The pleasure of x steven strict read. Stephen Strogatz

Mathematics is the most accurate and universal language of science, but is it possible to explain human feelings with the help of numbers? Love Formulas, Seeds of Chaos and Romantic Differential Equations - T&P publishes a chapter from the book "The Pleasure of X" by one of the best math teachers in the world, Steven Strogatz, published by Mann, Ivanov and Ferber.

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

The analysis below will be derisively ironic, but it touches on serious themes. Moreover, if the understanding of the laws of love can elude us, then the laws of the inanimate world are now well studied. They take the form of differential equations describing how interrelated variables change from moment to moment depending on their current values. Such equations may not have much to do with romance, but at least they can shed light on why, in the words of another poet, "the path of true love has never been smooth." To illustrate the method of differential equations, suppose that Romeo loves Juliet, but in our version of the story, Juliet is a windy sweetheart. The more Romeo loves her, the more she wants to hide from him. But when Romeo cools off 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 unfortunate lovers? How does love absorb them and leave them over time? This is where differential calculus comes to the rescue. By making equations summarizing the waxing and waning of Romeo and Juliet's feelings, and then solving them, we can predict the course of the couple's relationship. The final 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 come to this conclusion, I assumed that Romeo's behavior could be modeled with a differential equation,

which describes how his love ® changes in the next moment (dt). According to this equation, the number of changes (dR) is directly proportional (with a proportionality factor 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 grows in direct proportion to how much Juliet loves him. This assumption of a linear relationship is emotionally implausible, but it makes it possible to greatly simplify the solution of the equation.

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

The negative sign before the constant b reflects that her love cools down 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 that, all the necessary parameters will be set. We can use a computer to move forward slowly, step by step, changing the values of R and J according to the differential equations described above. In fact, with the help of the fundamental theorem of integral calculus, we can find the solution analytically. Because the model is simple, integral calculus produces a couple of exhaustive formulas that tell us how much Romeo and Juliet will love (or hate) each other at any given time in the future.

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

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

The model can be made more realistic in many ways. For example, Romeo may respond 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 will cool his feelings. Or refers to another type of guys who love to suffer - that's why he loves her.

Add to these scenarios two more behaviors of Romeo - he responds to Juliet's affection either by strengthening or weakening his own affection - and you will see that there are four different behaviors in love relationships. My students and the students of Peter Christopher's group at Worcester Polytechnic Institute suggested naming these types as follows: the Hermit or Evil Misanthrope for the Romeo who cools his feelings and withdraws from Juliet, and the Narcissistic Fool and Flirtatious Fink for the one who warms up his ardor, but rejected by Juliet. (You can come up with your own names for all of these types.)

Although the examples given are fantastic, the types of equations that describe them are very informative. They are the most powerful tools ever created by mankind for understanding the material world. Sir Isaac Newton used differential equations to discover the secrets of planetary motion. With the help of these equations, he combined the terrestrial and celestial spheres, showing that the same laws of motion apply to both.

Almost 350 years after Newton, mankind came to understand that the laws of physics are always expressed in the language of differential equations. This is true for the equations describing the flows 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 mysteries of the universe and realized its great significance, he published it as a Latin anagram. In a free translation, it sounds like this: "It is useful to solve differential equations."

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

In mathematics, we call such a 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 have been able to solve it. Later it turned out that the problem of three bodies contains the seeds of chaos, that is, in the long run, their behavior is unpredictable.

Newton knew nothing about the dynamics of chaos, but according to his friend Edmund Halley, he complained that the three-body problem gave him a headache and kept him awake so often that he would never think about it again.

Here I am with you, Sir Isaac.

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

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 life, 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 border 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 already know: numbers, ratios, shapes, changes and infinity - but at the same time we will consider each of them in more depth, in its modern incarnation.

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 and corporate networks, for private and public use, without the written permission of the copyright owner.

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

* * *

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

Foreword

In order to clarify what I mean by the life of numbers and their behavior, which we cannot control, let's go back to the Furry Paws Hotel. Suppose that Humphrey was just about to deliver the order, but then the penguins from another room unexpectedly called him and also asked for the same amount of fish. How many times does Humphrey have to shout the word "fish" after receiving two orders? If he didn't know anything about numbers, he would have to scream as many times as there are total 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 has mastered it, he will proudly say that he needs six plus six (or, if he is a poser, twelve) fish.

This is the same creative process as when we just came up with numbers. Just as numbers make counting easier than listing them one at a time, addition makes it easier to calculate any amount. At the same time, the one who makes the calculation develops as a mathematician. Scientifically, this thought can be formulated as follows: the use of 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 out of our control. Logic leaves us no choice here. In this sense, mathematics always includes both invention, so discovery: we inventing concepts, but open their consequences. As will become clear in the following chapters, in mathematics our freedom lies in the ability to ask questions and persistently seek answers to them, but without inventing 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 shovel them when performing calculations in spreadsheets when filling out tax returns and preparing annual reports. This side of arithmetic seems to many to be important from a practical point of view, but completely bleak.

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

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

You will hardly see anything unusual here. The way it is. Until we start manipulating numbers, they look pretty much the same. The game starts when we receive a task.

For example, let's look at sets that have 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, because 4 = 2 × 2 and 9 = 3 × 3. We get these numbers by squaring some other number (i.e., by squaring the stones).

Here is a problem that has a larger number of 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, then we will always have an extra stone.

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

If we extend these rules to numbers after 10, and consider 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 would make 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. Similarly, 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. Nothing can be laid out of them, except to arrange them in the form of a simple line (one row). These strange stubborn people are famous prime numbers.

So we see that numbers can have bizarre structures that give them a certain character. But in order to imagine the full range of their behavior, one must 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 at 1:

1 + 3 + 5 + 7 = 16

1 + 3 + 5 + 7 + 9 = 25

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

The key to it will be the observation that odd numbers can be represented as equilateral corners, the successive imposition of which on top of each other forms a square!

A similar way of reasoning is presented in another recently published book. Yoko Ogawa's charming novel The Housekeeper and the Professor follows a shrewd but uneducated young woman and her ten-year-old son. A woman has been hired to care for an elderly mathematician who, due to a head injury, only retains information about the last 80 minutes of his life in his short-term memory. 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 how: by asking about her shoe size or date of birth, and having small talk with her about her expenses. The professor also has a special liking for the housekeeper's son, whom he calls Ruth (Root - root), because the boy has a flat head on top, and this reminds him of the notation in mathematics for the square root √.

One day, the professor gives the boy a simple task - to 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. Can he find the answer without simple addition of numbers? Ruth kicks a chair and yells, "That's not fair!"

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

1 + 2 + 3 + … + 9 + 10 = 55,

Jul 25, 2017

The pleasure of X. An exciting 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 Pleasure of X. An Exciting Math Journey from One of the World's Best Teachers by Stephen Strogatz

Published in Russian for the first time.

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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.

The Pleasure of 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've missed something 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

* * *

This book is well complemented by:

Foreword

2. Stone arithmetic

About The Pleasure of X. An Exciting Math Journey from One of the World's Best Teachers by Stephen Strogatz

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