Maths Ed Ideas: Maths History

Showing posts with label Maths History. Show all posts

Thursday, 24 November 2022

The Most Dangerous Problem

The 'Most Dangerous Problem' is a story about the Collatz conjecture, one of the most famous (or to some, infamous) unsolved problems in mathematics. It is a story that will help students appreciate the beautiful hidden depths of mathematics, the difference between demonstration and proof, and the excitement that surprising and difficult problems invoke in the mathematics community. Moreover, it will encourage students to frame their own learning as part of the tradition in mathematics of pure intellectual pursuit for its own sake. It can be used by teachers to support their teaching of substitution, algorithmic thinking and iteration.

Downloads:

Lesson slides (.ppt format)
Lesson slides (.pdf format)

The Most Dangerous Problem

Whereas we are unsure of the definitive, exact origin of the problem, it has become traditionally and eponymously associated with Lothar Collatz (1910-1990), a German mathematician known for his work in numerical analysis, who posed this deceptively simple looking problem in 1937. 'Such simplicity, however,' in the words of Alex Bellos, 'stands in striking contrast to the difficulty of proving the conjecture itself.'

Indeed, proving Collatz's conjecture is widely considered to be completely out of our current reach, and provoked the prolific mathematician Paul Erdős to state that 'mathematics is not yet ready for such problems.' It has been described as 'uncrackable,' called 'the simplest impossible problem,' and monikored as 'the most dangerous problem in mathematics' because ― as mathematician Jeffrey Lagarias, the World's leading expert on the Collatz Conjecture states ― 'people become obsessed with it, and it really is impossible.' Mathematician Alex Kontorovich goes further and says that if 'someone actually admits in public that they're working on it, then there's something wrong with them!' (The cartoon from the wonderful xckd.com below will give you the general gist.)

Kontorovich even describes how some people have suggested that during the cold war people thought 'it was something invented by the Soviets to slow down U.S. science... because everybody [would be] sitting there twiddling their thumbs on this trivial thing that you can tell school children.' The problem has also been called Kakutani's problem, after the mathematician Shizuo Kakutani, who relayed the 'joke' that 'this problem was part of a conspiracy to slow down mathematical research in the U.S.'

Many of those who have worked on the Collatz conjecture have warned others to 'stay away,' but despite ― or perhaps because ― of this, the conjecture continues to intrigue, beguile and seduce the World's greatest mathematicians. One of the greatest, Terrence Tao, describes the conjecture as 'one of the most “dangerous” conjectures known [because it is] notorious for absorbing massive amounts of time from both professional and amateur mathematicians.' Tao went on to question why, if the Collatz conjecture is just 'a mere mathematical curiosity, with no obvious real-world applications, ... should we try to solve it?' You can see his beautiful response on slide 21 here, and this article describes Tao's subsequent work on it, and his remarkable discovery that the Collatz conjecture is 'almost' true for 'almost' all numbers, which is the closest we've got!

The Collatz Conjecture

In short, Collatz's conjecture is that the following algorithm...

Choose any positive integer you want.
If the number you chose is even, divide it in half.
If it is odd, multiply it by 3 then add 1 (hence the \(3x + 1\)).
Repeat this process.

...will eventually produce the number 1, regardless of which positive integer is chosen initially.

Most mathematicians believe this to be true. Indeed, by 2008 we had tested Collatz's algorithm for all numbers up to \(19 \times {2^{58}} = 5,476,377,146,882,523,136\) and found that they always, eventually reached 1. By 2018, the upper limit of numbers we had tested and shown to always reach 1 was \({2^{100000}-1}\), which has over 30,000 digits and so too large to write here, and which needed to apply the \(3x + 1\) computation 481,603 times and the \( \div 2\) computation 863,323 times. As of 2024, David Bařina's computer systems checked around 220 billion numbers per second (i.e. 1.361 light mm per number) to push this upper limit to \({1.5\times 2^{70}}\), or 1,770,887,431,076,116,955,136. But none of this, of course, proves that reaching 1 is always the case. And as yet, no-one has a clue how to prove it.

Examples

If we start say with 19...

19 is odd, so we multiply it by three to get 57, then add one to get 58
58 is even, so we halve it to get 29
29 is odd, so we multiply it by three to get 87, then add one to get 88
88 is even, so we halve it to get 44
44 is even, so we halve it to get 22
22 is even, so we halve it to get 11
11 is odd, so we multiply it by three to get 33, then add one to get 34
34 is even, so we halve it to get 17
17 is odd, so we multiply it by three to get 51, then add one to get 52
52 is even, so we halve it to get 26
26 is even, so we halve it to get 13
13 is odd, so we multiply it by three to get 39, then add one to get 40
40 is even, so we halve it to get 20
20 is even, so we halve it to get 10
10 is even, so we halve it to get 5
5 is odd, so we multiply it by three to get 15, then add one to get 16
16 is even, so we halve it to get 8
8 is even, so we halve it to get 4
4 is even, so we halve it to get 2
2 is even, so we halve it to get 1

If we carried on with 1...

1 is odd, so we multiply it by three to get 3, then add one to get 4
4 is even, so we halve it to get 2
2 is even, so we halve it to get 1

We end up in the cycle 4, 2, 1, 4, 2, 1, ... So, when starting with 19, the number 1 is reached after 20 steps. Of course, some numbers take longer to 'stop' than others (see here for a list of how long each number from 1 to 10,000 takes). And this provokes other questions, that may or may not be fruitful (that's one of the joys about mathematics, we don't know sometimes where it will take us). For example, is there a longest length of numbers before getting to 1? What is the highest number generated in the sequence before reaching 1?

Consider the sequence of 111 numbers generated when starting at 27, for example, before 1 is reached. The highest number is 9232, before it falls inexorably down to 1 and then "bounces" into the small loop 4, 2, 1, .... Plotting these 'hailstone' numbers, as they've been called, because 'a hailstone eventually becomes so heavy that it falls to ground,' maps out the beautiful journey of the sequence:

Notes, References & Links:

The Absurd Equation

The 'Absurd Equation' is a story about the work of Greek mathematician Diophantus, the problems he posed and the developments that they provoked. It s a story that will help students appreciate the diversity inherent in the development of mathematics; let them frame their own learning not just a function of that diversity, but also as an intrinsic part to the continuing story of mathematics. It can be used by teachers to support their teaching of setting up and solving algebraic equations to solve problems.

Diophantus and his Arithmetica

Diophantus was a Greek mathematician from Alexandria who lived in the 3rd century AD. He is known to us through his work Arithmetica, a series of texts containing over 100 mathematical problems that survived the destruction of the Library of Alexandria in 641AD.

Arithmetica has had a resounding influence on the development of mathematics, particularly on number theory and the solution of algebraic equations. It was translated into Arabic in the 10th century by the Persian mathematician Abu al-Wafa' Buzjani (بوژگانی) and spread through Europe after the Italian mathematician Rafael Bombelli's 1572 book L'Algebra, and the publication of Arithmetica in full in 1621 by French mathematician Claude Gaspar Bachet de Méziriac.

Arithmetica contains the earliest known use of algebraic symbolism, which had a huge influence on Islamic mathematics and the subsequent development of algebra as we know and love it. Indeed, the word Algebra itself comes from the Arabic ‏الجبر‎ (al-jabr), which itself came from the title of an early 9th century book Al-Kitab al-muhtasar fi hisab al-gabr wa-l-muqabala (The Compendious Book on Calculation by Completion and Balancing) by the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (c780–850). The development of algebra as a discipline independent of arithmetic and geometry by al-Khwārizmī was taken further by the French mathematician François Viète (1540-1603), and — with contributions to notation from Italian, German, Dutch and Welsh mathematicians — led to the algebra we take for granted today.

The Four Square Theorem

Diophantus was undoubtedly ahead of his time. He knew, for example, that every number can be written as the sum of at most four squares:

\[\begin{array}{c}\begin{align}1 &= {1^2}\\2 &= {1^2} + {1^2}\\ &\vdots \\43 &= {5^2} + {4^2} + {1^2} + {1^2}\\ &\vdots \\999,999,999 &= {30985^2} + {6319^2} + {3^2} + {2^2}\\ &\vdots \end{align}\end{array}\]

It was remarkable that Diophantus knew this, given that it took another 1500 years before it was proven — by Joseph-Louis Lagrange in 1770. Even Leonhard Euler, widely considered to be the greatest mathematician in history, and certainly the most prolific, was unable to prove that it was true. (You can use this applet to try some numbers out for yourself (maybe try your birthdate), and if you'd like to see and explore Lagrange's proof of the Four Square Theorem, this Wikipedia article isn't a bad place to start.)

As is always the case in mathematics, Diophantus' work provoked more questions that paved the way for more profound developments. For example, with respect to the four square theorem: what numbers can be written as the sum of four squares in only one way, excluding \({0^2}\)? (In fact, there are only 138 such numbers.) How many ways can each number be written as the sum of at most four squares (see here)? Do these numbers have certain characteristics? What numbers can be written as the sum of five squares, excluding \({0^2}\)? (In fact, only 12 numbers can't!) What about cubes...? And so on.

Arithmetica and Fermat's Last Theorem

Artihmetica is also renowned in the history of mathematics because it led to Fermat's Last Theorem, and the awe-inspiring story of its eventual proof by the Abel Prize Winner Sir Andrew Wiles (see Simon Singh's wonderful book, short video, and documentary). Fermat had a copy of Arithmetica and when working through it he wrote a tantalising note in the margin next to one problem: 'I have discovered a truly remarkable proof [but] this margin is too small to contain it.' Fermat was asserting that he could prove that there are no positive integers for which \({x^n} + {y^n} = {z^n}\) when n is greater than 2, but he died before he gave his proof. Fermat's son published his father's note in a 1670 edition of Arithmetica, after his father's death, and the mathematical world was subsequently transfixed for centuries.

The Absurd Equation

Diophantus was also the first Greek mathematician to recognise fractions as numbers. It may seem incredible to us to say this today, but before Diophantus, fractions simply weren't 'allowed' to be solutions to problems. Only positive integers were 'allowed'. This was because it was felt that numbers had to have a geometric sense; they were representations of lengths, areas, and volumes.

But the incredible Diophantus, the genius who opened the mathematical door for so many who followed, would not 'allow' negative numbers to be solutions to problems. He described the concept of \(\lambda \varepsilon \iota \psi \iota \varsigma \), meaning 'deficiency,' and the rules that 'deficiency multiplied by deficiency yields availability' (the product of two negative numbers is positive), and 'deficiency multiplied by availability yields deficiency' (the product of a negative and a positive number is negative). But even though he was happy to manipulate negative numbers in order to get to a solution, as a solution themselves Diophantus considered negative numbers 'useless'. By way of illustration he described the equation \(4 = 4x + 20\) as 'absurd,' because it leads to the 'useless' negative solution:

\[\begin{array}{c}\begin{align}4 &= 4x + 20\\ - 16 &= 4x\\ - 4 &= x\end{align}\end{array}\]

As a result, all of the problems in Arithmetica had positive integer solutions, and we now call problems of these type, i.e., that lead to equations whose only solutions of interest are integer solutions, Diophantine Equations.

The Problem of Diophantus' Age

Although we lack information about Diophantus’ life, we can work out his age upon his death from an algebraic problem inscribed on his tombstone. There are several versions of the epitaph, including this from Sir Thomas Little Heath's 1910 study of Diophantus:

'His boyhood lasted \(\frac{1}{6} \) of his life; his beard grew after \(\frac{1}{12}\) more; after \(\frac{1}{7}\) more he married, and his son was born five years later; the son lived to half his father’s age, and the father died four years after his son.'

\[\begin{array}{c}\begin{align}x &= \frac{x}{6} + \frac{x}{{12}} + \frac{x}{7} + 5 + \frac{x}{2} + 4\\x &= \frac{{14x}}{{84}} + \frac{{7x}}{{84}} + \frac{{12x}}{{84}} + 5 + \frac{{42x}}{{84}} + 4\\x &= \frac{{75x}}{{84}} + 9\\\frac{{9x}}{{84}} &= 9\\9x &= 756\\x &= 84\end{align}\end{array}\]

Notes, References & Links:

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 π^π 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:

Look and Say sequence

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...

Emirps

13, 17, 31, 37, 71, 73, 79, 97, ...

McNugget Numbers

0, 6, 9, 12, 15, 18, 20, 21, 24, 26, 27, 29, ...

Not McNugget Numbers

1, 2, 3, 4, 5, 7, 8, 10, 11, 13, ... 43

Dihedral calculator primes

2, 5, 11, 101, 181, 1181, 1811, 18181, ...

Home Primes

1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, ...

Zig numbers

1, 1, 5, 61, 1385, 50521, 2702765, 199360981, ...

Zag numbers

1, 2, 16, 272, 7936, 353792, 22368256, 1903757312, ..

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?

T

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 θ_N = 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 https://www.wired.com/2014/10/invention-equals-sign/

BBC Wales Arts. (2008, 15 December). Bertrand Russell. Retrieved February 2018 from http://www.bbc.co.uk/wales/arts/sites/bertrand-russell/

BHA Website. (n.d.). Plas Penrhyn, Wales. In Humanist Heritage. Retrieved February 2018 from http://humanistheritage.org.uk/articles/plas-penrhyn-wales/

Wednesday, 13 September 2017

Problem... Logical Deomnstrator

#Logic #Fractions #Sets #SetNotation #Venn #Probability
#CharlesStanhope #LogicalDemonstrator #MathsHistory

Thursday, 24 November 2022

The Most Dangerous Problem

The Most Dangerous Problem

The Collatz Conjecture

Examples

Tuesday, 22 November 2022

The Absurd Equation

Diophantus and his Arithmetica

The Four Square Theorem

Arithmetica and Fermat's Last Theorem

The Absurd Equation

The Problem of Diophantus' Age

Wednesday, 28 February 2018

On A Mathsy St David's Day

Wednesday, 13 September 2017

Problem... Logical Deomnstrator

Bridget Riley

Tony Robbin

Zaha Hadid Architects