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A pleasure to read x. The Pleasure of X. A fascinating excursion into the world of mathematics from one of the best teachers in the world - Stephen Strogatz. Steven StrogatzPleasure of X. A fascinating journey into the world of mathematics from one of the best teachers

A pleasure to read x. The Pleasure of X. A fascinating excursion into the world of mathematics from one of the best teachers in the world - Stephen Strogatz. Steven StrogatzPleasure of X. A fascinating journey into the world of mathematics from one of the best teachers

Jul 25, 2017

The Pleasure of X. A fascinating journey into the world of mathematics from one of the best teachers in the world Stephen Strogatz

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Title: The Pleasure of X. A fascinating journey into the world of mathematics from one of the best teachers in the world

About the book “The Pleasure of X: A Fascinating Journey into the World of Mathematics from One of the Best Teachers in the World” by Stephen Strogatz

This book can radically change your attitude towards mathematics. It consists of short chapters, in each of which you will discover something new. You will learn how useful numbers are for studying the world around you, you will understand the beauty of geometry, you will become acquainted with the grace of integral calculus, you will be convinced of the importance of statistics and you will come into contact with infinity. The author explains the fundamental math ideas simply and elegantly, giving brilliant examples that everyone can understand.

Published in Russian for the first time.

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Mathematics is the most accurate and universal language science, but is it possible to explain with the help of numbers human feelings? Formulas of love, seeds of chaos and romance differential equations- T&P publishes a chapter from the book “The Pleasure of X” by one of the best mathematics teachers in the world, Stephen Strogatz, published by Mann, Ivanov and Ferber.

In the spring, wrote Tennyson, imagination young man easily turns to thoughts of love. Unfortunately, a young man's potential partner may have his own ideas about love, and then their relationship will be full of the stormy ups and downs that make love so exciting and so painful. Some sufferers from unrequited love seek an explanation for these love swings in wine, others in poetry. And we will consult the calculus.

The analysis below will be tongue-in-cheek, but it touches on serious topics. Moreover, while understanding the laws of love may elude us, the laws of the inanimate world are now well studied. They take the form of differential equations that describe how interrelated variables change from moment to moment depending on their current values. Such equations may have little to do with romance, but they can at least shed light on why, in the words of another poet, “the path of true love never runs smooth.” To illustrate the method of differential equations, suppose that Romeo loves Juliet, but in our version of the story Juliet is a flighty lover. The more Romeo loves her, the more she wants to hide from him. But when Romeo grows cold towards her, he begins to seem unusually attractive to her. However, the young lover tends to reflect her feelings: he glows when she loves him, and cools down when she hates him.

What happens to our star-crossed lovers? How does love consume them and fade away over time? This is where differential calculus comes to the rescue. By creating equations that summarize the waxing and waning feelings of Romeo and Juliet, and then solving them, we can predict the course of the couple's relationship. The ultimate prognosis for her will be a tragically endless cycle of love and hate. At least a quarter of this time they will have mutual love.

To reach this conclusion, I assumed that Romeo's behavior could be modeled using a differential equation,

which describes how his love ® changes in the next moment (dt). According to this equation, the amount of change (dR) is directly proportional (with proportionality coefficient a) to Juliet's love (J). This relationship reflects what we already know: Romeo's love increases when Juliet loves him, but it also suggests that Romeo's love increases in direct proportion to how much Juliet loves him. This is an assumption linear dependence emotionally implausible, but it makes solving the equation much easier.

In contrast, Juliet's behavior can be modeled using the equation

The negative sign in front of the constant b reflects that her love is cooling as Romeo's love intensifies.

The only thing left to determine is their initial feelings (that is, the values of R and J at time t = 0). After this, all the necessary parameters will be set. We can use the computer to move forward slowly, step by step, changing the values of R and J according to the differential equations described above. In fact, using the fundamental theorem of integral calculus, we can find the solution analytically. Because the model is simple, integral calculus produces a pair of comprehensive formulas that tell us how much Romeo and Juliet will love (or hate) each other at any point in time in the future.

The differential equations presented above should be familiar to physics students: Romeo and Juliet behave as simple harmonic oscillators. Thus, the model predicts that the functions R (t) and J (t), describing the change in their ratios over time, will be sinusoids, each of them increasing and decreasing, but maximum values they don't match.

“The stupid idea to describe love relationships using differential equations came to me when I was in love for the first time and was trying to understand the incomprehensible behavior of my girlfriend.”

The model can be made more realistic in different ways. For example, Romeo may react not only to Juliet's feelings, but also to his own. What if he is one of those guys who is so afraid of being abandoned that he begins to cool his feelings. Or he belongs to another type of guy who loves to suffer - that’s why he loves her.

Add to these scenarios two more behaviors of Romeo: he responds to Juliet's affection by either increasing or weakening his own affection - and you will see that there are four different styles of behavior in a love relationship. My students and the students of Peter Christopher's group at Worcester Polytechnic Institute suggested calling representatives of these types like this: the Hermit or Evil Misanthrope for the Romeo who cools his feelings and distances himself from Juliet, and the Narcissistic Blockhead and Flirting Fink for the one who warms up his ardor, but rejected by Juliet. (You can think of proper names for all these types).

Although the examples given are fantastic, the types of equations that describe them are quite insightful. They represent the most powerful tools humanity has ever created for understanding material world. Sir Isaac Newton used differential equations to discover the secret of planetary motion. Using these equations, he combined earthly and celestial spheres, showing that the same laws of motion apply to both.

Almost 350 years after Newton, humanity has come to understand that the laws of physics are always expressed in the language of differential equations. This is true for the equations that describe the flow of heat, air and water, for the laws of electricity and magnetism, even for the atom, where quantum mechanics reigns.

In all cases, theoretical physics must find the correct differential equations and solve them. When Newton discovered this key to the secrets of the Universe and realized its great significance, he published it in the form of a Latin anagram. Loosely translated, it sounds like this: “It is useful to solve differential equations.”

The stupid idea to describe love relationships using differential equations came to me when I was in love for the first time and was trying to understand the incomprehensible behavior of my girlfriend. It was a summer romance at the end of my sophomore year of college. I then very much resembled the first Romeo, and she - the first Juliet. The cyclical nature of our relationship drove me crazy until I realized that we were both acting out of inertia, in accordance with a simple push-pull rule. But by the end of the summer, my equation began to fall apart, and I became even more confused. It turned out that an important event happened that I did not take into account: her ex-lover wanted her back.

In mathematics we call this problem the three-body problem. It is obviously unsolvable, especially in the context of astronomy, where it first arose. After Newton solved the differential equations for the two-body problem (which explains why the planets move in elliptical orbits around the Sun), he turned his attention to the three-body problem for the Sun, Earth, and Moon. Neither he nor other scientists were able to solve it. It was later discovered that the three-body problem contained the seeds of chaos, meaning that in the long term their behavior was unpredictable.

Newton knew nothing about chaos dynamics, but, according to his friend Edmund Halley, he complained that the three-body problem caused headache and keeps him awake so often that he won't think about it anymore.

Here I am with you, Sir Isaac.

In 2010, Steven Strogatz wrote a series of articles about the basics of mathematics for The New York Times. The articles caused a storm of delight. Each column became the most popular story in the newspaper and attracted hundreds of comments. Readers asked for more, and Stephen did not disappoint - this book appeared, which included both already published parts and completely new chapters.

Mathematics permeates everything in this world, including ourselves, but, unfortunately, few people understand this universal language well enough to appreciate its wisdom and beauty. Steven Strogatz is the math teacher you dreamed of in high school. A teacher who is able to ignite a spark of interest and instill a lifelong love for his subject. In this incredibly easy and fun book, he gives us all a second chance to get to know mathematics. In each short chapter, you'll discover something new, from why numbers are needed in the first place to topics such as geometry, integral calculus, statistics, and infinity. The author explains great mathematical ideas simply and elegantly, with brilliant examples that everyone can understand. This book is for everyone. Those who are little familiar with mathematics will become closely acquainted with it, and those who love mathematics will enjoy reading about the “queen of sciences.”

Preface

I have a friend who, despite his craft (he is an artist), is passionate about science. Whenever we get together, he talks enthusiastically about the latest advances in psychology or quantum mechanics. But as soon as we start talking about mathematics, he feels a trembling in his knees, which greatly upsets him. He complains that not only do these strange mathematical symbols defy his understanding, but sometimes he doesn't even know how to pronounce them.

In fact, the reason for his rejection of mathematics is much deeper. He will have no idea what mathematicians do in general and what they mean when they say that a given proof is elegant. Sometimes we joke that I just need to sit down and start teaching him from the very basics, literally 1 + 1 = 2, and go as deep into math as he can.

And although this idea seems crazy, this is exactly what I will try to implement in this book. I will guide you through all the main branches of science, from arithmetic to higher mathematics so that those who wanted a second chance could finally take advantage of it. And this time you won't have to sit at a desk. This book will not make you a math expert. But it will help you understand what this discipline studies and why it is so fascinating for those who understand it.

We'll explore how Michael Jordan's slam dunks can help explain basic calculus. I'll show you a simple and amazing way to understand the fundamental theorem of Euclidean geometry - the Pythagorean Theorem. We'll try to get to the bottom of some of life's mysteries, big and small: did Jay Simpson kill his wife; how to reposition a mattress so that it lasts as long as possible; how many partners need to be changed before getting married - and we will see why some infinities are larger than others.

Mathematics is everywhere, you just need to learn to recognize it. You can see the sine wave on the zebra's back, hear echoes of Euclid's theorems in the Declaration of Independence; what can I say, even in the dry reports that preceded the First World War, there are negative numbers. You can also see how new areas of mathematics influence our lives today, for example, when we search for restaurants using the computer or try to at least understand, or better yet, survive the frightening fluctuations of the stock market.

— Read online book “The Pleasure of X” by Stephen Strogatz —

A series of 15 articles under common name“Fundamentals of Mathematics” appeared online at the end of January 2010. In response to their publication, letters and comments poured in from readers of all ages, including many students and teachers. There were also simply inquisitive people who, for one reason or another, “lost the path” of comprehension mathematical science; now they felt they had missed something worthwhile and wanted to try again. I was especially pleased by the gratitude from my parents because, with my help, they were able to explain mathematics to their children, and they themselves began to understand it better. It seemed that even my colleagues and comrades, ardent admirers of this science, enjoyed reading the articles, except for those moments when they vied with each other to offer all sorts of recommendations for improving my brainchild.

Despite popular belief, there is a clear interest in mathematics in society, although little attention is paid to this phenomenon. All we hear about is fear of math, and yet many would love to try to understand it better. And once this happens, it will be difficult to tear them away.

This book will introduce you to the most complex and advanced ideas from the world of mathematics. The chapters are small, easy to read and not particularly dependent on each other. Among them are those included in that first series of articles in the New York Times. So, as soon as you feel a slight mathematical hunger, don’t hesitate to pick up the next chapter. If you want to understand in more detail a question that interests you, then at the end of the book there are notes with additional information and recommendations on what else you can read about this.

The Pleasure of X - Steven Strogatz (download)

(introductory version)

And finally, we suggest you watch an interesting video

The Joy of X

A Guided Tour of Math, from One to Infinity

Published with permission from Steven Strogatz, c/o Brockman, Inc.

All rights reserved. No part of the electronic version of this book may be reproduced in any form or by any means, including posting on the Internet or corporate networks, for private or public use without the written permission of the copyright owner.

Legal support for the publishing house is provided by the Vegas-Lex law firm.

* * *

This book is well complemented by:

Quanta

Scott Patterson

Brainiac

Ken Jennings

Moneyball

Michael Lewis

Flexible consciousness

Carol Dweck

Physics of the stock market

James Weatherall

Preface

I have a friend who, despite his craft (he is an artist), is passionate about science. Whenever we get together, he talks enthusiastically about the latest developments in psychology or quantum mechanics. But as soon as we start talking about mathematics, he feels a trembling in his knees, which greatly upsets him. He complains that not only do these strange mathematical symbols defy his understanding, but sometimes he doesn't even know how to pronounce them.

And although this idea seems crazy, this is exactly what I will try to implement in this book. I will guide you through all the major branches of science, from arithmetic to higher mathematics, so that those who wanted a second chance can finally take advantage of it. And this time you won't have to sit at a desk. This book will not make you a math expert. But it will help you understand what this discipline studies and why it is so fascinating for those who understand it.

To clarify what I mean by the lives of numbers and their behavior that we cannot control, let's go back to the Furry Paws Hotel. Suppose that Humphrey was just about to hand over the order, but then the penguins from another room unexpectedly called him and also asked for the same amount of fish. How many times must Humphrey shout the word "fish" after receiving two orders? If he didn't learn anything about numbers, he would have to scream as many times as there are penguins in both rooms. Or, using numbers, he could explain to the cook that he needed six fish for one number and six for another. But what he really needs is a new concept: addition. Once he's mastered it, he'll proudly say that he needs six plus six (or, if he's a poser, twelve) fish.

This is the same creative process as when we first came up with numbers. Just as numbers make counting easier than listing one at a time, addition makes it easier to calculate any amount. At the same time, the one who does the calculation develops as a mathematician. Scientifically, this idea can be formulated as follows: using the right abstractions leads to deeper insight into the essence of the issue and greater power in solving it.

Soon, perhaps, even Humphrey will realize that now he can always count.

However, despite such an endless perspective, our creativity always has some limitations. We can decide what we mean by 6 and +, but once we do, the results of expressions like 6 + 6 are beyond our control. Here logic will leave us no choice. In this sense, mathematics always includes both invention, so and opening: we invent concept, but open their consequences. As the following chapters will make clear, in mathematics our freedom lies in the ability to ask questions and persist in seeking answers without having to invent them ourselves.

2. Stone arithmetic

Like any phenomenon in life, arithmetic has two sides: formal and entertaining (or playful).

We studied the formal part at school. There they explained to us how to work with columns of numbers, adding and subtracting them, how to crunch them when doing calculations in spreadsheets when filling out tax returns and preparing annual reports. This side of arithmetic seems important to many from a practical point of view, but completely joyless.

You can become acquainted with the entertaining side of arithmetic only in the process of studying higher mathematics. However, it is as natural as a child's curiosity.

In the essay "The Mathematician's Lament," Paul Lockhart suggests studying numbers in more concrete examples than usual: he asks us to think of them as a number of stones. For example, the number 6 corresponds to the following set of pebbles:

You are unlikely to see anything unusual here. The way it is. Until we start manipulating the numbers, they look pretty much the same. The game begins when we receive a task.

For example, let's look at sets that contain from 1 to 10 stones and try to make squares out of them. This can only be done with two sets of 4 and 9 stones, since 4 = 2 × 2 and 9 = 3 × 3. We get these numbers by squaring some other number (that is, arranging the stones in a square).

Here is a task that has larger number solutions: you need to find out which sets will make a rectangle if you arrange the stones in two rows with an equal number of elements. Sets of 2, 4, 6, 8 or 10 stones are suitable here; the number must be even. If we try to arrange the remaining sets with an odd number of stones in two rows, we will invariably end up with an extra stone.

But all is not lost for these awkward numbers! If you take two such sets, then the extra elements will find a pair, and the sum will be even: odd number + odd number = even number.

If we extend these rules to numbers after 10, and assume that the number of rows in a rectangle can be more than two, then some odd numbers will allow such rectangles to be added. For example, the number 15 can form a 3 × 5 rectangle.

Therefore, although 15 is undoubtedly an odd number, it is a composite number and can be represented as three rows of five stones each. Likewise, any entry in the multiplication table produces its own rectangular group of pebbles.

But some numbers, like 2, 3, 5 and 7, are completely hopeless. You can't lay out anything from them except to arrange them in the form of a simple line (one row). These strange stubborn people are the famous prime numbers.

So we see that numbers can have weird structures that give them a certain character. But to understand the full range of their behavior, you need to step back from individual numbers and observe what happens during their interaction.

For example, instead of adding just two odd numbers, let's add all possible sequences of odd numbers, starting with 1:

1 + 3 + 5 + 7 = 16

1 + 3 + 5 + 7 + 9 = 25

Surprisingly, these sums always turn out to be perfect squares. (We already said that 4 and 9 can be represented as squares, and for 16 = 4 × 4 and 25 = 5 × 5 this is also true.) A quick calculation shows that this rule is also true for larger odd numbers and , apparently, tends to infinity. But what is the connection between odd numbers with their “extra” stones and the classically symmetrical numbers that form squares? By placing the pebbles correctly, we can make it obvious, which is the hallmark of an elegant proof.

The key to it is the observation that odd numbers can be represented as equilateral angles, the successive overlap of which forms a square!

A similar way of reasoning is presented in another recently published book. In Yoko Ogawa's charming novel The Housekeeper and the Professor is about a shrewd but uneducated young woman and her ten-year-old son. A woman was hired to care for an elderly mathematician whose short-term memory, due to a traumatic brain injury, only retains information about the last 80 minutes of his life. Lost in the present, alone in his squalid cottage, with nothing but numbers, the professor tries to communicate with the housekeeper the only way he knows: by asking about her shoe size or date of birth and making small talk with her about her expenses. The professor also takes a special liking to the housekeeper's son, whom he calls Ruth (Root), because the boy has a flat head on top, and this reminds him of the notation in mathematics square root √.

One day the professor offers the boy simple task– find the sum of all the numbers from 1 to 10. After Ruth carefully adds all the numbers together and returns with the answer (55), the professor asks him to look for an easier way. Will he be able to find the answer? without ordinary addition of numbers? Ruth kicks a chair and screams, “It’s not fair!”

Little by little, the housekeeper also gets drawn into the world of numbers and secretly tries to solve this problem herself. “I don’t understand why I’m so interested in a children’s puzzle that has no practical use,” she says. “At first I wanted to please the professor, but gradually this lesson turned into a battle between me and the numbers. When I woke up in the morning, the equation was already waiting for me:

1 + 2 + 3 + … + 9 + 10 = 55,

and it followed me around all day, as if it was burned into the retinas of my eyes, and there was no way I could ignore it.” There are several ways to solve the professor's problem (I wonder how many you can find). The professor himself suggests a method of reasoning, which we have already applied above. He interprets the sum from 1 to 10 as a triangle of pebbles, with one pebble in the first row, two in the second, and so on, up to ten pebbles in the tenth row.

This picture gives a clear idea of negative space. It turns out that it is only half full, which shows the direction of the creative breakthrough. If you copy a triangle made of pebbles, turn it over and combine it with an existing one, you get something very simple: a rectangle with ten rows of 11 pebbles in each, and total number stones will be 110.

Since the original triangle is half of this rectangle, the calculated sum of the numbers from 1 to 10 must be half of 110, that is, 55.

Representing a number as a group of pebbles may seem unusual, but it is actually as old as mathematics itself. The word "calculate" calculate) reflects this heritage and is derived from Latin calculus, meaning "pebble", which the Romans used when performing calculations. You don't have to be an Einstein (which means "one stone" in German) to enjoy manipulating numbers, but maybe being able to juggle pebbles will make it easier for you.

A slam dunk is a type of basketball shot in which a player jumps up and throws the ball through the hoop from top to bottom with one or two hands. Note translation

Jay Simpson is a famous American football player. He played the role of Detective Northberg in the famous “Naked Gun” trilogy. He was accused of murdering his ex-wife and her friend and was acquitted despite the evidence. Note translation

To get acquainted with the fascinating idea that numbers live own life, and mathematics can be considered an art form, see P. Lockhart, A Mathematician's Lament (Bellevue Literary Press, 2009). Note ed.: There are many translations of Lockhard’s essay “The Cry of a Mathematician” on the Russian Internet. Here is one of them: http://mrega.ru/biblioteka/obrazovanie/130-plachmatematika.html. Here and below, footnotes in curly brackets refer to the author's notes.

This famous phrase is taken from E. Wigner's essay The unreasonable effectiveness of mathematics in the natural sciences, Communications in Pure and Applied Mathematics, Vol. 13, No. 1, (February 1960), pp. 1–14. The online version is available at http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html.

For further thoughts on this topic, and whether mathematics was invented or discovered, see M. Livio, Is God a Mathematician? (Simon and Schuster, 2009) and R. W. Hamming, The unreasonable effectiveness of mathematics, American Mathematical Monthly, Vol. 87, No. 2 (February 1980).

For young readers who want to explore numbers and their structures, see H. M. Enzensberger, The Number Devil (Holt Paperbacks, 2000). Note ed.: Among the numerous Russian books about the beginnings of mathematics, non-standard approaches to its study, the development of mathematical creativity in children and similar topics consonant with the following chapters of the book, we will indicate the following for now: Pukhnachev Yu., Popov Yu. Mathematics without formulas. M.: JSC "Stoletie", 1995; Oster G. Problem book. Beloved guide to mathematics. M.: AST, 2005; Ryzhik V.I. 30,000 mathematics lessons: A book for teachers. M.: Education, 2003: Tuchnin N.P. How to ask a question? About mathematical creativity of schoolchildren. Yaroslavl: Verkh. - Volzh. book publishing house, 1989.

Excellent, but more complex examples visualizations of mathematical images are presented in R. B. Nelsen, Proofs without Words (Mathematical Association of America, 1997).

This book is well complemented by:

Quanta

Scott Patterson

Brainiac

Ken Jennings

Moneyball

Michael Lewis

Flexible consciousness

Carol Dweck

Physics of the stock market

James Weatherall

The Joy of X

A Guided Tour of Math, from One to Infinity

Stephen Strogatz

A fascinating journey into the world of mathematics from one of the best teachers in the world

Information from the publisher

Published in Russian for the first time

Published with permission from Steven Strogatz, c/o Brockman, Inc.

Strogatz, P.

The Pleasure of X. A fascinating journey into the world of mathematics from one of the best teachers in the world / Steven Strogatz; lane from English - M.: Mann, Ivanov and Ferber, 2014.

ISBN 978-500057-008-1

This book can radically change your attitude towards mathematics. It consists of short chapters, in each of which you will discover something new. You will learn how useful numbers are for studying the world around you, you will understand the beauty of geometry, you will become acquainted with the grace of integral calculus, you will be convinced of the importance of statistics and you will come into contact with infinity. The author explains fundamental mathematical ideas simply and elegantly, with brilliant examples that everyone can understand.

No part of this book may be reproduced in any form without the written permission of the copyright holders.

Legal support for the publishing house is provided by the Vegas-Lex law firm.

Preface

A series of 15 articles under the general title “Fundamentals of Mathematics” appeared online at the end of January 2010. In response to their publication, letters and comments poured in from readers of all ages, including many students and teachers. There were also simply curious people who, for one reason or another, “lost their way” in understanding mathematical science; now they felt that they had missed something worthwhile and would like to try again. I was especially pleased by the gratitude from my parents because, with my help, they were able to explain mathematics to their children, and they themselves began to understand it better. It seemed that even my colleagues and comrades, ardent admirers of this science, enjoyed reading the articles, except for those moments when they vied with each other to offer all sorts of recommendations for improving my brainchild.

This book will introduce you to the most complex and advanced ideas from the world of mathematics. The chapters are small, easy to read and not particularly dependent on each other. Among them are those included in that first series of articles in the New York Times. So, as soon as you feel a slight mathematical hunger, don’t hesitate to pick up the next chapter. If you want to understand the issue that interests you in more detail, then at the end of the book there are notes with additional information and recommendations on what else you can read about it.

For the convenience of readers who prefer a step-by-step approach, I have divided the material into six parts in accordance with the traditional order of studying topics.

Part I "Numbers" begins our journey with arithmetic in kindergarten And primary school. It shows how useful numbers can be and how magically effective they are in describing the world around us.

Part II, “Ratios,” shifts attention from the numbers themselves to the relationships between them. These ideas lie at the heart of algebra and are the first tools for describing how one thing affects another, showing the cause-and-effect relationship of a variety of things: supply and demand, stimulus and response - in short, all the kinds of relationships that make the world so rich and varied .

Part III “Figures” tells not about numbers and symbols, but about figures and space - the domain of geometry and trigonometry. These topics, along with the description of all observable objects through shapes, logical reasoning and proof, take mathematics to a new level of precision.

In Part IV, Time for a Change, we'll look at calculus, the most exciting and diverse branch of mathematics. Calculus makes it possible to predict the trajectory of planets, the cycles of tides and make it possible to understand and describe all periodically changing processes and phenomena in the Universe and within us. An important place in this part is given to the study of infinity, the pacification of which became a breakthrough that allowed calculations to work. Calculations helped solve many problems that arose in the ancient world, and this ultimately led to a revolution in science and modern world.

Part V, “The Many Faces of Data,” deals with probability, statistics, networks, and data science—still relatively new fields, born out of the less-always orderly aspects of our lives, such as opportunity and luck, uncertainty, risk, variability, chaos, interdependence. Using the right tools of mathematics and the appropriate types of data, we will learn to detect patterns in the flow of randomness.

At the end of our journey in Part VI, “The Limits of the Possible,” we will approach the limits of mathematical knowledge, the border region between what is already known and what is as yet elusive and unknown. We will again go through the topics in the order we are already familiar with: numbers, ratios, figures, changes and infinity - but at the same time we will look at each of them in more depth, in its modern incarnation.

I hope that all the ideas described in this book will seem fascinating to you and will make you exclaim more than once: “Wow!” But you always have to start somewhere, so let's start with a simple but fascinating activity like counting.

1. Number Basics: Fish Addition

The best demonstration of number concepts I have ever seen (the clearest and funniest explanation of what numbers are and why we need them) was in an episode of the popular children's show Sesame Street called 123: Counting Together "(123 Counter with Me). X...