Alex’s Adventures in Numberland
Dispatches from the Wonderful World of Mathematics
By: Alex Bellos
All of the ways you have never considered mathematics, considered.
Notes on the the first 7 chapters. With more to follow.
Intro
Recidivist = a convicted criminal who reoffends repeatedly
John Major = a prime minister of Great Britain who served as leader of the Conservative party from 1990 to 1997
The saying “tongue in cheek” implies that whatever saying or phrase it is referencing is not seriously intended and should not be taken at face value (originally it was in contempt but as since evolved to be more humorous)
To win a competition that determines the number of cones along a particular stretch of roadway (assuming they are placed at basically equidistant intervals) simply calculate: number of cones = length of road ÷ distance between cones
A degree in mathematics and philosophy! One foot in science and one in liberal arts… the author decided to enter journalism which meant superficially abandoning the former and embracing the latter
“entering the world of maths as an adult was very different from entering it as a child” (requirements to pass exams and the institutional pressure is quite effective at draining some of the enjoyment from the discipline)
ethnomathematics = the study of how different culture approach math and how it is shaped by religion
Experience as a foreign correspondent was abstractedly transferred to ‘Numberland’
India invented zero
How casinos use probability in their favor
Japan’s most numerate chimpanzee
The hundred year campaign to introduce two new number to our ten-number system
Why was Britain the first nation to mint a heptagonal coin?
The maths behind Sudoku
Interesting point: unlike humanities that are in a permanent state of reinvention, and applied sciences that are constantly being experimented with towards refinement, mathematics does not age
Pythagoras and Euclid are the oldest names we study at school and their theorems hold just as much validity now as they did before
“abstract mathematical thought is one of the great achievements of the human race, and arguably of all human progress.” – Alex Bellos
Chapter Zero: A Head for Numbers
How to solve a Rubik’s Cube
The Munduruku language: an indigenous group of about 7000 people live in small villages spread across the rainforest in the Brazilian Amazon and speak a language with no tenses, no plurals, no words for numbers beyond five! How?
No one knows for certain, but numbers are probably no more than 10,000 years old
If someone asks you to do an inventory of your family, look at them incredulously and say “j’ai une grande famille”
When King David counted his own people he was punished with three days of pestilence and 77,000 deaths…
When numbers are spread out evenly on a ruler the scale is called linear
When numbers get closer together as they get larger the scale is called logarithmic
We are all born conceiving numbers in a logarithmic way
Exact numbers provide us with a linear framework that contradicts our logarithmic intuition
If you think about it, the higher numbers get and the further away from our reality they become, the more they blur together. Millionaire? Billionaire? The difference between 9 billion and 10 billion… no big deal right
Clever Hans, the equine Einstein
the incredible stories of avian intelligence
The first non-human to count with Arabic numerals, the great ape Ai
The aspect of numbers that represent quantity is called cardinality
The convenient feature of numbers that orders them in succession is called ordinality
In school the above two features are taught without being discerned, and we naturally slip between them
The investigation of animals’ mystery of numbers is an active academic pursuit
Interestingly, all creatures seem to be born with brain that have a predisposition for maths… numerical competence is crucial to survival in the wild if you think about it, being able to determine which plants or resources are more bountiful, or knowing how many predators are in a certain place
How do ants find their way around? It may have less to do with sight and much more to do with a proficiency at counting strides, or an internal pedometer of sorts
Chinese characters for numbers, and what kind of difference that makes for learning to manipulate them
What the heck does the phrase “cheek-by-jowl” mean?
Our ready-built capacity to understand exact numbers known as the exact number module (we understand the exact number of items in small collections, and by adding to these collections one by one, we can learn to understand how bigger numbers behave)
dyscalculia = number blindness, when someone’s number sense is defective
An electrode and how it is tiny enough to slide through the brain without causing damage or pain (however it is against ethical guidelines to be used in humans unless for therapeutic reasons like the treatment of epilepsy)
Chapter One: The Counter Culture
The very strange way that Lincolnshire shepards would count their sheep
The trick of a good base system is having a base that is large enough to express bigger numbers without having to say too much, but not so large that we have to memorize a bunch of stuff
Until a few hundred years ago, no manual of arithmetic was complete without diagrams of finger-counting. That is how reliant we were on our hands and fingers and toes as inspirational mechanisms for number sense
Various body tally systems
How the Chinese system can put the count of the whole world in your hand
Why might 12 be considered superior to 10? Divisibility: 12 can be divided by 2,3,4, and 6 while 10 can only be divided by 2 and 5.
The multi-divisibility of 12 also explains the utility of imperial measure (one foot = 12 inches) this is quite handy for carpenters, tailors, etc
Divisibility is also relevant to multiplication tables, the easiest tables to learn in any base are the ones of numbers that divide that base (for example, with 10, the ones that end in 5 or 0)
The campaign for base 12 should not be conflated with the crusade for the metric system over the imperial measure
The duodecimal system
The simplest form of numerical notation is the tally: Incas kept count by tying knots on ropes, cave dwellers painted marks on rocks, etc
Humans can instantly tell the difference between one item and two, between two items and three, but beyond four it gets a bit more difficult
Around 8000 BC, a practice of using small clay pieces with markings to keep record of things like livestock spread throughout the ancient world
In 4000 BC in Sumer (present-day Iraq) the token system evolved into a script in which a reed was pressed into clay and so numbers were first represented by circles or fingernail shapes
The short failed history of decimal time
Base two = binary which is usually expressed using 0s and 1s
Each extra zero on the end in binary represents multiplication by two (100 = 4)
“Freedom is the freedom to say two plus two equals four” – Winston Smith from George Orwell’s 1984
Mathematical truths cannot be influenced by culture or ideology… right?
Between 10 and 20 English is a mess in the lack of consistency for forming the names of our numbers
Asian children find it easier to learn to count than Europeans because the naming of their numbers is much more logical: eleven in ten one and twelve is ten two etc
Some European languages such as Welsh follow the Chinese example
In Japan, language is recruited as an ally in ease of mathematical understanding. Words and phrases are modified in order to make their multiplication tables (kuku) easier to learn
In French, quatre-vingts, or four twenties, suggests that ancestors used to use a base twenty system
Chapter Two: Behold!
Pythagoras is the most famous name in mathematics, entirely due to his theorem about triangles. He also made the discovery of square numbers
Pythagoras made a discovery related to music when he was walking past a smithy and heard hammers clinking; he noticed the pitch changed depending on the weights of the anvils. He then began to investigate the relationship between the pitch of a vibrating string and its length
He made a realization: if the length of a string is halved, the pitch increases by an octave
The crest of the Brotherhood was the pentagram
Now we don’t worship numbers, but before it reflected the scale of the wonderment that surrounded discoveries of abstract mathematical knowledge at the time
Pythagoras believed in reincarnation and was probably a vegetarian
Numerology is now basically an established dish on the “buffet of modern mysticism” (giving numbers spiritual and lottery like significance)
Pythagoras’s Theorem: for any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides OR a² + b² = c² (where c is the hypotenuse)
In Egypt, rope-stretching was the most popular way to form the right triangles that were needed to construct pyramids and arrange bricks
Hypotenuse comes from the greek word “stretched against”
Annairzi’s theorem that derives from a repeating pattern
Euclid and his “watertight system of mathematical truths” all preserved in his book The Elements
Euclid began by carving the 2D space we are familiar with into the family of shapes that we know as polygons which are all made only from straight lines
His mission was to investigate what 3D shapes can be made from joining identical polygons together: the tetrahedron, the cube, the octahedron, the icosahedron, and the dodecahedron (the quintet became known as the Platonic solids)
There are four different ways to define the center of a triangle, and they all represent different points unless it is an equilateral triangle in which they all coincide
The orthocenter = intersection of lines from each vertex that meet the opposing sides perpendicularly. These are called altitudes
The circumcenter = the intersection of the perpendiculars drawn from the halfway points of the sides of the triangle and these neat lines will always meet, regardless of the triangle
The centroid = the intersection of the lines that go from the vertex to the midpoints of opposing lines
The midcircle = the intersection of the lines that go through the midpoints of each side and are also the intersections of the sides and the altitudes
In 1767 Leonard Euler proved that for all triangles, these four types of centers are always on the same line. This was mind-blowing, and exposed a dazzling harmony that would have awed Pythagoras
Ancient Athens was given a challenge by the gods that cannot be solved with Euclidean tools: given a cube, construct a second cube that has twice the volume! This is the Delian Problem
The other problems of classic antiquity are the squaring of the circle (constructing a square that has the same area as a given circle), and the the trisection of an angle (the construction of an angle that is a third of a given angle)
The most sacred object in Islam is a Platonic solid the Ka’ba or Cube is the black palladium at the centre of Mecca’s Sacred Mosque… pilgrims walk anticlockwise around this during the Hajj
Mathematics plays more of a role in Islam than in any other major religion: Muslims must face Mecca during daily prayer wherever they are in the world, and before the advancement of GPS technology, this relied on complicated astronomical calculations, putting Islamic science ahead of everyone else for almost a thousand years
What does it mean to tessellate?
Any four-sided shape can produce periodic tessellations
Periodicity can be defined as the capacity to lift up a piece of a pattern and move it along to another position so that the pattern still lines up perfectly with the original pattern
What would it be like to be a cosmologist?
In the 1980s physicists and chemists discovered a new type of crystal that had previously not existed; it was a tiny structure that displayed a non periodic pattern. These became classified as quasicrystals
Hinduism also uses geometry to illustrate the divine: Mandalas are symbolic representations of deities and the cosmos
Origami the Japanese art of paper-folding evolved from the custom of Japanese farmers thanking the gods at harvest time by making an offering of some of their crops on a piece of paper
The Menger sponge
When some man named Erik Demaine was 17 he and his collaborators proved that it is possible to create any shape with straight sides by folding a piece of paper and making just one cut
Chapter Three: Something About Nothing
Above one thousand Indians introduce a comma after every two digits while the rest of the world’s convention is every three
There are an estimated 10⁸⁰ atoms in the universe
The smallest measurable unit of time is known as Planck time
If you take the units of Planck time since the Big Bang and multiply that number by the number of atoms in the universe than you have the number of unique positions of every particle since the Big Bang
The Indians were not the first to introduce a place-holder, that was probably the Babylonians
Greeks were able to make mathematical discoveries without a zero because they had had a spatial understanding of mathematics
Indian usage of numbers was spread verbally to the Arabic world where it was later, erroneously, adopted by the Western world and the credit was attributed to them
Some people consider the “ingenious” use of a placeholder value like 0 to be the Devil’s work in early stages
The Tata Institute of Fundamental Research in Bombay is where Aja worked!
Chapter Four: Life of Pi
Jedediah Buxton was a wonder boy who amazed the locals of the Derbyshire farm community with his ability to make astonishing arithmetical computations =
The ability to calculate rapidly has no correlation with mathematical insight or creativity
When 2 is multiplied by itself 13 times, the answer ends in 2
When 3, 4, 5, 6, 7, 8, and 9 are multiplied by themselves 13 times, the same happens respectively
The ratio of a circle’s circumference to its diameter is always the same no matter how big or small the circle
The earliest approximation of pi came from the Babylonians who used a value of 3 1/8
Pi hunters were those determined to find the number’s true value: they constructed polygons of 3072 sides to approximate it to a whopping 5 decimal places 3.14159, but his son went one decimal further with a polygon of 12,288 sides giving 3.141592
From there they went off, the Dutch used a polygon with 60 x 2 ²⁹ to find 20 decimals. A daring declaration of ‘whoever wants to, can come closer’ accompanying the breakthrough
Leibniz discovered the alternating and infinite series of unit fractions of odd numbers for approximating by using calculus in the 17th century
Newton and Leibniz argued over who had first discovered the calculus until later on it was realized that an Indian mathematician Madhava actually beat them both out by 200 years
In 1947 pi was known to 808 decimal places before computers changed the game
Natural numbers: begin with 1 and count upwards
Integers are more useful with the addition of 0 and the negative numbers
Fractions are equivalent to ratios between integers
How many rational numbers are there between 0 and 1? An infinite amount
When a rational number is written as a decimal fraction it has a finite amount of digits, or the expansion will repeat itself in the same pattern forever
When a number is irrational, its decimal expansion never repeats… like pi
A transcendental number is an irrational number that CANNOT be described by an equation with a finite number of terms
The transcendence of pi proves that a circle cannot be squared
One of the most pointless yet entertaining feuds in British intellectual life was between Thomas Hobbes (who published in his Leviathan a faulty proof that the circle could be squared, and John Wallis (who published a pamphlet pointing out his errors)
; σ’ ε ρ τ υ θ ι ο π α σ δ φ γ η ξ κ λ ζ χ ψ ω β ν μ
; Σ Ε Ρ Τ Υ Θ Ι Ο Π Α Σ Δ Φ Γ Η Ξ Κ Λ Ζ Χ Ψ Ω Β Ν Μ
Apparently some man named Captain Fox, while recovering from battle wounds inflicted during the American Civil War, threw a piece of wire 1100 times on a board of parallel lines to derive pi to 2 decimal places
As of the time this book was published, Akira Haraguchi, a 60-year-old retired engineer holds the current world record for reciting digits of pi: in 2006, to 100,000 places, in 16h28min, with 5min breaks every 2h to eat rice balls
“how I need a drink, alcoholic in nature, after the heavy lectures involving quantum mechanics. All of thy geometry, Herr Planck, is fairly hard” = a popular mnemonic in England to remember the digits of pi. The number of letters in each word correlates with the number of pi
Pi was named in 1706 in a book titled: A new Introduction to the Mathematics, for the Use of some Friends who have neither Leisure, Convenience, nor, perhaps, Patience, to search into so many different Authors, and turn over so many tedious Volumes, as is unavoidably required to make but tolerable the process in the Mathematics
π did not become standard notation until it was adopted by Leonhard Euler in 1736
The book the Indian Clerk that I read in summer of 2016 is about the Indian mathematician Srinivasa Ramanujan who was self-taught, and wrote a letter to a Cambridge professor G.H. Hardy about his discovery of an equation that approximated pi extraordinarily efficiently and accurately. He died at 32
Breaking records in pi calculations is a good way to test the processing capacity and reliability of computers. It is essential in quality-testing supercomputers
The story of pi mimics its circular nature: the most ancient and simplest ratio in math has been enlisted as a massively important tool on the frontline of computer technology
The Feynman point in pi = at the 762nd decimal point where one digit is repeated six times consecutively (999999)
Random numbers are needed in industry and commerce (for example when sample groups need to be selected… the better the random number generator, the more representative the sample)
Britain was not immune to the counter-cultural uprisings occurring around the world in 1968
The 50p piece was introduced around this time in design to replace the ten-shilling note and shift from imperial to decimal currency (it had an unorthodox, multilateral-curve heptagon shape)
A circle can be defined as a curve for which every point is equidistant from a fixed point, or the center
As counterintuitive as it might sound, if you had a roller made of 50p cross sections, and tried to roll something like a book across it, the book would travel smoothly without bobbing up and down
Chapter Five: The X-Factor
Mathematicians tend to like magic tricks (in the sense that they can be very fun and often conceal interesting theory)
Choose any three-digit number in which the first and last digits differ by at least two, ex: 846. Then reverse the number (648) and subtract the smaller from the larger 846-648 = 198. Then add this number to its reverse: 891+198 = 1089. 1089 will always be the sum you get regardless of the three-digit number you choose, as long as you meet those initial conditions
So what is this seemingly magical, reoccurring 1089?
The Egyptian hieroglyph for addition was a pair of legs walking from right to left, while subtraction was a pair of legs walking left to right
Algebra is the generic term for the maths of equations, in which numbers and operations are written as symbols
The root of algebrista is the Arabic al-jabr which refers to crude surgical techniques as well as restoration or reunion
The line of thought worked simply: consider A = B - C. Restoration is the process of returning the equation to A + C = B (in other words a negative term can be made positive by resetting it on the other side of the equal sign)
Whatever you do to one side of an equation, you must do to the other
Between the 15th and 16th centuries mathematical sentences moved from rhetorical to symbolic expression, and slowly, words were replaced with letters
René Descartes published his Discourse on Method in which he applied mathematical reasoning to human thought (he started by doubting all of his beliefs, and after stripping everything away he was left with only one certainty: that he existed) AKA the Cartesian plane
The father of algebra, Diaphantus, died at age 84
All of science relies on the language of equations
Pythagorean triples are any numbers that satisfy a²+b²=c², however, there haven’t been any numbers found to satisfy a³+b³=c³ or a⁴+b⁴=c⁴
The logarithm was a massively important invention in the early 17th century (thought up by Scottish mathematician John Napier)
Logarithmic scales are also useful for representing large quantities on graphs or scaling things because numbers get closer together the higher they are
Throughout the 1950s and 1960s the Curta (a small precision machine for all arithmetic calculations) was the only pocket sized calculator in existence that could produce exact answers
Just as you can play with the relationship between circles and ellipses with (x/n)² + (y/n)² = 1, so can you experiment with x^n + y^n = 1, where as n -> ∞, the shape on a cartesian plane resembles more and more a square
Chapter Six: Playtime
The idea that numbers can entertain is as old as maths itself
Mathematical riddles, rhymes, and games are now collectively known as recreational maths
A landmark event in the history of recreational maths is said to have taken place by the banks for the Yellow River in China around 2000 BC when Emperor You saw a turtle creep out of the water. It was a divine turtle with black and white dots on its underbelly that formed a magical pattern called lo shu
The bridge problem: in a former Prussian capital there are seven bridges that crossed a river connecting to the city. People wanted to know if it would be possible to make a journey across all seven bridges without crossing any bridge more than once
Euler created a graph to solve the puzzle and show it was impossible by creating a graph in which each landmass was represented with a dot, or node, and each bridge by a line or a link
The London Underground borrows the idea that came out of working with this puzzle… it doesn’t matter the information about the exact position of the bridges, but how they were connected
Euler’s theorem launched the graph theory and presaged the development of topology (an area of maths that studies the properties of objects that do not change when the object is squeezed, twisted, or stretched)
Is it part of the human condition to have an intuitive ability to solve puzzles? Perhaps even a form of existential therapy! We know there is usually a predetermined solution, or solutions, within grasp, yet we still get the satisfaction of coming there on our own
The geometrical vanish
The Rubik’s cube came from Cold War, Eastern Europe and had a sexy resonance to its name, design, and origins
There are records for solving the Rubik’s cube one-handed, blindfolded, on a rollercoaster, under water, with chopsticks, while balancing on a unicycle, and during freefall
The most mathematically interesting and relevant is how to solve the cube using the fewest moves possible
“Maths is not sums, calculations, and formulae. It is pulling things apart to understand how things work”
