \frac{6}{5} \times \frac{6}{7} \times ...\]

    Newton's approximation.  Sir Isaac Newton derived this formula in 1666, using it to calculate pi to 16 places 'using 22 terms of the series' (in Pickover, 2008).  In recognition of his obsessive calculations in the face of pi's irrationality, Newton wrote, 'I am ashamed to tell you to how many figures I carried these computations, having no other business at the time'.  (Play with Newton's approximation using this interactive Wolfram Demonstration.)
    \[\pi  \approx \frac{3}{4}\sqrt 3  + 24\int\limits_0^{\frac{1}{4}} {\sqrt {x - {x^2}} } dx\]giving
    \[\pi  \approx \frac{3}{4}\sqrt 3  + 24\left( {\frac{1}{{12}} - \frac{1}{{5 \times {2^5}}} - \frac{1}{{28 \times {2^7}}} - \frac{1}{{72 \times {2^9}}} - ...} \right)\]

    Machin's formula.  Published in William Jones' 1706 book Synopsis palmariorum matheseos, the same book where \(\pi \) was first used to denote the ratio of a circle's circumference to its diameter, Machin's rapidly converging series allowed \(\pi \) to be computed to 100 decimal places.  (To see how Machin's formula converges to \(\pi \), play with this interactive Wolfram Demonstration.)

    \[\frac{\pi }{4} = 4{\tan ^{ - 1}}\left( {\frac{1}{5}} \right) - {\tan ^{ - 1}}\left( {\frac{1}{{239}}} \right)\]Or,
    \[\frac{\pi }{4} = 4{\cot ^{ - 1}}\left( 5 \right) - {\cot ^{ - 1}}\left( {239} \right)\]

    The Basel Problem.  A famous problem (as outlined here by Marianne Freiberger in Plus magazine) named after the home town of the renowned Bernoullis who worked on it, and famously solved by the great Leonarhd Euler in 1735, giving the surprising \(\pi \)-related result (see this accessible outline by Chris Sangwin).  Watch this superb video, from Grant Sanderson with Ben Hambrecht, giving a new take and beautiful proof of the problem, using light!
    \[\frac{{{\pi ^2}}}{6} = \frac{1}{{{1^2}}} + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \frac{1}{{{4^2}}} + ...\]

    \(\pi \) birthdays
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    At some point on \(\pi \) day:
    • A 3 year old born on 21 January 2015 will be \(\pi \) years old.  
    • A 9 year old born on 30 April 2008 will be \({\pi ^2}\) — or \(\pi \)\(\pi \) — years old.  
    • A 12 year old born on 19 August 2005 will be 4\(\pi \) — or \(\left\lceil \pi  \right\rceil \pi \) — years old.  
    • A 15 year old born on 28 June 2002 will be 5\(\pi \) — or \(\left\lceil \pi  \right\rceil \pi \) + \(\pi \) — years old.  
    • A 36 year old born on 26 September 1981 will be \({\pi ^\pi }\) years old.
      • Including Serena Williams, American tennis player with the most Grand Slam wins of any player in the Open Era, and Christina Milian, American singer, songwriter and actress.
    If you teach in a primary, make sure to find students in your school born on 30 April 2008 (Y5) and make a fuss of their \(\pi \)\(\pi \) birthday.  They may not know what \(\pi \) is yet, but this is a lovely way to sew a seed of interest and give them something to take home and surprise their parents with.  If you teach in a secondary, make sure to find students in your school born on 19 August 2005 (Y8) and 28 June 2002 (Y11) and make a fuss of their 4\(\pi \) and 5\(\pi \) — or their \(\left\lceil \pi  \right\rceil \pi \) and \(\left\lceil \pi  \right\rceil \pi \) + \(\pi \) — birthdays respectively.  And of course, if you have any 36 year old colleagues born on 26 September 1981, make sure you make a fuss of their \({\pi ^\pi }\) birthdays!  (See here for my post with a calendar for the whole of 2018 outlining pi-related birthdays.)

    \(\pi \)-day birthdays
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    The fact that Albert Einstein was born on \(\pi \)-day 1879 in Ulm, Germany, is well-known, and rightly celebrated.  Less known, however, and less celebrated, is the fact that the renowned Polish mathematician, Wacław Sierpiński, was born on \(\pi \)-day just \(\left\lceil \pi  \right\rceil \pi \) years later in 1882 in Warszawa, Poland.

    Sierpiński made groundbreaking contributions to set theory, number theory, functions and topology, and has a number of well-known fractal objects named after him that are interesting to explore with students when exploring iterative procedures and nth term generalisations, as well as, of course, when playing around with the idea of infinity.  Three of these Sierpiński fractal objects are the Sierpiński carpet, the Sierpiński triangle (also referred to as the Sierpiński sieve or gasket), and the Sierpiński curve (including the Sierpiński arrowhead curve).

    The first seven iterations of the Sierpiński carpet and Sierpiński triangle are shown in the images below.  An interesting problem to consider with students is to find the area of the carpet and triangle after 1, 2, 3, ...n iterations.

    Taking the carpet above as an example, assuming the original square has a side length of 1 and thus an area of 1, after the 1st iteration the shaded area is \({\raise0.5ex\hbox{$\scriptstyle 8$}
    \lower0.25ex\hbox{$\scriptstyle 9$}}\) of 1.  Visualising this 1st iteration as being made up of 8 smaller shaded squares, i.e. each with a side length of \({\raise0.5ex\hbox{$\scriptstyle 1$}
    \lower0.25ex\hbox{$\scriptstyle 3$}}\), we can see that the area of each of these smaller squares after the subsequent 2nd iteration is \({\raise0.5ex\hbox{$\scriptstyle 8$}
    \lower0.25ex\hbox{$\scriptstyle 9$}}\) of \({\raise0.5ex\hbox{$\scriptstyle 1$}
    \lower0.25ex\hbox{$\scriptstyle 9$}}\), but there are 8 of these squares, thus meaning that after the second iteration the area of the original starting square that is shaded is \({\raise0.5ex\hbox{$\scriptstyle 8$}
    \lower0.25ex\hbox{$\scriptstyle 9$}} \times {\raise0.5ex\hbox{$\scriptstyle 8$}
    \lower0.25ex\hbox{$\scriptstyle 9$}} = {\left( {{\raise0.5ex\hbox{$\scriptstyle 8$}
    \lower0.25ex\hbox{$\scriptstyle 9$}}} \right)^2}\).  And so on, such that after iterations, the area of the original square \({A_n}\) is

    \[{A_n} = {\left( {\frac{8}{9}} \right)^n}\]
    Students could ponder what the area of the carpet is exactly, when \(n = \infty \), i.e. after an infinite number of iterations (this problem was solved in 2012 by Yaroslav Sergeyev, as revealed in this paper).

    The greatest \(\pi \)-day birthday in history
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    In 1592, it is possible (probable perhaps) that someone was \(\pi \) years old, at \(\pi \) o'clock precisely, on what we now know as \(\pi \) day.  Or to put it another way, it is likely that at least one person was 3.141592653... years old on 3/14/1592 (mm/dd form) at 3:14am and 15.92653... seconds (or at 3.141592653... am).  So, when was this person born?

    (Click here for the problem download, a solution think through, and the name of someone who would have been \(\pi \) years old at some point on \(\pi \) day in 1592.)

    \(\pi \) o'clock
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    You might like to try out this popular problem from my blog with students.  The clock face implies for students what is meant by \(\pi \) o’clock in the problem, but perhaps it could mean something else, e.g. 3.141592... may suggest 3 hours and 0.141519... of an hour, rather than the 3:14 and 15.92 seconds the clock implies.  How accurate could students calculate the angle, i.e. will they use \(\pi \) to 4, 6, 8, 10... decimal places?

    \(\pi \) is the 73rd day of the year
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    73 is a permutable (or anagrammatic) prime, a prime that can have its digits' rearranged (in base 10) in any permutation and still be a prime number. 73 is also an emirp, a number whose reverse, 37, is also prime — a property also evident in terms of the respective ordinal positions of 73 and its emirp partner, 73 being the 21st prime while 37 is the 12th prime.  73 is a Sophie Germain prime and is palindromic in binary 1001001 (interestingly, all Fermat primes and Mersenne primes are subsets of the binary palindromic primes).  73 is also an octal palindrome 111 and the only octal prime repunit.  73 is, moreover, and for some of the reasons given here, Sheldon Cooper's favourite integer in The Big Bang Theory — as was first referenced in the show’s 73rd episode.  (Jim Parsons, incidentally, the actor who plays Sheldon Cooper, was born in 1973.)  73 is also the number that marks when English-speaking children have learned the rules of counting sufficiently to overcome the cognitive need for memorisation, thus implying that once you can count to 73 in English, you can count forever (see this post for more detail about 73).

    \(\pi \) primes
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    A pi-prime (sequence A005042 in the OEIS) is a prime number made up of the initial digits of the decimal expansion of \(\pi \).  To date we have found the first four pi-primes (sequence A060421), with another four found to be probable.  The first four pi-primes are:

    The fifth, a probable prime, 16,208 digits long, took four and a half months to compute:


    Play the \(\pi \)-day Lottery
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    This is (part of) the UK Lotto lottery ticket I have bought for the 2018 \(\pi \)-day draw:

    As you can see, the first set of numbers are the first twelve digits of \(\pi \) {31, 41, 59, 26, 53, 58}, and, moreover, the maximum number of digits from the start of \(\pi \) that can be used in the UK Lotto (six numbers from 1 to 59 are drawn).  The second set of numbers are the numbers 1 to 6 {1, 2, 3, 4, 5, 6}.  The third set of numbers are a randomly generated set {18, 58, 41, 11, 10, 31}.

    Suggested explorations/diversions with/for students:
    • Find the probability of each separate set of numbers winning.
    • Assuming one of the sets of numbers drawn in full, and thus wins the jackpot, which set would you expect to win the most money from?  
      • Whilst betting on the numbers 1, 2, 3, 4, 5 and 6 does not reduce your probability of winning, it does reduce the amount of money you are likely to win.  This is because, perhaps surprisingly, the numbers 1, 2, 3, 4, 5 and 6 are selected by around 10,000 people each week, thus meaning that the winning jackpot would be shared by 10,000.
      • Similarly, maybe a few mathsy people will buy a ticket with the first set of numbers on \(\pi \)-day, therefore reducing the amount of money you would win if drawn because of the shared jackpot.
    • Find the combinations of all 'wins' you could possibly make with all three sets of numbers.
      • You 'win' —  relatively of course —  in the Lotto if you match 6 numbers (Jackpot), or if you match 5 and the 'Bonus Ball', or if you match 5, 4, 3, or 2.  (Visit the Lotto website for more detail.
    • What is the least amount of initial digits from \(\pi \) that can be used in the UK Lotto?
    • Find all the combinations of initial \(\pi \) digits that could buy a lottery ticket in the UK Lotto.
      • Find the probability that a string of the initial digits of \(\pi \) will win the Lottery.

    \(\pi \) as the Prime Counting Function
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    In 1909 Edmund Landau used the symbol \(\pi \) to describe Prime Counting Function \(\pi \left( x \right)\), which gives the number of primes not exceeding a given number \(x\).  \(\pi \) used in this sense has nothing to do with \(\pi \) used in Jones' sense, i.e. \(\pi \) as Archimedes' constant, \(\pi \) as the ratio of a circle's circumference to its diameter.  By way of illustration, \(\pi \left( 7 \right) = 4\) because the number of primes less than or equal to the number 7 is 4 (i.e. 2, 3, 5 and 7).

    Suggested explorations/diversions with/for students:
    • What is the largest n you can find the value of \(\pi \left( n \right)\) for
      • You may want to use the 'Sieve of Eratosthenes'
      • The largest n for which a value of \(\pi \left( n \right)\) has been computed (by Staple 2015, as part of his Masters' degree) is \(\pi \left( {{{10}^{25}}} \right)\) = 1,699,246,750,872,437,141,327,603, which took 40,000 computing hours to find.
    • Find the first five Ramanujan primes.
      • A Ramanujan prime is the smallest number \({R_n}\) such that \(\pi \left( x \right) - \pi \left( {x/2} \right) \ge n\) for all \(x \ge {R_n}\).  

    Fake proofs for \(\pi \) 
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    These two now well-known fake 'proofs' are always interesting to share with students, in a 'surely this can't be right' sense.  Firstly, this 'proof' that \(\pi \) = 4 always engages students visually, and is a nice conceit when thinking about gradients and derivatives (play with this Wolfram Demonstration for \(\pi \) = 4).  Secondly, this following proof — which, on the face of it, seems fine to many — is a nice conceit when working with students on algebraic manipulation. 

    \[\begin{array}{c}\begin{align}x &= \frac{{\pi  + 3}}{2}\\2x &= \pi  + 3\\2x\left( {\pi  - 3} \right) &= \left( {\pi  + 3} \right)\left( {\pi  - 3} \right)\\2\pi x - 6x &= {\pi ^2} - 3\pi  + 3\pi  - 9\\2\pi x - 6x &= {\pi ^2} - 9\\9 - 6x &= {\pi ^2} - 2\pi x\\9 - 6x + {x^2} &= {\pi ^2} - 2\pi x + {x^2}\\{\left( {3 - x} \right)^2} &= {\left( {\pi  - x} \right)^2}\\3 - x &= \pi  - x\\3 &= \pi \end{align}\end{array}\]
    Suggested explorations/diversions with/for students:
    • What is wrong with the 'proof' above?
    • What happens when you 'fix' it?
      • In this case, the problem comes from moving from \({\left( {3 - x} \right)^2} = {\left( {\pi  - x} \right)^2}\) to \(3 - x = \pi  - x\).  The 'proof' only provides the positive square root, which produces  the erroneous result.  
    • Make up you own fake \(\pi \) proof.

      Memorising \(\pi \)
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      Daniel Tammet, essayist and novelist, is described as an autistic savant.  Born with high-functioning autism, Daniel famously memorised and recited \(\pi \) to 22,514 places on \(\pi \)-day in 2004, taking him just over five hours to do so.  This clip, lasting just over five minutes, is an excerpt from a longer documentary on Daniel, 'The Boy With The Incredible Brain'.

      For more of Daniel:

      Suresh Kumar Sharma of India recited 70,030 digits of pi on 21 October 2015, taking 17 hours and 14 minutes to do so.  See this article about the 'secret' to memorizing pi to such huge amounts of digits.  And see this article by @alexbellos in The Guardian about Akira Haraguch, who holds the unofficial World Record having recited pi to 100,000 digits in October 2006, over 16 hours.  It is fun to hold a class or school competition, linked to the 'self-referential stories' below perhaps, to see which student can recite \(\pi \) to the greatest amount of decimal places.

      Self-referential stories for \(\pi \)
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      Arguably the best known \(\pi \) mnemonic, constructed by British astronomer Sir James Jeans, is: "How I want a drink, alcoholic of course, after the heavy chapters involving quantum mechanics", the number of letters in each word corresponding to the respective digit in the decimal expansion of  \(\pi \).  This from Michael Keith in 1986 gives \(\pi \) to 356 places.

      The Music of \(\pi \)
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      This song, 'Pi', by Kate Bush from her 2005 album 'Aerial', is an ode to a 'sweet and gentle and sensitive man, with an obsessive nature and deep fascination for numbers, and a complete infatuation for the calculation of pi'.  Throughout the song Kate sings this number:


      You will note, however, that whilst the 55th digit of π is 0, Kate sings 3 and then 1, before getting back on track and singing the next 24 digits correctly.  She then, however, completely misses out the next 22 digits of π before singing the next 37 digits correctly.


      I wonder if she sings the song live?

      Other links to explore 
      (film, audio, websites, applets, articles, papers)
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        Notes, References & Links:

        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.

        Click to jump to:

        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.

        The OEIS
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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: