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

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

The pleasure of X

An amusing trip 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 / Stephen 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 the fundamental math ideas simply and elegantly, giving 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

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.

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 that they had missed something O great, 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.

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.

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.

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

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.

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.

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. Computing helped solve many problems that arose in the ancient world, and this ultimately led to a revolution in science and the modern world.

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...

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

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 only get acquainted with the entertaining side of arithmetic in the process of studying higher mathematics {3}. However, it is as natural as a child's curiosity {4}.

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. {5}

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,

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

The pleasure of X

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 / Stephen 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

One day in May last year, I was sitting as an assistant on test work in mathematics in 10th grade. Bored, I took the “extra” version of the work from the teacher’s table and began to solve it. The work was done in the Unified State Exam format in mathematics, which I finished studying back in 1989, after graduating from high school. However, without much effort I managed to solve 11 tasks in part B- more than many who wrote the paper that day. One of the students +Yulia Soboleva , watched with surprise as I decided, and then came up to me:

— This is the first time I have seen an assistant who is not a math teacher sit and solve. Sorry for the question, but has this been useful to you in any way in your life?

The tenth grader’s question did not stump me. The fact is that with mathematics at school I had a love without reciprocity: in the sense that mathematics loved me, and I loved her.- No. That is, mathematics was always easy for me, there were no problems, I also remember all my mathematics teachers with warmth... But I didn’t like mathematics, and that’s all! This is how it happens. And, having entered humanitarian university(I am a history teacher by training), I suddenly began to acutely feel the lack of mathematics. It began to seem to me that I was becoming stupid by leaps and bounds. And therefore, on 1— In 2 years, to fill this void, I myself (!) took and solved collections of Olympiad problems, and solved the entire textbook for the senior year in a new way. AND- oh, miracle! Clarity of mind and logical thinking began to gradually return. And then, already in the 3rd year,I read L. Carroll's book "The Logic Game" (thanks Sergei Mikhelson), I became interested in logic and the need to study mathematics somehow disappeared. And when, a couple of years after graduation, I started teaching economics, mathematics became firmly established in my mind.- Problems need to be solved somehow.
Why did I write all this? Such a long preface is intended to explain why I gladly accepted the offer +Natalia Shanina, assistant project manager of the publishing house +Mann, Ivanov and Ferber, take the book “The Pleasure of X” for review (such a verbal pun).
I liked the book from the first pages: I love it when they show beauty mathematics. I also love it when there are patterns in simple things. Therefore, already in the first chapter, I was shocked by the discovery: if we add odd numbers in succession, then in the sum we will get the squares of numbers corresponding to the number of odd numbers taken in the series. Then- that odd numbers form corners from which you can make a square, like this, for example:

As I read the book, I made new discoveries for myself. Having a love for different algorithms (I strive to derive an algorithm even in some creative and near-creative processes), I could not help but notice a simple algorithm for squaring numbers up to 50. I liked it so much that I even sketched it in a notebook.

Geometric solution quadratic equations made me delighted: it seemed like I had never experienced difficulties in solving them, but, meanwhile, the discriminant and root formulas seemed something abstract. But if you add geometry, everything becomes obvious and understandable.

What about the tasks? Oh, these tasks that require not so much mathematics as logic and attentiveness. Who among you has not encountered problems like: “If you turn on a cold water tap, the bathtub will fill in half an hour; if you turn on a hot water tap, it will fill in an hour. How long will it take to fill the bathtub when both taps are turned on?” The apparent simplicity of the task usually leads to the answer “45 minutes.” The answer, of course, is incorrect. But can you explain why the correct answer is- "20 minutes"? And do it in different ways? But the author of the book does it brilliantly.

Even reading those sections of the book that turned out to be difficult for me (well, I don’t remember mathematics to such an extent) was easy. I didn’t understand everything, but I enjoyed reading it in this case too. Because the author sees a specific application in everything mathematical laws in the surrounding reality. Statistics, oncology, even choosing a marriage partner - there are traces of mathematics everywhere. And this quote especially touched me: “Back in the days before Google existed, searching the web was a hopeless endeavor.”.

Only two things bothered me while reading.

Well, I don’t like reading in electronic format. Moreover, in the case of mathematics, you immediately want to solve/calculate something. If I were reading a paper book, I would write directly on the margins and free pages - books from the publishing house +Mann, Ivanov and Ferber published in such a way that they initially assume that there will be readers who will not only read the book, but also write in it.
The book contains a large number of notes. The publisher traditionally leaves in the text of the book only references with brief information, and makes detailed notes in the form of endnotes. For me, this reading format is inconvenient (and in electronic format it is doubly inconvenient). I don't like jumping back and forth through the book. And reading the notes after reading the main text is illogical. In the end I just looked through them with my eyes. Although they deserve to be part of the main text: they are written interestingly, in the same style as the text of the book.

I would recommend this book not only to mathematics lovers, but also to high school and college students. To provide an understanding of some things that seem too abstract in a school or university course. Well, and math teachers, of course. Here +Natalia Lvova I've already read (review). I would really like to recommend this book and +Diana Sonina, but - alas and ah! — the daughter follows the same path as her mother. Mathematics is easy, it's a winner municipal Olympiad, but what they do with their math teacher with degrees in research work(with whom she has won prizes more than onceat various conferences), solving Olympiad problems for high school students, is difficult for me to understand. But at the same time he doesn’t even want to hear about mathematics. Necessary- he does, but without pleasure.Meanwhile, when answering my student’s question about how mathematics has been useful to me in life, in addition to some pragmatic things, I always have an answer in store: you need to study well at school, including in order to be able to help your own children study. But my daughter doesn’t really need my help.- she copes on her own. Therefore, the question remains open: why, given excellent starting conditions - good teacher, good ability in the subject, are there children who do not like mathematics? I was discussing this the other day with +Marina Kurvits, I’m ready to discuss this with other “mathematician acquaintances” -+Jüri Kurvits And +Ljudmilla Rozhdestvenskaja. What is the reason? I nIs it necessary to somehow change the situation? For me it was resolved in my youth. But I am still haunted by the thought that, having not fallen in love with mathematics earlier, I missed some opportunities in my life...

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