The pleasure of x by steven strogatz. Steven StrogatzPleasure of X

One of the May days of last year, I sat as an assistant on a math test in grade 10. Bored, I took an "extra" version of the work from the teacher's table and began to solve it. The work was done in the format of the Unified State Examination in mathematics, which I finished studying back in 1989, having graduated from high school. However, without much effort, I managed to solve 11 tasks in part B.- more than many who wrote the work 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 see an assistant who is not a math teacher sit and decide. Sorry for the question, but did it somehow come in handy in your life?

The tenth grader's question didn't baffle me. The fact is that with mathematics at school I had 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 it! That's how it happens. And, having entered a liberal arts university (I am a history teacher by education), I suddenly began to acutely feel the lack of mathematics. It began to seem to me that I was becoming stupid not by the day, but by the hour. Therefore, on 1-2 courses, in order to fill this void, she (!) took and solved collections of Olympiad problems, solved the entire textbook for the graduating class in a new way. And— oh, miracle! Clarity of mind and logical thinking began to gradually return. And then, studying already in the 3rd year,read L. Carroll's book "The Logic Game" (thanks Sergei Michelson), became interested in logic and the need for mathematics classes somehow disappeared. And when, a couple of years after graduation, I started teaching economics, mathematics firmly settled 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, publishing house +Mann, Ivanov and Ferber, take the book "The Pleasure of X" for review (such a verbal pun turned out).
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 the simple. Therefore, already in the first chapter, I was shocked by the discovery: if we add consecutively odd numbers, then in total 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 note a simple algorithm for squaring numbers up to 50. I liked it so much that I even sketched it in a notebook.

The geometric method of solving quadratic equations delighted me: it seemed that I never experienced difficulties in solving them, but, meanwhile, the discriminant and root formulas seemed to be something abstract. But, if you add geometry, everything becomes obvious and understandable.

What about tasks? Oh, these tasks that require not so much mathematics as logic and attentiveness. Who among you has not met puzzles like: "If you turn on the faucet with cold water, then the bath will be filled in half an hour, if with hot water, then in an hour. How long will it take to fill the bath when both faucets are turned on?" The apparent simplicity of the task usually leads to the answer "45 minutes". The answer, of course, is wrong. 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 in such a volume) was easy. I didn’t understand everything, but I enjoyed reading it even in this case. Because the author sees in everything a concrete application of mathematical laws in the surrounding reality. Statistics, oncology, even the choice of a partner in marriage - there are traces of mathematics everywhere. And this quote was especially touching: "Back in the days before Google didn't exist, searching the web was a hopeless endeavor".

There were only two things that got in the way.

Well, I do not like to read in electronic format. Moreover, in the case of mathematics, you immediately want to solve / calculate something. If I read a paper book, I would write directly on the margins and free pages - books of 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 has a lot of notes. The publisher traditionally leaves in the text of the book only links with brief information, and makes detailed notes in the form of endnotes. For me, this reading format is inconvenient (and doubly inconvenient in electronic format). I don't like jumping back and forth in a book. And reading notes after reading the main text is illogical. In the end, I just looked them over. Although they deserve to be part of the main text: they are written in an interesting way, in the same style as the text of the book.

I would recommend this book not only to math lovers, but also to high school students and 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 already read (review). I would like to recommend this book and +Diana Sonina but - alas! The daughter follows the same path as the mother. Mathematics is easy, she is a winner of the municipal Olympiad, and what they do with their mathematics teacher with degrees in research work (with which she 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- does, but without pleasure.And, meanwhile, when answering my student’s question about how mathematics was 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 their children learn. But my daughter doesn't really need my help.- handles herself. That is why the question remains open: why, with excellent starting conditions - a good teacher, good abilities in the subject, there are children who do not like mathematics? Discussed this the other day with +Marina Kurvits, ready to discuss this with other "familiar mathematicians" -+Juri Kurvits and +Ljudmilla Rozhdestvenskaja. What is the reason? I nis there any way to change the situation? Here I have it 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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The joy of X

A Guided Tour of Math, from One to Infinity

Stephen Strogatz

pleasure from X

An exciting 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.

Strogats, P.

pleasure from X. An exciting journey into the world of mathematics from one of the best teachers in the world / Stephen Strogatz; per. from English. - M. : Mann, Ivanov and Ferber, 2014.

ISBN 978-500057-008-1

This book is able to 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, understand the beauty of geometry, get acquainted with the elegance of integral calculus, see the importance of statistics and get in touch with infinity. The author explains fundamental mathematical 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 of the publishing house is provided by the law firm "Vegas-Lex"

Foreword

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

In fact, the reason for his dislike of mathematics is much deeper. He will never understand what mathematicians generally do and what they mean when they say that this proof is elegant. Sometimes we joke that I should just sit down and start teaching him from the very basics, literally from 1 + 1 = 2, and go into mathematics as much as he can.

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

We'll learn how Michael Jordan's slam dunks can help explain the basics of calculus. I will 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 shift the mattress so that it lasts as long as possible; how many partners need to be changed before a wedding is played - 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 sinusoid on the back of a zebra, you can 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 affect our lives today, for example, when we look for restaurants using a computer or try to at least understand, or better yet, survive the frightening fluctuations in the stock market.

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, among whom were many students and teachers. There were also simply inquisitive people who, for one reason or another, “lost their way” in comprehension of mathematical science; now they feel like they missed something. about and would like to try again. I was particularly pleased with the gratitude from my parents for the fact that 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 kinds of recommendations for improving my offspring.

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

This book will introduce you to the most complex and advanced ideas from the world of mathematics. The chapters are short, easy to read, and don't really depend 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, do not hesitate to take on 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 topics.

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

Part II "Ratios" shifts attention from the numbers themselves to the relationships between them. These ideas are at the heart of algebra and are the first tools for describing how one affects the other, showing the causal relationship of a variety of things: supply and demand, stimulus and reaction - in short, all kinds of relationships that make the world so diverse and rich. .

Part III "Figures" is 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 forms, through logical reasoning and proof, raise mathematics to a new level of precision.

In Part IV "Time of Change" we will look at calculus - the most impressive and multifaceted area of \u200b\u200bmathematics. Calculus makes it possible to predict the trajectory of the 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 devoted to the study of infinity, the pacification of which was 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.

Part V "Many Faces of Data" deals with probability, statistics, networks and data processing - these are still relatively young fields, generated by the not always ordered aspects of our lives, such as opportunity and luck, uncertainty, risk, volatility, randomness, interdependence. Using the right math tools and the right data types, we'll learn to spot patterns in a stream 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 boundary area between what is already known and what is still elusive and not known. We will again go through the topics in the order we are already familiar with: numbers, ratios, shapes, changes and infinity - but at the same time we will consider each of them in more depth, in its modern incarnation.

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The pleasure of X. An exciting journey into the world of mathematics from one of the best teachers in the world
Stephen Strogatz

Published in Russian for the first time.

Stephen Strogatz

The pleasure of X. An exciting journey into the world of mathematics from one of the best teachers in the world

Steven Strogatz

A Guided Tour of Math, from One to Infinity

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

How useful are numbers for studying the world around us, what is the beauty of geometry, how elegant are integral calculuses, and how important is statistics? Steven Strogatz talks about all this in his book The Pleasure of X. The author explains fundamental mathematical ideas simply and elegantly, giving examples that everyone can understand. the site publishes one of the chapters of the book published by the Mann, Ivanov and Ferber publishing house.

Statistics has suddenly become trendy. With the advent of the Internet, e-commerce, social media, the human genome sequencing project, and the rise of digital culture in general, the world has become flooded with data. Marketers study our tastes and habits. Intelligence services collect information about our location, emails and phone calls. Sports statisticians juggle numbers to decide which players to buy, who to recruit, and who to bench. Everyone strives to combine the dots into a graph and discover a pattern in the chaotic accumulation of data.

Not surprisingly, these trends are reflected in learning. "Let's get down to the statistics," admonishes Greg Mankiw, an economist at Harvard University, in a New York Times column.

“The high school math curriculum devotes too much time to traditional topics such as Euclidean geometry and trigonometry. These mental exercises, useful for an ordinary person, however, are of little use in everyday life. It would be much more useful for students to learn more about probability theory and statistics.” David Brooks goes even further. In his article on disciplines that deserve attention for getting a decent education, he writes: “Take statistics. You will see, it turns out that knowing what a standard deviation is will be very useful to you in life.

It is quite possible, and it is also good to understand what distribution is. This is the first thing I intend to talk about. And I would like to focus on it, because this is one of the main lessons of statistics: things seem hopelessly random and unpredictable when considered individually, but in the aggregate they reveal regularity and predictability.

You may have seen a demonstration of this principle at some science museum (if not, videos can be found online). A typical exhibit is a contraption called a Galton board, which is somewhat like a pinball machine, only without the flippers. Inside it at regular intervals are even rows of pins.

Galton board

The experiment begins with hundreds of balls being launched into the top of the Galton board. When they fall, they collide with the pins and with equal probability bounce either to the right or to the left, and then they are distributed at the bottom of the board, falling into compartments of the same width. The height of the column of balls shows the probability with which the ball can be in a given place. Most of the balls are placed approximately in the middle, there are already fewer on the sides, and even fewer on the edges.

In general, the picture is extremely predictable: the balls always form a distribution in the form of a bell, although it is impossible to predict where each individual ball will end up.

How do individual accidents turn into general patterns? But that's how randomness works. In the middle column, the most balls have accumulated because, before rolling down, many of them will make approximately the same number of jumps to the right and left, and as a result they will be somewhere in the middle. Several single balls located along the edges form distribution tails - these are the balls that, when colliding with the pins, always bounced in the same direction. Such bounces are unlikely, which is why there are so few balls around the edges.

Just as the location of each ball is determined by the sum of many random events, so many phenomena in this world are the result of many small circumstances and also obey the bell curve. This is how insurance companies operate. They can accurately name the number of their clients who die each year. However, they do not know who exactly will not be lucky this time.

Or take, for example, the height of a person. It depends on countless accidents related to genetics, biochemistry, nutrition and environment. Therefore, it is likely that, when considered together, the height of adult men and women will be a bell-shaped curve.

In a blog post titled "False Data People Report About Themselves Online," dating site statistics OkCupid recently posted a graph of the growth of their customers, or rather the values they reported. It was found that the growth rates of both sexes, as expected, form a bell-shaped curve. Surprisingly, however, both distributions were skewed to the right by about two inches from the expected values.

Strogats S. Pleasure from H. - M. : Mann, Ivanov and Ferber, 2014.

Thus, either the height of customers surveyed by OkCupid is above average, or they add a couple of inches to their height when describing themselves online.

An idealized version of these bell curves is what mathematicians call a normal distribution. This is one of the most important concepts in statistics, which has a theoretical justification. It can be shown that the normal distribution arises from the addition of a large number of small random factors, each of which acts independently of the others. And many things happen that way.

But not all. And this is the second point to which I would like to draw attention. The normal distribution is not as ubiquitous as it seems. For a hundred years, and especially in the last few decades, scientists and statisticians have noted the existence of many phenomena that deviate from this curve and follow their own schedule. It is curious that such types of distributions are practically not mentioned in textbooks on elementary statistics, and if they occur, they are usually considered as some kind of pathology.

This is strange. I will try to explain that many phenomena of modern life make more sense if these "pathological" distributions are understood. This is the new normal. Take, for example, the distribution of city sizes in the United States. Instead of clustering around some average bell curve, the vast majority of cities are small and therefore cluster on the left side of the graph.

Strogats S. Pleasure from H. - M. : Mann, Ivanov and Ferber, 2014.

And the larger the population of the city, the less often such cities are found. In other words, in the aggregate, the distribution will be an L-shaped curve rather than a bell curve.

And there is nothing surprising in this. Everyone knows that there are much fewer megacities than small towns. Although it is not so obvious, the sizes of cities follow a simple beautiful distribution - if you look at them on a logarithmic scale.

We will assume that the difference between two cities is the same if their population differs by the same number of times (just as any two piano keys separated by an octave always differ twice in frequency). And we will do the same on the vertical axis.

Strogats S. Pleasure from H. - M. : Mann, Ivanov and Ferber, 2014.

Now the data is on a curve that is almost a perfect straight line. Based on the properties of logarithms, it is easy to deduce that the original L-shaped curve is a power dependence, which is described by a function of the form

where x is the city's population, y is the number of cities that have that size, c is a constant, and the exponent a (power-law exponent) determines the negative slope of the straight line.

Power distributions have some illogical, from the point of view of traditional statistics, properties. For example, unlike a normal distribution, their modes, medians, and means do not match due to the skewed, skewed shape of the L-shaped curves.

President Bush benefited greatly from this, declaring in 2003 that the tax cut saved each family an average of $1,586. Although mathematically correct, here he to his advantage took as a basis the average deduction, which hid the huge deductions of hundreds of thousands of dollars received by the 0.1% of the richest population in the country. It is known that the “tail” on the right side of the income distribution follows a power law, and in such a situation, the use of the average value is misleading, because it is far from its real value. In reality, most families received less than $650 back. In this distribution, the median is much smaller than the mean.

This example demonstrates the most important property of power-law distributions: they have "heavy tails" compared to at least the small "fluid tails" of a normal distribution. Large tails like this, though rare, are more common in data distributions than regular bell curves.

On Black Monday, October 19, 1987, the Dow Jones Industrial Average fell 22%. Compared to the usual level of volatility in the stock market, this drop was more than twenty standard deviations. According to traditional statistics (which use the normal distribution), such an event is almost impossible: its probability is less than one in 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 (10 to the power of 50). However, this happened - because the price fluctuations in the stock market did not follow a normal distribution.

Distributions with a "heavy tail" are better suited to describe them. This happens with earthquakes, fires and floods, making it difficult for insurance companies to manage risk.

The same mathematical model describes the number of deaths in wars and terrorist attacks, as well as other, much more peaceful things, such as the number of words in a novel or the number of sexual partners a person has.

Although the adjectives used to describe long tails do not cast them in a very favorable light, "tailed" distributions proudly carry their tails. Bold, heavy and long? Yes it is. But in this case, show me which one is normal?

This book is well complemented by:

Quanta

Scott Patterson

Brainiac

Ken Jennings

moneyball

Michael Lewis

Flexible mind

Carol Dweck

The Physics of the Stock Market

James Weatherall

The Joy of X

A Guided Tour of Math, from One to Infinity

Stephen Strogatz

An exciting 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.

Strogats, P.

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

ISBN 978-500057-008-1

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

Legal support of the publishing house is provided by the law firm "Vegas-Lex"

Foreword

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, among whom were many students and teachers. There were also simply inquisitive people who, for one reason or another, “lost their way” in comprehension of mathematical science; now they feel like they missed something worthwhile and would like to try again. I was particularly pleased with the gratitude from my parents for the fact that 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 kinds of recommendations for improving my offspring.

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 topics.

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

I hope you find all the ideas in this book exciting and will make you say, “Well, well!” more than once. But you always have to start somewhere, so let's start with a simple but fascinating action like counting.

1. Number Basics: Adding Fish

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

Portal for the student. Self-training

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