Wednesday, 30 November 2022

10 Mathsy Picture Books

This post is a list of mathsy picture books intended to be shared and enjoyed together, at home, or indeed school. All the books listed have had the seal of approval from my own children, who have devoured them, and demanded to take many into school for show and tell! They are books that are not only beautiful in terms of their content and presentation, not only captivating in terms of both the writing and illustration, but also in terms of the discussions they invoke with children, both mathematical, civic and humanistic. They are books that make no attempt to teach mathematics, but by celebrating mathematics and mathematicians, and, indeed, the mathematicians in all of us, frame mathematics as the quiet protagonist. They are books that do not propagate the exclusivist view of the exceptional that for many is associated with mathematics, but rather tell stories that will embolden young minds and invoke a sense of possibility and awe. Whilst they are all founded in and framed by mathematics, they reveal deeper cultural stories, gently exposing children to themes of mindset, resilience, equality and inclusion. Publishers' recommended ages are provided, but I would not hesitate to share these stories with the younger, or, indeed, older. 




Quick Links

  • The Boy Who Dreamed of Infinity
  • Nothing Stopped Sophie
  • The Boy Who Loved Math
  • Maryam’s Magic
  • Ada's Ideas
  • Molly and the Mathematical Mysteries
  • 365 Penguins
  • A Hundred Billion Trillion Stars
  • Infinity and Me
  • Hello Numbers! What Can You Do?



  • The Boy Who Dreamed of Infinity: A Tale of the Genius Ramanujan

    Written by Amy Alznauer  |  Illustrated by Daniel Miyares  |  Recommended for ages 4-8  |  For more information see this review on the Goodreads website   See this page from teachingbooks.net for author interviews and lots of great activities to explore the book further with children  |  See also the book's page on the author's website with more great activities and videos to explore the book further  |  And see this more in depth review on Math Book Magic

    This is a beautifully written and exquisitely illustrated book recounting the captivating and inspiring story of Indian mathematician Srinivasa Ramanujan (1887-1920). Ramanujan, with virtually no formal training in mathematics, independently solved mathematical problems that were once considered unsolvable. He is considered an 'intuitive mathematical genius,' whose 'discoveries have influenced several areas of mathematics,' most notably in number theory and infinite series.

    The story describes Ramanujan's search to unlock the 'secrets of numbers,' alongside his search to find someone who understood him and his work, which came to an end after he wrote to English mathematician Godfrey Harold Hardy in 1913, aged 26. Hardy recognised Ramanujan's genius and arranged as a result for Ramanujan to travel to ― and attend ― the University of Cambridge as a research student. The author gently and lyrically tells the story of Ramanujan's early life as a boy exploring his mathematical ideas with chalk on the temple floors in his home city of Erode, south east India, through his struggles with school and up to his contact with Hardy, and ends with his journey to England. Complemented by the dreamy, watercolour illustrations of Daniel Miyares, Ramanujan's genius, struggles and resilience flow gorgeously from each page, encouraging discussions as human as they are mathematical.

    After the story the author recounts more of Ramanujan's life and achievements, as well as describing her mathematician father George Andrews' discovery of Ramanujan’s lost notebook in Cambridge University's Wren Library, and how it changed their lives forever. You can also watch the author talk about this in this video released by publisher Candlewick Press.  

    Grown Ups might be interested in Robert Kanigel's fascinating biography of Ramanujan: 'The Man Who Knew Infinity: Life of the Genius Ramanujan,' and the 2015 film adaptation 'The Man Who Knew Infinity,' starring Dev Patel and Jeremy Irons.

    Click here to return to Quick Links at the top of the page




    Nothing Stopped Sophie: The Story of Unshakable Mathematician Sophie Germain

    Written by Cheryl Bardoe  |  Illustrated by Barbara McClintock  |  Recommended for ages 4-8  |  See this review on the Goodreads website   See this page from teachingbooks.net for author interviews and lots of great activities to explore the book further with children  |  See this short but typically insightful account of Germain and her legacy from the renowned science and mathematics author Simon Singh

    This is a wonderfully written and illustrated book recounting the remarkable story of the trailblazing eighteenth century French mathematician Marie-Sophie Germain (1776-1831). Germain was, in short, an extraordinary mathematician whose story is, in more ways than one, captivating. Forbidden to attend University, Germain famously taught herself, reading Newton, Euler and others. She corresponded with preeminent mathematicians of the time, most notably Adrien-Marie Legendre, Joseph-Louis Lagrange and Carl Friedrich Gauss. But to circumnavigate the prejudices of 19th-century French society that threatened to impede her way in mathematics, Germain assumed a man's identity, writing under the nom de plume of a Monsieur LeBlanc. In the words of my incredulous daughter, 'What?! She had to pretend to be a boy to do maths that boys couldn't do?!' She was eventually revealed as a mathématicienne to Gauss, who described Germain as 'a superior genius'.

    From having her candles and clothes confiscated by her parents to discourage her from studying such an 'unfeminine subject,' to having to pretend to be a man to have her work recognised, the difficulties Germain faced and how she overcame them are affectionately recounted by author Cheryl Bardoe, and guided along the way by Barbara McClintock's joyful paintings and collages. 

    Germain was inspired by Ernst Chladni's beautiful, awe-inspiring 'figures' to formulate a mathematical theory of elastic surfaces that described Chladni's experimental observations. (This wonderful little video by Steve Mould shows these amazing patterns.) Despite many setbacks and after years of work, Germain eventually succeeded and published her Récherches sur la théorie des surfaces élastiques in 1821, and in doing so winning the prix extraordinaire from the esteemed Académie des sciences

    Grown Ups might be interested in Dora Musielak's fictional chronicle of Germain's 'coming of age of a teenager learning mathematics on her own, growing up during the most turbulent years of the French Revolution': 'Sophie's Diary: A Mathematical Novel.'

    Click here to return to Quick Links at the top of the page 




    The Boy Who Loved Math: The Improbable Life of Paul Erdős

    Written by Deborah Heiligman  |  Illustrated by LeUyen Pham  |  Recommended for ages 5-8  |  For more information see this review on the Goodreads website   See this page from teachingbooks.net for author interviews and lots of great activities to explore the book further with children  |  See also the book's page on the author's website with more great activities and videos to explore the book further  |  And see this more in depth review and exploration from The Marginalian

    This is a beautifully written and playfully illustrated book recounting the story of the eccentric Hungarian genius and one of the most prolific mathematicians of the twentieth century Paul Erdős (1913-1996). Author Deborah Heiligman tenderly recounts Erdős' remarkable life, from the prodigy's struggles at school, through the love and care he received from his mother who nurtured Erdős burgeoning love of mathematics, to the social awkwardness he exhibited as an adult, his insatiable appetite to 'do math,' and his famous collaborations with mathematicians around the world. The much loved 'Magician from Budapest,' who became affectionately known as 'Uncle Paul,' is brought to life with LeUyen Pham's wonderful illustrations warmly incorporating some of Erdős' mathematical obsessions.

    Grown Ups might be interested in Paul Hoffman's superb biography of Erdős: 'The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth'. If you want to delve deeper into some of the mathematics that Erdős was inspired by and worked on, you might be interested in 'Proofs from THE BOOK,' a book of mathematical proofs by Martin Aigner and Günter Ziegler, which is dedicated to Erdős, who often referred to 'The Book' in which God keeps the most elegant proof of each mathematical theorem.

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    Maryam’s Magic: The Story of Mathematician Maryam Mirzakhani

    Written by Megan Reid  |  Illustrated by Aaliya Jaleel  |  Recommended for ages 8-10  |  For more information see this review on the Goodreads website   See this page from teachingbooks.net for author interviews and lots of great activities to explore the book further with children  |  And see this more in depth review and exploration from Rhapsody in Books

    This brilliant and beautifully illustrated book recounts the story Iranian mathematician Maryam Mirzakhani (1977-2017), the first woman to win the Fields Medal (i2014), the most prestigious award in mathematics, 'for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces'. (The Fields Medal is awarded only once every four years, and you can view this short video from the International Mathematical Union announcing and describing Mirzakhani's award.)

    The story is one that celebrates creativity and curiosity. Author Megan Reid affectionately describes Mirzakhani's love of reading as a little girl, her love of stories and of school, but her dislike of maths! With Aaliya Jaleel's crisp, joyous and fascinating illustrations, the book shows how Mirzakhani's love of mathematics burgeoned and flourished after her teacher introduced her to geometry when she was 12 ― and how she learned that shapes have their own magical and often untold stories, stories which, moreover, she could tell with mathematics. 

    On announcing her tragic death in 2014, Stanford University recounted how Mirzakhani famously described herself as a 'slow' mathematician, and how she was 'resolute and fearless in the face of problems others would not, or could not, tackle.' The University describes how Mirzakhani worked on problems by 'doodl[ing] on large sheets of white paper, scribbling formulas on the periphery of her drawings,' which led her young daughter to describe 'her mother at work as “painting.”' Megan Reid's beautiful writing and Aaliya Jaleel's joyous illustrations capture this sense of mathematics as painting in a way that young readers can, quite wonderfully, connect with.

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    Ada's Ideas: The Story of Ada Lovelace, the World's First Computer Programmer

    Written and Illustrated by Fiona Robinson  |  Recommended for ages 6-9  |  See this review on the Goodreads website   See this page from teachingbooks.net for author interviews and lots of great activities to explore the book further with children  |  See this interview with the author about the book and its design  |  And see this more in depth review and exploration from Rhapsody in Books

    This wonderfully written and lovingly illustrated book recounts the story mathematician Ada Lovelace (1815-1852). Ada, Countess of Lovelace, was the daughter of romantic poet Lord Byron and Anne Isabella, the11th Baroness Wentworth and Baroness Byron. Lovelace is regarded as the world’s first computer programmer, having had her work and achievements only relatively recently recognised, and as a result has become an icon for women in technology today. (We celebrate Ada's achievements  and those of women in science, technology, engineering and maths ― every year on 'Ada Lovelace Day'.)

    The story is a portrait of Ada, her young life and her friendship with mathematician Charles Babbage (who called Ada 'the Enchantress of Numbers') that inspired Ada to create what has become widely acknowledged as the World's first computer program. Ada's parents separated when when she was only a month old, and her father left England forever four months later. Ada's mother encouraged Ada's study of mathematics in part as a way to counter the 'madness' of Ada's father, Lord Byron, and his love of poetry. With evocative, flourishing writing and exquisite, whimsical 3D artwork, Fiona Robinson beautifully tells Ada's story, framing her visionary work on Babbage's Analytical Engine in the context of the time, and against the backdrop and challenges her unusual upbringing presented.

    Grown Ups might be interested in Oxford University's Bodleian library's blog dedicated to Lovelace.  Rachel Thomas at Plus magazine has written another excellent and accessible article here, and you may also enjoy this superb BBC documentary about Lovelace by mathematician Hannah FryIf you want to delve a little deeper, consider 'Untangling the Tale of Ada Lovelace', an excellent biographical blog post by Stephen Wolfram. You can read a short, fascinating interview with Professor Ursula Martin, Professor of Computer Science at the University of Oxford about Lovelace's papers here and you may also the book by Martin, and her colleagues Christopher Hollings and Adrian Rice: 'Ada Lovelace: The Making of a Computer Scientist.'

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    Molly and the Mathematical Mysteries: Ten Interactive Adventures in Mathematical Wonderland

    Written by Eugenia Cheng  |  Illustrated by Aleksandra Artymowska  |  Recommended for ages 8-12  |  For more information see this review on the Goodreads website  |  See this page from teachingbooks.net for author interviews and activities to explore the book further with children  |  And see this more in depth review and exploration on Math Book Magic

    This brilliant and beautifully presented interactive book charts the young girl Molly's journey of discovery as she ventures into a curious, puzzling world. Following a trail of clues, the reader is encouraged to help Molly solve them to continue her adventure. Author, concert pianist, artist and mathematician Eugenia Cheng engages the reader wonderfully and ― with Aleksandra Artymowska's fantastic illustrations and intricately constructed interactive devices ― makes us think! A world of Latin squares and logical paradoxes, of M.C. Escher–style settings, where insides and outsides are inverted, and where different types of symmetry and fractals are revealed, the 'implausible but not impossible' world that Molly journeys through exposes children to the important idea that mathematics is about imagination, it isn't just about numbers.

    Grown Ups might be interested in the author's books about mathematics for adults. You will also find a curation of lots of other mathsy sources on her website, including resources for non-specialists, a fantastic set of videos, and more.

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    365 Penguins

    Written by Jean-Luc Fromental  |  Illustrated by Joëlle Jolivet  |  Recommended for ages 5-7  |  For more information see this review on the Goodreads website   See this page from teachingbooks.net for author interviews and activities to explore the book further with children

    This funny, playful and uniquely illustrated book tells the story of a family's strange gifts from their ecologist Uncle Victor. With Global Warming revealed as the ultimate theme, author Jean-Luc Fromental's comical writing and Joëlle Jolivet's bold and striking illustrations combine to recount the travails of the family as they struggle to cope with the penguins that are delivered to them daily over the course of a year, and how they cleverly use arithmetic and properties of number to help them cope with diminishing space to house the penguins, food, and increasing costs.  

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    A Hundred Billion Trillion Stars

    Written by Seth Fishman  |  Illustrated by Isabel Greenberg  |  Recommended for ages 8-10  |  For more information see this review on the Goodreads website   See this page from teachingbooks.net for author interviews and lots of great activities to explore the book further with children  |  And see this more in depth review and exploration from Rhapsody in Books

    This friendly and adorably illustrated book playfully tells the story of some 'big, enormous, gigantic, humongous, incredible numbers,' encouraging children's appreciation of the vastness of the universe. From estimating the amount of trees on Earth, and rabbits!, to estimating how many breaths we take in a lifetime and comparing the collective weights of humans and ants, author Seth Fishman's highly accessible writing combined with Isabel Greenberg's vibrant illustrations gently illuminate the gargantuan numbers of our Universe. The book encourages more questions, gently revealing the enormity of the task to appreciate just how huge some numbers are, and, indeed, why we should bother!

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    Infinity and Me

    Written by Kate Hosford  |  Illustrated by Gabi Swiatkowska  |  Recommended for ages 6-9  |  For more information see this review on the Goodreads website   See this page from teachingbooks.net for author interviews and lots of great activities to explore the book further with children  |  See this page on the author's website for an interesting backstory  |  And see this more in depth review and exploration from The Marginalian

    This utterly wonderful and heart-warming story recounts young Uma's contemplation of infinity. And all whilst wearing her red shoes. Author Kate Hosford's precise writing and the glorious images of renowned illustrator Gabi Swiatkowska combine to produce, in The Marginalian's words, 'an infinitely delightful parable of the inescapable humanity we bring to even the most intellectually ambitious inquiries.' In exploring a concept that is inherently fascinating to children, and which humanity has been grappling with for millennia, young Uma seeks the loving help of the people in her life, who respond with life-affirming and analogies that lead Uma to a heartening resolution. 

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    Hello Numbers! What Can You Do?: An Adventure Beyond Counting

    Written by Edmund Harriss and Houston Hughes  |  Illustrated by Brian Rea  |  Recommended ages 3-6  |  For more information see this review on the Goodreads website   See the Publisher's page  |  And see this review and exploration from ChalkDust Magazine

    The authors, mathematician Edmund Harriss who researches the Geometry of Tilings and Patterns, and the poet Houston Hughes, combine gloriously in this story of the numbers from zero to five. With its energetic Dr Seuss like rhyme, and Brian Rea's wonderful, colour-coded illustrations, the book reveals the story of these important numbers, hinting ― with each 'new one' revealing new personalities ― at deeper mathematics. Encouraging number play and exploration, the progressing story encourages the young reader to count, and induces thinking about properties of number, symmetry, angles, shapes, and more. 

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    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:

    Tuesday, 22 November 2022

    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: