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Overview
Thinking in Numbers is the book that Daniel Tammet, mathematical savant and bestselling author, was born to write. In Tammet's world, numbers are beautiful and mathematics illuminates our lives and minds. Using anecdotes, everyday examples, and ruminations on history, literature, and more, Tammet allows us to share his unique insights and delight in the way numbers, fractions, and equations underpin all our lives.
Inspired variously by the complexity of snowflakes, Anne Boleyn's eleven fingers, and his many siblings, Tammet explores questions such as why time seems to speed up as we age, whether there is such a thing as an average person, and how we can make sense of those we love. His provocative and inspiring new book will change the way you think about math and fire your imagination to view the world with fresh eyes.
Product Details
| ISBN-13: | 9780316187367 |
|---|---|
| Publisher: | Little, Brown and Company |
| Publication date: | 07/15/2014 |
| Edition description: | Reprint |
| Pages: | 304 |
| Sales rank: | 203,664 |
| Product dimensions: | 5.50(w) x 8.10(h) x 0.90(d) |
About the Author
Read an Excerpt
Thinking In Numbers
On Life, Love, Meaning, and Math
By Daniel Tammet
Little, Brown and Company
Copyright © 2014 Daniel TammetAll rights reserved.
ISBN: 978-0-316-18737-4
CHAPTER 1
FAMILY VALUES
In a smallish London suburb where nothing much ever happened, my family gradually became the talk of the town. Throughout my teens, wherever I went, I would always hear the same question, "How many brothers and sisters do you have?"
The answer, I understood, was already common knowledge. It had passed into the town's body of folklore, exchanged between the residents like a good yarn.
Ever patient, I would dutifully reply, "Five sisters, and three brothers."
These few words never failed to elicit a visible reaction from the listener: brows would furrow, eyes would roll, lips would smile. "Nine children!" they would exclaim, as if they had never imagined that families could come in such sizes.
It was much the same story in school. "J'ai une grande famille" was among the first phrases I learned to say in Monsieur Oiseau's class. From my fellow students, many of whom were single sons or daughters, the sight of us siblings together attracted comments that ranged all the way from faint disdain to outright awe. Our peculiar fame became such that for a time it outdid every other in the town: the one-handed grocer, the enormously obese Indian girl, a neighbor's singing dog, all found themselves temporarily displaced in the local gossip. Effaced as individuals, my brothers, sisters, and I existed only in number. The quality of our quantity became something we could not escape. It preceded us everywhere: even in French, whose adjectives almost always follow the noun (but not when it comes to une grande famille).
With so many siblings to keep an eye on, it is perhaps little wonder that I developed a knack for numbers. From my family I learned that numbers belong to life. The majority of my math acumen came not from books but from regular observations and day-to-day interactions. Numerical patterns, I realized, were the matter of our world. To give an example, we nine children embodied the decimal system of numbers: zero (whenever we were all absent from a place) through to nine. Our behavior even bore some resemblance to the arithmetical: over angry words, we sometimes divided; shifting alliances between my brothers and sisters combined and recombined them into new equations.
We are, my brothers, sisters, and I, in the language of mathematicians, a "set" consisting of nine members. A mathematician would write:
S = {Daniel, Lee, Claire, Steven, Paul, Maria, Natasha, Anna, Shelley}
Put another way, we belong to the category of things that people refer to when they use the number nine. Other sets of this kind include the planets in our solar system (at least, until Pluto's recent demotion to the status of a non- planet), the squares in a game of x's and o's, the players in a baseball team, the muses of Greek mythology, and the Justices of the U.S. Supreme Court. With a little thought, it is possible to come up with others, including:
{February, March, April, May, August, September, October, November, December} where S = the months of the year not beginning with the letter J.
{5, 6, 7, 8, 9, 10, Jack, Queen, King} where S = in poker, the possible high cards in a straight flush.
{1, 4, 9, 16, 25, 36, 49, 64, 81} where S = the square numbers between 1 and 99.
{3, 5, 7, 11, 13, 17, 19, 23, 29} where S = the odd primes below 30.
There are nine of these examples of sets containing nine members, so taken together they provide us with a further instance of just such a set.
Like colors, the commonest numbers give character, form, and dimension to our world. Of the most frequent—zero and one—we might say that they are like black and white, with the other primary colors—red, blue, and yellow—akin to two, three, and four. Nine, then, might be a sort of cobalt or indigo: in a painting it would contribute shading, rather than shape. We expect to come across samples of nine as we might samples of a color like indigo—only occasionally, and in small and subtle ways. Thus a family of nine children surprises as much as a man or woman with cobalt-colored hair.
I would like to suggest another reason for the surprise of my town's residents. I have alluded to the various and alternating combinations and recombinations between my siblings. In how many ways can any set of nine members divide and combine? Put another way, how large is the set of all subsets?
{Daniel} ... {Daniel, Lee} ... {Lee, Claire, Steven} ... {Paul} ... {Lee, Steven, Maria, Shelley} ... {Claire, Natasha} ... {Anna}...
Fortunately, this type of calculation is very familiar to mathematicians. As it turns out, we need only to multiply the number two by itself, as many times as there are members in the set. So, for a set consisting of nine members the answer to our question amounts to: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512.
This means that there existed in my hometown, at any given place and time, 512 different ways to spot us in one or another combination. 512! It becomes clearer why we attracted so much attention. To the other residents, it really must have seemed that we were legion.
Here is another way to think about the calculation that I set out above. Take any site in the town at random, say a classroom or the municipal swimming pool. The first "2" in the calculation indicates the odds of my being present there at a particular moment (one in two—I am either there, or I am not). The same goes for each of my siblings, which is why two is multiplied by itself a total of nine times.
In precisely one of the possible combinations, every sibling is absent (just as in one of the combinations we are all present). Strange as it may sound, we can even define those sets containing no objects. Mathematicians call them an "empty set." Where sets of nine members embody everything we can think of, touch, or point to when we use the number nine, empty sets are all those that are represented by the value zero. So while a Christmas reunion in my hometown can bring together as many of us as there are Justices on the U.S. Supreme Court, a trip to the moon will unite only as many of us as there are pink elephants, four-sided circles, or people who have swum the breadth of the Atlantic Ocean.
Our mind uses sets when we think and when we perceive just as much as when we count. Our possible thoughts and perceptions about these sets can range almost without limit. Fascinated by the different cultural subdivisions and categories of an infinitely complex world, the Argentine writer Jorge Luis Borges offers a mischievously tongue-in-cheek illustration in his fictional Chinese encyclopedia entitled The Celestial Emporium of Benevolent Knowledge:
Animals are classified as follows: (a) those that belong to the Emperor; (b) embalmed ones; (c) those that are trained; (d) suckling pigs; (e) mermaids; (f) fabulous ones; (g) stray dogs; (h) those that are included in this classification; (i) those that tremble as if they were mad; (j) innumerable ones; (k) those drawn with a very fine camel's-hair brush; (l) et cetera; (m) those that have just broken the flower vase; (n) those that at a distance resemble flies.
Never one to forgo humor in his texts, Borges here also makes several thought- provoking points. First, though a set as familiar to our understanding as that of "animals" implies containment and comprehension, the sheer number of its possible subsets actually swells toward infinity. With their handful of generic labels ("mammal," "reptile," "amphibious," etc.), standard taxonomies conceal this fact. To say, for example, that a flea is tiny, parasitic, and a champion jumper is only to begin to scratch the surface of all its various aspects.
Second, defining a set owes more to art than it does to science. Faced with the problem of a near endless number of potential categories, we are inclined to choose from a few—those most tried and tested within our particular culture. Western descriptions of the set of all elephants privilege subsets like "those that are very large," and "those possessing tusks," and even "those possessing an excellent memory," while excluding other equally legitimate possibilities such as Borges's "those that at a distance resemble flies," or the Hindu "those that are considered lucky."
Memory is a further example of the privileging of certain subsets (of experience) over others, in how we talk and think about a category of things. Asked about his birthday, a man might at once recall the messy slice of chocolate cake that he devoured, his wife's enthusiastic embrace, and the pair of fluorescent green socks that his mother presented to him. At the same time, many hundreds, or thousands, of other details likewise composed his special day, from the mundane (the crumbs from his morning toast that he brushed out of his lap) to the peculiar (a sudden hailstorm on the mid-July afternoon that lasted several minutes). Most of these subsets, though, would have completely slipped his mind.
Returning to Borges's list of subsets of animals, several of the categories pose paradoxes. Take, for example, the subset (j): "innumerable ones." How can any subset of something—even if it is imaginary, like Borges's animals—be infinite in size? How can a part of any collection not be smaller than the whole?
Borges's taxonomy is clearly inspired by the work of Georg Cantor, a nineteenth- century German mathematician whose important discoveries in the study of infinity provide us with an answer to this paradox.
Cantor showed, among other things, that parts of a collection (subsets) as great as the whole (set) really do exist. Counting, he observed, involves matching the members of one set to another. "Two sets A and B have the same number of members if and only if there is a perfect one-to-one correspondence between them." So, by matching each of my siblings and myself to a player on a baseball team, or to a month of the year not beginning with the letter J, I am able to conclude that each of these sets is equivalent, all containing precisely nine members.
Next came Cantor's great mental leap: in the same manner, he compared the set of all natural numbers (1, 2, 3, 4, 5 ...) with each of its subsets such as the even numbers (2, 4, 6, 8, 10 ...), odd numbers (1, 3, 5, 7, 9 ...), and the primes (2, 3, 5, 7, 11 ...). Like the perfect matches between each of the baseball team players and my siblings and me, Cantor found that for each natural number he could uniquely assign an even, an odd, and a prime number. Incredibly, he concluded, there are as "many" even (or odd, or prime) numbers as all the numbers combined.
Reading Borges invites me to consider the wealth of possible subsets into which my family "set" could be classified, far beyond those that simply point to multiplicity. All grown up today, some of my siblings have children of their own. Others have moved far away, to the warmer and more interesting places from where postcards come. The opportunities for us all to get together are rare, which is a great pity. Naturally I am biased, but I love my family. There is a lot of my family to love. But size ceased long ago to be our defining characteristic. We see ourselves in other ways: those that are studious, those that prefer coffee to tea, those that have never planted a flower, those that still laugh in their sleep ...
Like works of literature, mathematical ideas help expand our circle of empathy, liberating us from the tyranny of a single, parochial point of view. Numbers, properly considered, make us better people.
CHAPTER 2
ETERNITY IN AN HOUR
Once upon a time I was a child who loved to read fairy tales. Among my favorites was "The Magic Porridge Pot" by the Brothers Grimm. A poor, good-hearted girl receives from a sorceress a little pot capable of spontaneously concocting as much sweet porridge as the girl and her mother can eat. One day, after eating her fill, the mother's mind goes blank and she forgets the magic words "Stop, little pot."
So it went on cooking and the porridge rose over the edge, and still it cooked on until the kitchen and whole house were full, and then the next house, and then the whole street, just as if it wanted to satisfy the hunger of the whole world.
Only the daughter's return home, and the requisite utterance, finally brings the gooey avalanche to a belated halt.
The Brothers Grimm introduced me to the mystery of infinity. How could so much porridge emerge from so small a pot? It got me thinking. A single flake of porridge was awfully slight. Tip it inside a bowl and one would probably not even spot it for the spoon. The same held for a drop of milk, or a grain of sugar.
What if, I wondered, a magical pot distributed these tiny flakes of porridge and drops of milk and grains of sugar in its own special way? In such a way that each flake and each drop and each grain had its own position in the pot, released from the necessity of touching. I imagined five, ten, fifty, one hundred, one thousand flakes and drops and grains, each indifferent to the next, suspended here and there throughout the curved space like stars. More porridge flakes, more drops of milk, more grains of sugar are added one after another to this evolving constellation, forming microscopic Big Dippers and minuscule Great Bears. Say we reach the ten thousand four hundred and seventy-third flake of porridge. Where do we include it? And here my child's mind imagined all the tiny gaps—thousands of them—between every flake of porridge and drop of milk and grain of sugar. For every minute addition, further tiny gaps would continue to be made. So long as the pot magically prevented any contact between them, every new flake (and drop and granule) would be sure to find its place.
Hans Christian Andersen's "The Princess and the Pea" similarly sent my mind spinning toward the infinite, but this time, an infinity of fractions. One night, a young woman claiming to be a princess knocks at the door of a castle. Outside, a storm is blowing and the pelting rain musses her clothes and turns her golden hair black. So sorry a sight is she that the queen of the castle doubts her story of high birth. To test the young woman's claim, the queen decides to place a pea beneath the bedding on which the woman will sleep for the night (princesses being most delicate creatures!). Her bed is piled to a height of twenty mattresses. But in the morning the woman admits to having hardly slept a wink.
The thought of all those tottering mattresses kept me up long past my own bedtime. By my calculation, a second mattress would double the distance between the princess's back and the offending pea. The tough little legume would therefore be only half as prominent as before. Another mattress reduces the pea's prominence to one-third. But if the young princess's body is sensitive enough to detect one-half of a pea (under two mattresses) or one-third of a pea (under three mattresses), why would it not also be sensitive enough to detect one-twentieth? In fact, possessing limitless sensitivity (this is a fairy tale after all), not even one-hundredth, or one-thousandth or one-millionth of a pea could be tolerably borne.
Which brings us back to the Brothers Grimm and their tale of porridge. For the princess, even a pea felt infinitely big; for the poor daughter and her mother, even an avalanche of porridge reduced to the infinitesimally small.
"You have too much imagination," my father said when I shared these thoughts with him. "You always have your nose in some book." My father kept a pile of paperbacks and regularly bought the weekend papers, but he was never a particularly enthusiastic reader. "Get outdoors more—there's no good being cooped up in here."
Hide-and-seek in the park with my brothers and sisters lasted all of ten minutes. The swings held my attention for about as long. We walked the perimeter of the lake and threw breadcrumbs out onto the grimy water. Even the ducks looked bored.
Games in the yard offered greater entertainment. We fought wars, cast spells, and traveled back in time. In a cardboard box we sailed along the Nile; with a bedsheet we pitched a tent in the hills of Rome. At other times, I would simply stroll the local streets to my heart's content, dreaming up all manner of new adventures and imaginary expeditions.
(Continues...)
Excerpted from Thinking In Numbers by Daniel Tammet. Copyright © 2014 Daniel Tammet. Excerpted by permission of Little, Brown and Company.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
Preface ix
1 Family Values 3
2 Eternity in an Hour 11
3 Counting to Four in Icelandic 21
4 Proverbs and Times Tables 33
5 Classroom Intuitions 44
6 Shakespeare's Zero 55
7 Shapes of Speech 63
8 On Big Numbers 74
9 Snowman 85
10 Invisible Cities 94
11 Are We Alone? 105
12 The Calendar of Omar Khayyám 117
13 Counting by Elevens 126
14 The Admirable Number Pi 134
15 Einstein's Equations 148
16 A Novelist's Calculus 161
17 Book of Books 174
18 Poetry of the Primes 183
19 All Things Are Created Unequal 195
20 A Model Mother 206
21 Talking Chess 219
22 Selves and Statistics 232
23 The Cataract of Time 247
24 Higher than Heaven 255
25 The Art of Math 263
Acknowledgments 273