Alex's Adventure in Numberland Highlights

by Alex Bellos

When numbers are spread out evenly on a ruler, the scale is called linear. When numbers get closer as they get larger, the scale is called logarithmic.* It turns out that the logarithmic approach is not exclusive to Amazonian Indians. We are all born conceiving numbers this way.

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universal intuition; it is a product of culture. The precedence of approximations and ratios

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The precedence of approximations and ratios over exact numbers, Pica suggests, is due to the fact that ratios are much more important for survival in the wild than the ability to count.

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Yet what about the difference between a billion gallons of water and ten billion gallons of water? Even though the difference is enormous, we tend to see

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It is unlikely we would confuse one pint of beer and ten pints of beer. Yet what about the difference between a billion gallons of water and ten billion gallons of water?

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Likewise, the terms millionaire and billionaire are thrown around almost as synonyms – as if there is not so much difference between being very rich and very, very rich. Yet a billionaire is a thousand times richer than a millionaire. The higher numbers are, the closer together they feel.

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The higher numbers are, the closer together they feel.

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Yet maybe in our dependence on linearity we have gone too far in stifling our own logarithmic intuition. Perhaps, said Pica, this is a reason why so many people find maths difficult.

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Researchers had previously assumed that the intuitive ability to discriminate amounts does not contribute much to how good a student is at tasks such as solving equations and drawing triangles. Yet this study found a strong correlation between a talent at reckoning and success in formal maths. The better one’s approximate number sense, it seems, the higher one’s chance of getting good grades.

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Yet this study found a strong correlation between a talent at reckoning and success in formal maths.

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Yet this study found a strong correlation between a talent at reckoning and success in formal maths. The better one’s approximate number sense, it seems, the higher one’s chance of getting good grades.

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a condition called dyscalculia, or number blindness, in which one’s number sense is defective. It occurs in an estimated 3–6 percent of the population. Dyscalculics do not ‘get’ numbers the way most people do.

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The most common bases throughout history have been five, ten and twenty, and there is an obvious reason why. These numbers are derived from the human body. We have five fingers on one hand, so five is the first obvious place to take a breath when counting upwards from one. The next natural pause comes at two hands, or ten fingers, and after that at hands and feet, or twenty fingers and toes.

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an ingenious – if rather overly intricate – technique allows you to count up to one less than ten billion – 9,999,999,999. Each finger has nine imaginary points – three on each crease line, as marked on the diagram opposite. These points on the right little finger represent the digits 1 to 9. The points on the right fourth finger take us from 10 to 90. The right middle finger goes from 100 to 900, and so on, with each new finger representing the next power of ten. It is therefore possible to count every single person on Earth with only your fingers, which is one way to have the whole world in your hands.

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In this Chinese system, each finger has nine points, representing the digits 1 to 9 for each order of magnitude, so the right hand can express any number up to 105– 1 when the other hand touches the relevant points. Swapping hands, the numbers continue to 1010– 1. A ‘zero’ point is not needed on any finger, since when there are no values relating to that finger it is simply left alone by the other hand.

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They can be uttered in less than a quarter of a second, so in a two-second span a Chinese-speaker can rattle through nine of them. English number words, by contrast, take just under a third of a second to say (thanks to the frankly cumbersome ‘seven’, with two syllables, and the extended syllable ‘three’), so our limit in two seconds is seven.

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Since The Elements, logical reasoning has been the gold standard of all human enquiry.

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Oliver Byrne, whose day job was Surveyor of Her Majesty’s Settlements in the Falkland Islands, rewrote Euclid in colour.

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The most complex of these is the Sri Yantra, a figure made up of five triangles pointing down and four pointing up, all overlapping a central point, or bindu.

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A guru of modern origami is Robert Lang, who as well as advancing the theory behind paper-folding has turned the pastime into a sculptural art form. A former NASA physicist, Lang has pioneered the use of computers in designing fold patterns to create new and increasingly complex figures. His original figures include bugs, scorpions, dinosaurs and a man playing a grand piano. The fold patterns are almost as beautiful as the finished design.

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It is worth taking a breath to consider the view. There are an estimated 1080 atoms in the universe. If we take the smallest measurable unit of time – known as Planck time, which is a second divided into 1043 parts – then there have been about 1060 units of Planck time since the Big Bang.

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Owing to its ease of use, the Indian method spread to the Middle East, where it was embraced by the Islamic world, which accounts for why the numerals have come to be known, erroneously, as Arabic.

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By one more than the one before All from 9 and the last from 10 Vertically and Cross-wise Transpose and Apply If the Samuccaya is the Same it is Zero If One is in Ratio the Other is Zero By Addition and by Subtraction By the Completion or Non-Completion Differential Calculus By the Deficiency Specific and General The Remainders by the Last Digit The Ultimate and Twice the Penultimate By One Less than the One Before The Product of the Sum All the Multipliers

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another example: 8×7. Again, the first digit can be derived in any one of the four ways: 8 + 7 – 10 = 5, or –2 – 3 + 10 = 5, or 8 – 3 = 5 or 7 – 2 = 5. The second digit is the product of the digits in the second column, –2×–3

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3 7 6 8 5 2 Step 1: We start with the right column: 6×2 = 12

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Only a few great mathematicians have demonstrated lightning-calculator skills, and many mathematicians have surprisingly poor arithmetic.

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Ten decimal places is sufficient to calculate the circumference of the Earth to within a fraction of a centimetre.

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With 39 decimal places, it is possible to compute the circumference of a circle surrounding the known universe to within an accuracy of a radius of a hydrogen atom.

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As more and more digits were found in pi, one thing seemed pretty clear: the numbers obeyed no obvious pattern.

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Yet unfortunately for Pythagoras, there are numbers that cannot be expressed in terms of fractions, and – rather embarrassingly for him – it is his own theorem that leads us to one.

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Numbers that cannot be written as fractions are called irrational.

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Lindemann’s discovery was a milestone for number theory. It also settled, once and for all, what was probably the most celebrated unsolved problem in mathematics: whether or not it was ple to square the circle. In order to explain how it did this, however, I need to introduce the formula that says that the area of a circle is r2, where r is the radius. (The radius is the distance from the centre to the side, or half the diameter.) A visual proof of why this is true is an instance where a pie is the best metaphor for pi. Imagine you have two same-sized circular pies, a white one and a grey one, as below in A. The circumference of each pie is pi times the diameter, or pi times twice the radius, or 2pr. When sliced into equal segments the pieces can be rearranged, as in B with quarter segments, or as in C with ten segments. In both cases the length of the side remains 2pr. If we keep on slicing smaller and smaller segments, then the shape would eventually become a rectangle, as in D, with sides r and 2r. The area of the rectangle – which is the area of the two pies – is therefore 2r2, so the area of one pie is r2.

Alex Bellos, Alex’s Adventures in Numberland, loc. 2249-2260


Hobbes replied to Wallis’s comments with an addendum to his book entitled Six Lessons to the Professors of Mathematics. Wallis countered with Due Correction for Mr Hobbes in School Discipline for not saying his Lessons right. Hobbes followed this with Marks of the Absurd Geometry, Rural Language, Scottish Church Politics and Barbarisms of John Wallis. This led to Wallis’s Hobbiani Puncti Dispunctio! or the Undoing of Mr Hobbes’s Points. The quarrelling lasted almost a quarter of a century, until Hobbes’s death in 1679.

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An early adopter is said to have been a certain Captain Fox in the American Civil War, who, while recovering from battle wounds, threw a piece of wire eleven hundred times on a board of parallel lines and managed to derive pi to 2 decimal places.

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How I need a drink, alcoholic in nature, after the heavy lectures involving quantum mechanics. All of thy geometry, Herr Planck, is fairly hard. ‘How’ has 3 letters, ‘I’ has 1, ‘need’ has 4, and so on. Among numbers, only pi has

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A well-known English one is credited to the astrophysicist Sir James Jeans: How I need a drink, alcoholic in nature, after the heavy lectures involving quantum mechanics. All of thy geometry, Herr Planck, is fairly hard. ‘How’ has 3 letters, ‘I’ has 1, ‘need’ has 4, and so on.

Alex Bellos, Alex’s Adventures in Numberland, loc. 2336-2338


It begins as a pastiche of Edgar Allan Poe: One; A poem A Raven Midnights so dreary, tired and weary, Silently pondering volumes extolling all by-now obsolete lore. During my rather long nap – the weirdest tap! An ominous vibrating sound disturbing my chamber’s antedoor. ‘This,’ I whispered quietly, ‘I ignore.’

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algebra. Start by choosing any three-digit number in which the first and last digits differ by at least two – for example, 753. Now, reverse this number to get 357. Subtract the smaller from the larger: 753 – 357 = 396. Finally, add this number to its reverse: 396 + 693. The sum you get is 1089. Try it again, with a different number, say 421. 421 – 124 = 297 297 + 792 = 1089

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recurrent 1089 is almost instantly unravelled when the

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Consider the number 614. This is equal to 600 + 10 + 4. In fact, any three-digit number written abc can be written 100a + 10b + c (note: abc in this case is not a×b×c). So, let’s call our initial number abc, where a, b and c are single digits. For the sake of convenience, make a bigger than c. The reverse of abc is cba which can be expanded as 100c + 10b + a. We are required to subtract cba from abc to give an intermediary result. So abc – cba is: (100a + 10b + c) – (100c + 10b + a) The two b terms cancel each other out, leaving an intermediary result of: 99ac, or 99(a – c) At a basic level algebra doesn’t involve any special insight, but rather the application of certain rules. The aim is to apply these rules until the expression is as simple as possible. The term 99(a – c) is as neatly arranged as it can be. Since the first and last digits in abc differ by at least 2, then a – c is either 2, 3, 4, 5, 6, 7 or 8. So, 99(a – c) is one of the following: 198, 297, 396, 495, 594, 693 or 792. Whatever three-figure number we started with, once we have subtracted it from its reverse, we have an intermediary result that is one of the above eight numbers. The final stage is to add this intermediary number to its reverse. Let’s repeat what we did before and apply it to the intermediary number. We’ll call our intermediary number def, which is 100d + 10e + f. We want to add def to fed, its reverse. Looking closely at the list of possible intermediary numbers above, we see that the middle number, e, is always 9. And also that the first and third numbers always add up to 9, in other words d + f = 9. So, def + fed is: 100d + 10e + f + 100f + 10e + d Or: 100(d + f ) + 20e + d + f Which is: (100 × 9) + (20 × 9) + 9 Or: 900 + 180 + 9 Hey presto! The total is 1089, and the riddle is laid bare.

Alex Bellos, Alex’s Adventures in Numberland, loc. 2675-2698


Consider the number 614. This is equal to 600 + 10 + 4. In fact, any three-digit number written abc can be written 100a + 10b + c (note: abc in this case is not a×b×c). So, let’s call our initial number abc, where a, b and c are single digits. For the sake of convenience, make a bigger than c. The reverse of abc is cba which can be expanded as 100c + 10b + a. We are required to subtract cba from abc to give an intermediary result. So abc – cba is: (100a + 10b + c) – (100c + 10b + a) The two b terms cancel each other out, leaving an intermediary result of: 99ac, or 99(a – c) At a basic level algebra doesn’t involve any special insight, but rather the application of certain rules. The aim is to apply these rules until the expression is as simple as possible. The term 99(a – c) is as neatly arranged as it can be. Since the first and last digits in abc differ by at least 2, then a – c is either 2, 3, 4, 5, 6, 7 or 8. So, 99(a – c) is one of the following: 198, 297, 396, 495, 594, 693 or 792. Whatever three-figure number we started with, once we have subtracted it from its reverse, we have an intermediary result that is one of the above eight numbers. The final stage is to add this intermediary number to its reverse. Let’s repeat what we did before and apply it to the intermediary number. We’ll call our intermediary number def, which is 100d + 10e + f. We want to add def to fed, its reverse. Looking closely at the list of possible intermediary numbers above, we see that the middle number, e, is always 9. And also that the first and third numbers always add up to 9, in other words d + f = 9. So, def + fed is: 100d + 10e + f + 100f + 10e + d Or: 100(d + f ) + 20e + d + f Which is: (100 × 9) + (20 × 9) + 9 Or: 900 + 180 + 9 Hey presto! The total is 1089, and the riddle is laid bare.

Alex Bellos, Alex’s Adventures in Numberland, loc. 2675-2698


He noticed that he could multiply by adding lengths of this ruler. If a compass was placed with the left spike at 1, and the right at a, then when the left spike was moved to b, the right spike pointed to a ×

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= 1×1×1 1 = 1 512 = 8×8×8 8 = 5 + 1 + 2 4913 = 17×17×17 17 = 4 + 9 + 1 + 3 5832 = 18×18×18 18 = 5 + 8 + 3 + 2 17,576 = 26×26×26 26 = 1 + 7 + 5 + 7 + 6 19,683 = 27×27×27 27 = 1 + 9 + 6 + 8 + 3

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It is a biannual homage to the man who revolutionized recreational mathematics in the second half of the last century. Martin Gardner, now 93 years old, wrote a monthly maths column in Scientific American between 1957 and 1981. This

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Second, every natural number above 1 can be written as a unique product of imes. In other words, every number is equal to a unique set of prime numbers multiplied by each other. For example, 221 is 13×17. The next number, 222, is 2×3×37. The one after that, 223, is prime, so produced only by 223×1, and 224 is 2×2×2×2 × 2 × 7. We could carry on for ever and each number could be winnowed down to a product of primes in only one possible way. For example, a billion is 2×2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5. This characteristic of numbers is known as the fundamental theorem of arithmetic,

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In other words, whatever 233-digit number you choose, if you follow the steps of multiplying all the digits together according to the rules for persistence, you will get to a single-digit number in 11 steps or fewer.

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39 27 14 4 and 39 is the smallest number that reduces in three steps.

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Consider that for a moment: Conway’s powertrain is such a lethal machine that it annihilates every number in the universe apart from 2592 and 24547284284866560000000000 – two seemingly unrelated, fixed points in the never-ending expanse of numbers.

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In fact, though, the numbers are generated using the following simple rule: ‘subtract if you can, otherwise add’. To get the nth term, we take the previous term and either add or subtract n from it. The rule is that subtraction must be used unless that results in either a negative number or in a number that is already in the sequence. Here’s how the first eight terms are calculated.

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It was only in 1636 that Pierre de Fermat discovered the second set of amicable numbers: 17,296 and 18,416.

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GIMPS was one of the first ‘distributed computing’ projects and has been one of the most successful.

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and $250,000 for the first prime with a billion digits. If you plot the largest-known primes discovered since 1952 on a graph with a logarithmic scale against the time of discovery, they fall on what is almost a straight line.

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But, bizarrely, the harmonic series is divergent, a decelerating but unstoppable snail. After 100 terms of the series, the total has only just passed 5. After 15,092,688,622,113,788,323,693,563,264,538, 101,449,859,497 terms, the total exceeds 100 for the first time.

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In order to achieve an overhang of 50 blocks, we would need a tower of 15×1042 blocks – which would be much higher than the distance from here to the edge of the observable universe.

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I’ll use the example of 666. Between 1 and 1000 the number 666 occurs once. Between 1 and 10,000 it occurs 20 times, and between 1 and 100,000 it occurs 300 times. In other words, the percentage occurrence of 666 is 0.1 percent in the first 1000 numbers, 0.2 percent in the first 10,000 and 0.3 percent in the first 100,000.

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This seems fairly straightforward now, but in the seventeenth century the idea that random events that haven’t yet taken place can be treated mathematically was a momentous conceptual breakthrough.

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The above argument is known as Pascal’s Wager. It can be summarized as follows: if there is the slightest probability that God exists, it is overwhelmingly worthwhile to beliee in Him. This is because if God doesn’t exist a non-believer has nothing to lose, but if He does exist a non-believer has everything to lose. It’s a no-brainer. Be a Christian, go on, you might as well.

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By Pascal’s argument, it is worthwhile to believe that this cat made from green cheese exists,

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By Pascal’s argument, it is worthwhile to believe that this cat made from

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By Pascal’s argument, it is worthwhile to believe that this cat made from green cheese exists, which is, of course, absurd.

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For us to be able to understand in mathematical terms the statement that there is a 1 in anything chance of God existing, there must be a possible world where God does in fact exist. In other words, the premise of the argument presupposes that, somewhere, God exists.

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The best deal to be found in a casino is at the craps table. The game originated from a French variant of an English dice-rolling game. Players throw two dice and the outcome depends on which numbers land and how they add up. In craps, your chances of winning are 244 out of 495 possible outcomes, or 49.2929 percent, giving an expected loss of just 14.1p per £10 bet.

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Another way of looking at an expected loss is to consider it in terms of payback percentage. If you bet £10 at craps, you can expect to receive about £9.86 back.

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In fact, we could rephrase the birthday paradox as the statement that for a 365-sided dice, after 23 throws it will be more likely than not that the dice will have landed on the same side twice.

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Imagine a drunkard’s walk in three dimensions. Call it the buzz of the befuddled bee. The bee starts at a point suspended in space and then flies in a straight line in a random direction for a fixed distance. The bee stops, dozes, then buzzes off in another random direction for the same distance. And so on. What is the chance of the bee eventually buzzing back into the spot where it started? The answer is only 0.34, or about a third. It was weird to realize that in two dimensions the chance of a drunkard walking back into the lamp-post was an absolute certainty, but it seems even weirder to think that a bee buzzing for ever is very unlikely ever to return home.

Alex Bellos, Alex’s Adventures in Numberland, loc. 4876-4881


Events that are random under one set of information might well become non-random under a greater set of information. This is turning a maths problem into a physics one. A coin flip is random because we don’t know how it will land, but flipped coins obey Newtonian laws of motion.

Alex Bellos, Alex’s Adventures in Numberland, loc. 4950-4952


Events that are random under one set of information might well become non-random under a greater set of information. This is turning a maths problem into a physics one. A coin flip is random because we don’t know how it will land, but flipped coins obey Newtonian laws of motion. If we knew exactly the speed and angle of the flip, the density of the air and any other relevant physical data, we would be able to calculate exactly the face on which it would land.

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Two of them, Fischer Black and Myron Scholes, created the Black-Scholes formula indicating how to price financial derivatives – Wall Street’s most famous (and infamous) equation.

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His follow-up book, Beat the Market, helped transform securities markets.

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Yet my poor forecasting record did not diminish my belief that I was indeed an expert in baguette-assessing. It was, I reasoned, the same self-delusion displayed by sports and financial pundits who are equally unable to predict random events, and yet build careers out of it. Perhaps

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In ‘Cutting a Round Cake on Scientific Principles’ Galton marked intended cuts as broken straight lines, and cuts as solid lines. This method minimizes exposing the insides of the cake to become dry, which would happen if one cuts a slice in the traditional (and, he concludes, ‘very faulty’) way. In the second and third stages the cake is to be held together with an elastic band.

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Pascal’s can be constructed much more simply than by working out the distributions of randomly falling balls through a Victorian bean machine. Start with a 1 in the first row, and under it place two 1s so as to make a triangle shape. Continue with subsequent rows, always placing a 1 at the beginning and end of the rows. The value of every other position is the sum of the two numbers above it.

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There is only one combination of three items: mango, lychee, banana. If we want to select only two fruits, we can do this in three different ways: mango and lychee, mango and banana, lychee and banana. There are also only three ways of taking the fruit individually, which is each fruit on its own. The final option is to select zero fruit, and this can happen in only one way. In other words, the number of combinations of three different fruits produces the string 1, 3, 3, 1 – the third line of Pascal’s triangle.

Alex Bellos, Alex’s Adventures in Numberland, loc. 5385-5389


The coefficients of the individual terms are the rows of Pascal’s triangle. (x + y)2 = x2 + 2xy + y2 (x + y)3 = x3 + 3x2y + 3xy2 + y3 (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4

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economist Nassim Nicholas Taleb’s position in his bestselling book The Black Swan is that we have tended to underestimate the size and importance of the tails in distribution curves.

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We saw from the quincunx that random errors are distributed normally. So, the more random errors we can introduce into measurement, the more likely it is that we will get a bell curve from the data – even if the phenomenon being measured is not normally distributed. When the normal distribution is found in a set of data, this could simply be because the measurements have been gathered too shambolically.

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‘Everybody believes in the [bell curve]: the experimenters because they think it can be proved by mathematics; and the mathematicians because they believe it has been established by observation.’

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who in 1911 set up the world’s first university statistics department, at University College London.

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Regression to the mean is not a complicated idea. All it says is that if the outcome of an event is determined at least in part by random factors, then an extreme event will probably be followed by one that is less extreme.

Alex Bellos, Alex’s Adventures in Numberland, loc. 5548-5550


Yet despite its simplicity, regression is not appreciated by most people. I would say, in fact, that regression is one of the least grasped but most useful mathematical concepts you need for a rational understanding of the world. A surprisingly large number of simple misconceptions about science and statistics boil down to a failure to take regression to the mean into account.

Alex Bellos, Alex’s Adventures in Numberland, loc. 5550-5552


Player B can be genuinely praised for having a 60–40 score ratio on average over many games, but praising him for any sequence of five baskets in a row is no different from praising the talent of a coin flipper who gets five consecutive heads.

Alex Bellos, Alex’s Adventures in Numberland, loc. 5584-5586


The rules, or axioms, of a system are the statements that are accepted without proof, so mathematicians always try to make them as simple and self-evident as possible.

Alex Bellos, Alex’s Adventures in Numberland, loc. 5616-5617


Euclid proved all 465 theorems of The Elements with only five axioms, which are more commonly known as his five postulates: 1. There is a straight line from any point to any point. 2. A finite straight line can be produced in any straight line. 3. There is a circle with any centre and any radius. 4. All right angles are equal to one another. 5. If a straight line falling on two given straight lines makes the interior angles on the same side less than two right angles, the two given straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.

Alex Bellos, Alex’s Adventures in Numberland, loc. 5617-5623


For any given line and a point not on that line: On a flat surface there is one and only one parallel line through that point. On a spherical surface there are zero parallel lines through that point.* On a hyperbolic surface there is an infinite number of parallel lines through

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In his book on Cantor, Everything and More, David Foster Wallace writes: ‘The Mentally Ill Mathematician seems now in some ways to be what the Knight Errant, Mortified Saint, Tortured Artist, and Mad Stist have been for other eras: sort of our Prometheus, the one who goes to forbidden places and returns with gifts we can all use but he alone pays for.’

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Some numbers are squares, such as 1, 4, 9 and 16, and some are not squares, such as 2, 3, 5, 6, 7, etc. 2. The totality of all numbers must be greater than the total of squares, since the totality of all numbers includes squares and non-squares. 3. Yet for every number, we can draw a one-to-one correspondence between numbers and their squares, for example: 4. So, there are, in fact, as many squares as there are numbers. Which is a contradiction, since we have said, in point 2, that there are more numbers than squares. Galileo’s

Alex Bellos, Alex’s Adventures in Numberland, loc. 5866-5873


It turns out that the irrational numbers are so densely packed that there are more of them in any finite interval on the number line than there are fractions on all of the number line.

Alex Bellos, Alex’s Adventures in Numberland, loc. 6013-6014


Mathematics, as I wrote in the opening chapters of this book, emerged as part of man’s desire to make sense of his own environment. By making notches on wood, or counting with fingers, our ancestors invented numbers. This was helpful for farming and trade, and ushered us into ‘civilization’. Then, as mathematics developed, the subject became less about real things and more about abstract ones. The Greeks introduced concepts such as a point and a line, and the Indians invented zero, which opened the door to even more radical abstractions like negative numbers. While these concepts were at first counter-intuitive, they were assimilated quickly and we now use them on a daily basis. By the end of the nineteenth century, however, the umbilical cord linking mathematics to our own experience snapped once and for all. After Riemann and Cantor, maths lost its connection to any intuitive appreciation of the world.

Alex Bellos, Alex’s Adventures in Numberland, loc. 6045-6051