Professor Stewart's Incredible Numbers epub. Professor Stewart's incredible numbers. Life, the universe and...

The world is built on the power of numbers.
Pythagoras

Even in early childhood, we learn to count, then at school we get an idea of ​​the unlimited number series, the elements of geometry, fractional and irrational numbers, and we study the principles of algebra and mathematical analysis. The role of mathematics in modern knowledge and modern practical activity is very great.

Without mathematics, progress in physics, engineering, and production organization would be impossible.
Number is one of the basic concepts of mathematics, allowing one to express the results of counting or measurement. We need numbers to regulate our entire lives. They surround us everywhere: house numbers, car numbers, dates of birth, checks...

Ian Stewart, a world-famous popularizer of mathematics and the author of many fascinating books, admits that numbers have fascinated him since early childhood, and “to this day he is fascinated by numbers and learns more and more new facts about them.”

The heroes of his new book are numbers. According to the English professor, each of them has its own individuality. Some of them play a major role in many areas of mathematics. For example, the number π, which expresses the ratio of the circumference of a circle to its diameter. But, as the author believes, “even the most modest number will have some unusual property.” So, for example, it is impossible to divide by 0 at all, and “somewhere at the very foundation of mathematics, all numbers can be derived from zero.” The smallest positive integer is 1. It is the indivisible unit of arithmetic, the only positive number that cannot be obtained by adding smaller positive numbers. We start counting from 1; no one has any difficulty multiplying by 1. Any number when multiplied by 1 or divided by 1 remains unchanged. This is the only number that behaves this way.
The publication opens with a brief overview of numerical systems. The author shows how they developed in the context of changing human ideas about numbers. If mathematical knowledge in the distant past was used to solve everyday problems, today practice poses increasingly complex problems for mathematics.
Each chapter of the book talks about one "interesting number." There are chapters “0”, “√2”, “-1”... Reading Ian Stewart’s book, you really begin to understand how amazing the world of numbers is! Of course, a reader without some mathematical knowledge may find Professor Stewart's Incredible Numbers difficult to understand. The publication is addressed, rather, to those who strive to become an erudite, or want to show off their knowledge. But, if you love mathematics and want to learn about, for example, super-mega large numbers or mega-small ones, this book is for you.

Having dealt with the numbers 1 to 10, we'll take a step back and look at 0.
Then take another step back to get −1.
This opens up a whole world of negative numbers for us. Also shows new uses for numbers.
Now they are needed not only for counting.

0. Is nothing a number or not?

Zero first appeared in systems for recording numbers and was intended precisely for this purpose - for recording, that is, designation. Only later was zero recognized as an independent number and allowed to take its place - the place of one of the fundamental components of the mathematical number system. However, zero has many unusual, sometimes paradoxical properties. In particular, it is impossible to divide anything by 0 in any reasonable way. And somewhere deep down, at the very foundation of mathematics, all numbers can be derived from 0.

Number system structure

In many ancient cultures, the symbols for 1, 10, and 100 were not related to each other in any way. The ancient Greeks, for example, used the letters of their alphabet to represent the numbers 1 to 9, 10 to 90, and 100 to 900. This system is potentially fraught with confusion, although it is usually easy to determine from the context what exactly a letter stands for: the actual letter or number. But, in addition, such a system made arithmetic operations very difficult.

Our way of writing numbers, when the same digit means different numbers, depending on its place in the number, is called positional notation (see Chapter 10). This system has very serious advantages for counting on paper “in a column”, and this is how, until recently, most calculations in the world were carried out. With positional notation, the main thing you need to know is the basic rules for adding and multiplying ten symbols 0–9. These patterns also apply when the same numbers are in other positions.
Eg,
23 + 5 = 28   230 + 50 = 280   2300 + 500 = 2800.

However, in ancient Greek notation the first two examples look like this:
κγ + ε = κη     σλ + ν = σπ,
and there are no obvious similarities between them.

However, positional notation has one additional feature that appears in particular in the number 2015: the need for a null character. In this case, he says that there are no hundreds in the number. In Greek notation there is no need for a null character. In the number σπ, say, σ means 200 and π means 80. We can be sure that there are no units in the number simply because there are no unit symbols α - θ in it. Instead of using the null character, we simply do not write any single characters in the number.

If we tried to do the same in the decimal system, 2015 would become 215, and we would not be able to tell what exactly the number meant: 215, 2150, 2105, 2015, or maybe 2,000,150. Early versions of the positional system used a space , 2 15, but the space is easy to miss, and two spaces in a row is just a slightly longer space. So there is confusion and it is always easy to make mistakes.

A Brief History of Zero

Babylon

The Babylonians were the first among world cultures to come up with a symbol that meant “there is no number here.” Let us remember (see Chapter 10) that the basis of the Babylonian number system was not 10 but 60. In early Babylonian arithmetic, the absence of the component 60 2 was indicated by a space, but by the 3rd century. BC e. they invented a special symbol for this. However, the Babylonians do not seem to have considered this symbol to be a real number. Moreover, at the end of the number this symbol was omitted, and its meaning had to be guessed from the context.

India

The idea of ​​positional notation of numbers in a base 10 number system first appeared in the Lokavibhaga, a Jain cosmological text of 458 AD, which also uses Shunya(meaning "emptiness") where we would put a 0. In 498, the famous Indian mathematician and astronomer Aryabhata described the positional system of writing numbers as "place after place, each 10 times larger in magnitude." The first known use of a special symbol for the decimal digit 0 dates back to 876 in an inscription at the Chaturbhuja Temple at Gwalior; this symbol represents - guess what? Small circle.

Mayan

The Central American Mayan civilization, which reached its peak somewhere between 250 and 900 AD, used a base-20 number system and had a special symbol to represent zero. In fact, this method dates back much earlier and is believed to have been invented by the Olmecs (1500–400 BC). In addition, the Mayans actively used numbers in their calendar system, one of the rules of which was called the “long count.” This meant counting the date in days after the mythical date of creation, which, according to the modern Western calendar, would have been August 11, 3114 BC. e. In this system, the symbol for zero is absolutely necessary, since without it it is impossible to avoid ambiguity.

Is zero a number?

Until the 9th century. zero was considered convenient symbol for numerical calculations, but was not considered a number in itself. Probably because it was not used for counting.

If they ask how many cows you have - and you do have cows - you will point to each of them in turn and count: “One, two, three...” But if you don’t have any cows, you will not point to some cow and say: “Zero,” because you have nothing to point to. Since 0 is never counted, it is obviously not a number.

If this position seems strange to you, then it should be noted that even earlier “one” was also not considered a number. In some languages, the word "number" also means "several" or even "many". In almost all modern languages ​​there is a distinction between singular and plural. Ancient Greek also had a “dual” number, and when talking about two objects or persons, special forms of words were used. So in this sense, “two” was also not considered the same number as all the others. The same is observed in several other classical languages ​​and even in some modern ones, such as Scottish Gaelic or Slovenian. Traces of these same forms are visible in English, where “both” ( both) and "all" ( all) - different words.

As the zero symbol became more widely used, and as numbers began to be used for more than just counting, it became clear that in many respects the zero behaved just like any other number. By the 9th century. Indian mathematicians already considered zero to be a real number, and not just a symbol that conveniently represents spaces between other symbols for the sake of clarity. Zero was freely used in everyday calculations.

On the number line, where the numbers 1, 2, 3... are written in order from left to right, no one has any problem with where to put the zero: to the left of the 1. The reason is quite obvious: adding 1 to any number shifts it by one step to the right. Adding 1 to 0 shifts it by 1, so a 0 should be placed where one step to the right gives a 1. Which means one step to the left of a 1.

The recognition of negative numbers finally secured zero's place in the series of real numbers. No one argued that 3 is a number. If we accept that −3 is also a number and that adding two numbers always produces a number, then the result of 3 + (−3) must be a number. And the number is 0.

Unusual properties

I said "in many ways, zero behaves just like any other number." In many, but not all. Zero is a special number. It must be special because it is a single number neatly squeezed between positive and negative numbers.

It is clear that adding 0 to any number will not change that number. If I have three cows and I add one more to them, then I will still have three cows. Admittedly, there are strange calculations like this:

One cat has one tail.
No cat has eight tails.
Therefore, adding:
One cat has nine tails.

This little joke plays on different interpretations of the negation “No”.

From this special property of zero it follows that 0 + 0 = 0, which means −0 = 0. Zero is the opposite of itself. This is the only such number, and this happens precisely because on the number line zero is sandwiched between positive and negative numbers.

What about multiplication? If we consider multiplication as sequential addition, then
2 × 0 = 0 + 0 = 0
3 × 0 = 0 + 0 + 0 = 0
4 × 0 = 0 + 0 + 0 + 0 = 0,
and therefore
n× 0 = 0
for any number n. By the way, this also makes sense in financial matters: if I put three times zero rubles into my account, then in the end I will not put anything there. Again, zero is the only number that has this property.

In arithmetic m × n equals n × m for all numbers n And m. This agreement implies that
0 × n = 0
for anyone n, despite the fact that we cannot add “zero times” by n.

What's wrong with division? Dividing zero by a non-zero number is simple and clear: the result is zero. Half of nothing, a third or any other part of nothing is nothing. But when it comes to dividing a number by zero, the strangeness of zero comes into play. What is, for example, 1:0? We define m : n like a number q, for which the expression is true q × n = m. So 1:0 is what it is q, for which q× 0 = 1. However, such a number does not exist. Whatever we take as q, we get q× 0 = 0. And we will never get units.

The obvious way to solve this problem is to take it for granted. Division by zero is prohibited because it makes no sense. On the other hand, before fractions were introduced, the expression 1:2 didn't make sense either, so maybe we shouldn't give up so quickly. We could try to come up with some new number that would allow us to divide by zero. The problem is that such a number violates the basic rules of arithmetic. For example, we know that 1 × 0 = 2 × 0, since both are equal to zero individually. Dividing both sides by 0, we get 1 = 2, which is frankly ridiculous. So it seems reasonable to simply not allow division by zero.

Numbers from nothing

The mathematical concept that is perhaps closest to the concept of “nothing” can be found in set theory. A bunch of- this is a certain set of mathematical objects: numbers, geometric figures, functions, graphs... A set is defined by listing or describing its elements. “The set of numbers 2, 4, 6, 8” and “the set of even numbers greater than 1 and less than 9” define the same set, which we can form by enumerating: (2, 4, 6, 8),
where the curly braces () indicate that the elements of a set are contained within.

Around 1880, the German mathematician Cantor developed detailed set theory. He was trying to understand some of the technical aspects of mathematical analysis related to function breakpoints - places where a function makes unexpected jumps. The structure of multiple discontinuities played an important role in his answer. In this case, it was not individual gaps that mattered, but their entirety. Cantor was really interested in infinitely large sets in connection with analysis. He made a serious discovery: he found out that infinities are not the same - some of them are larger, others are smaller (see chapter ℵ 0).

As I mentioned in the section "What is a number?", another German mathematician, Frege, picked up Cantor's ideas, but he was much more interested in finite sets. He believed that with their help it was possible to solve a global philosophical problem related to the nature of numbers. He thought about how sets are related to each other: for example, how many cups are related to many saucers. The seven days of the week, the seven dwarves, and the numbers 1 to 7 line up perfectly with each other so that they all define the same number.

Which of the following sets should we choose to represent the number seven? Frege, answering this question, did not mince words: all at once. He defined number as the set of all sets corresponding to a given set. In this case, no set is preferred, and the choice is made unambiguously, and not randomly or arbitrarily. Our symbols and number names are just convenient shortcuts for these gigantic sets. The number seven is a set everyone sets equivalent to gnomes, and this is the same as the set of all sets equivalent to days of the week or the list (1, 2, 3, 4, 5, 6, 7).

It's probably unnecessary to point out that this is a very elegant solution conceptual problem does not give us anything concrete in terms of a reasonable system for representing numbers.

When Frege presented his ideas in the two-volume work The Fundamental Laws of Arithmetic (1893 and 1903), many thought that he had solved the problem. Now everyone knew what the number was. But just before the publication of the second volume, Bertrand Russell wrote a letter to Frege that said (I paraphrase): “Dear Gottlob, consider the set of all sets that do not contain themselves.” It's like a village barber who shaves those who don't shave themselves; With such a definition, a contradiction arises. Russell's paradox, as it is now called, showed how dangerous it is to assume that all-encompassing sets exist (see chapter ℵ 0).

Mathematical logic experts tried to solve the problem. The answer turned out to be strictly the opposite of Frege's “broad thinking” and his policy of lumping all possible sets into one heap. The trick was to choose exactly one of all possible sets. To determine the number 2, it was necessary to construct a standard set with two elements. To define 3, you can use a standard set with three elements, and so on. The logic here does not go in cycles if these sets are first constructed without using numbers explicitly, and only then assign numeric symbols and names to them.

The main problem was the choice of standard sets to use. They had to be defined in an unambiguous and unique way, and their structure had to somehow relate to the counting process. The answer came from a very specific set known as the empty set.

Zero is a number, the basis of our entire number system. Consequently, it can be used to count the elements of a certain set. What many? Well, it should be a set with no elements. It is not difficult to come up with such a set: let it be, for example, “the set of all mice weighing more than 20 tons each.” In mathematical language, this means that there is a set that does not have a single element: the empty set. In mathematics, it is also easy to find examples: the set of prime numbers that are multiples of 4, or the set of all triangles with four vertices. These sets look different - one contains numbers, the other contains triangles - but in fact they are the same set, since such numbers and triangles do not actually exist and it is simply impossible to distinguish between the sets. All empty sets contain exactly the same elements: namely, none. Therefore, the empty set is unique. The symbol for it was introduced by a group of scientists working under the common pseudonym Bourbaki in 1939, and it looks like this: ∅. Set theory needs the empty set in the same way that arithmetic needs the number 0: if you include it, everything becomes much simpler.

Moreover, we can determine that 0 is the empty set.

What about the number 1? It is intuitively clear that here we need a set consisting of exactly one element, and a unique one. Well... the empty set is unique. Thus, we define 1 as a set whose only element is the empty set: in symbolic language (∅). This is not the same as the empty set because this set has one element, whereas the empty set does not. I agree, this single element is an empty set, it happened so, but still this element is present in the set. Think of the set as a paper bag with elements. An empty set is an empty package. A set whose only element is the empty set is a package that contains another package, the empty one. You can see for yourself that this is not the same thing - there is nothing in one package, and there is a package in the other.

The key step is to determine the number 2. We need to uniquely obtain a specific set with two elements. So why not use the only two sets we've mentioned so far: ∅ and (∅)? Therefore we define 2 as the set (∅, (∅)). And this, according to our definitions, is the same as 0, 1.

Now a general pattern begins to emerge. Let's define 3 = 0, 1, 2 - a set with three elements that we have already defined. Then 4 = 0, 1, 2, 3; 5 = 0, 1, 2, 3, 4 and so on. Everything, if you look at it, goes back to the empty set. Eg,
3 = {∅, {∅}, {∅, {∅}}}
4 = {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}}.

You probably don't want to see what the number of gnomes looks like.

The building materials here are abstractions: the empty set and the act of forming a set by enumerating its elements. But the way these sets relate to each other leads to the creation of a strict framework for a number system, in which each number represents a special set that (intuitively) has exactly that number of elements. And the story doesn't end there. Having defined the natural numbers, we can use similar set theory tricks to define negative numbers, fractions, real numbers (infinite decimals), complex numbers, and so on, all the way to the latest ingenious mathematical concept in quantum theory.

So now you know the terrible secret of mathematics: at its foundation lies nothingness.

-1. Less than nothing

Can a number be less than zero? Counting cows won't do anything like that, unless you imagine "virtual cows" that you owe to someone. In this case, you have a natural extension of the numerical concept that will make life much easier for algebraists and accountants. At the same time, surprises await you: a minus for a minus gives a plus. Why on earth?

Negative numbers

Having learned to add numbers, we begin to master the reverse operation: subtraction. For example, 4 − 3 in the answer gives the number that, when added to 3, gives 4. This is, of course, 1. Subtraction is useful because without it it is difficult for us, for example, to know how much money we will have left if we initially had 4 rubles, but we spent 3 rubles.

Subtracting a smaller number from a larger one causes virtually no problems. If we spent less money than we had in our pocket or wallet, then we still have something left. But what happens if we subtract a larger number from a smaller one? What is 3 − 4?

If you have three 1 ruble coins in your pocket, then you will not be able to take four such coins out of your pocket and give them to the cashier at the supermarket. But today, with credit cards, anyone can easily spend money they don’t have, not only in their pocket, but also in their bank account. When this happens, a person gets into debt. In this case, the debt would be 1 ruble, not counting bank interest. So in a certain sense 3 − 4 is equal to 1, but another 1: a unit of debt, not money. If 1 had its opposite, it would be exactly like this.

To distinguish debt from cash, it is customary to prefix the number with a minus sign. In such a recording
3 − 4 = −1,
and we can consider that we have invented a new type of number: negative number.

History of Negative Numbers

Historically, the first major extension of the number system was fractions (see Chapter ½). The second were negative numbers. However, I intend to deal with these types of numbers in reverse order. The first known mention of negative numbers is in a Chinese document from the Han Dynasty (202 BC - 220 AD) called The Art of Counting in Nine Sections (Jiu Zhang Xuan Shu).

This book used a physical “helper” for counting: counting sticks. These are small sticks made of wood, bone or other material. To represent numbers, sticks were laid out in certain shapes. In the unit digit of a number, the horizontal line means “one” and the vertical line means “five”. The numbers in the hundredth place look the same. In the tens and thousands digits, the directions of the sticks are reversed: the vertical one means “one”, and the horizontal one means “five”. Where we would put 0, the Chinese simply left a space; however, the space is easy to miss, in which case the rule about changing directions helps avoid confusion if, for example, there is nothing in the tens section. This method is less effective if the number contains several zeros in a row, but this is a rare case.

In The Art of Counting in Nine Sections, sticks were also used to represent negative numbers, and in a very simple way: they were colored black rather than red. So
4 red sticks minus 3 red ones equals 1 red stick,
But
3 red sticks minus 4 red sticks equals 1 black stick.

Thus, the black stick figure represents debt, and the size of the debt corresponds to the red stick figures.

Indian mathematicians also recognized negative numbers; in addition, they compiled consistent rules for performing arithmetic operations with them.

The Bakhshali manuscript, dating from around the 3rd century, contains calculations with negative numbers, which can be distinguished from others by the + sign in places where we would use -. (Mathematical symbols have changed many times over time, sometimes in such a way that it’s easy for us to get confused by them.) The idea was picked up by Arab mathematicians, and from them it gradually spread throughout Europe. Until the 17th century European mathematicians usually interpreted a negative answer as proof that the problem in question had no solution, but Fibonacci already understood that in financial calculations they could represent debts. By the 19th century negative numbers no longer frightened mathematicians and baffled them.

Writing Negative Numbers

Geometrically, it is convenient to represent numbers as points on a line going from left to right and starting at 0. We have already seen that this number line there is a natural continuation that includes negative numbers and goes in the opposite direction.

Performing addition and subtraction on the number line is very convenient and simple. For example, to add 3 to any number, you need to move three steps to the right. To subtract 3, you need to move 3 steps to the left. This action gives the correct result for both positive and negative numbers; for example, if we start with −7 and add 3, we will move 3 steps to the right and get −4. The rules for performing arithmetic operations for negative numbers also show that adding or subtracting a negative number gives the same result as subtracting or adding the corresponding positive number. So to add -3 to any number, we need to move 3 steps to the left. To subtract −3 from any number, you need to move 3 steps to the right.

Multiplication involving negative numbers is more interesting. When we first learn about multiplication, we think of it as repeated addition. Eg:
6 × 5 = 5 + 5 + 5 + 5 + 5 + 5 = 30.

The same approach suggests that when multiplying 6 × −5 we should proceed similarly:
6 × −5 = −5 + (−5) + (−5) + (−5) + (−5) + (−5) = −30.

Further, one of the rules of arithmetic states that multiplying two positive numbers gives the same result regardless of the order in which we take the numbers. So, 5 × 6 must also equal 30. It is, because
5 × 6 = 6 + 6 + 6 + 6 + 6 = 30.

So it seems reasonable to adopt the same rule for negative numbers. Then −5 × 6 is also equal to −30.

What about −6 × −5? There is less clarity on this issue. We can't write in a row minus six times −5, and then add them. Therefore, we have to consistently address this issue. Let's see what we already know.

6 × 5 = 30
6 × −5 = −30
−6 × 5 = −30
−6 × −5 =?

At first glance, many people think that the answer should be −30. Psychologically, this is probably justified: the whole action is permeated with a spirit of “negativity,” so the answer should probably be negative. Probably the same feeling lies behind the stock phrase: “But I didn’t do anything.” However, if you Nothing didn’t do it, which means you should have done “nothing”, that is something. Whether such a remark is fair depends on the rules of grammar you use. An extra negation can also be considered as an intensifying construction.

In the same way, what will be equal to −6 × −5 is a matter of human agreement. When we come up with new numbers, there is no guarantee that the old concepts will apply to them. So mathematicians could decide that −6 × −5 = −30. Strictly speaking, they might have decided that multiplying -6 by −5 would produce a purple hippopotamus.

However, there are several good reasons why −30 is a poor choice in this case, and all of these reasons point in the opposite direction - towards the number 30.

One reason is that if −6 × −5 = −30, then this is the same as −6 × 5. Dividing both by −6, we get −5 = 5, which contradicts everything we have already said about negative numbers .

The second reason is because we already know: 5 + (−5) = 0. Take a look at the number line. What is five steps to the left of the number 5? Zero. Multiplying any positive number by 0 produces 0, and it seems reasonable to assume that the same applies to negative numbers. So it makes sense to think that −6 × 0 = 0. Therefore
0 = −6 × 0 = −6 × (5 + (−5)).

According to the usual rules of arithmetic, this is equal to
−6 × 5 + −6 × −5.

On the other hand, if we chose −6 × -5 = 30, we would get
0 = −6 × 0 = −6 × (5 + (−5)) = −6 × 5 + (−6) × −5 =
= −30 + 30 = 0,
and everything would fall into place.

The third reason is the structure of the number line. By multiplying a positive number by −1, we turn it into the corresponding negative number; that is, we rotate the entire positive half of the number line by 180°, moving it from right to left. Where should the negative half go, in theory? If we leave it in place, we get the same problem, because −1 × −1 is −1, which is equal to −1 × 1, and we can conclude that −1 = 1. The only reasonable alternative is exactly this Or rotate the negative part of the number line by 180°, moving it from left to right. This is neat because now multiplying by −1 completely reverses the number line, reversing the order of the numbers. It follows from this, as night follows day, that a new multiplication by −1 will rotate the number line by 180° once again. The order of the numbers will again be reversed, and everything will return to where it started. So, −1 × −1 is where −1 ends up when we rotate the number line, which is 1. And if we decide that −1 × −1 = 1, then it follows directly that −6 × −5 = 30.

The fourth reason is the interpretation of a negative amount of money as debt. In this variant, multiplying a certain amount of money by a negative number gives the same result as multiplying it by the corresponding positive number, except that the real money turns into debt. On the other side, subtraction, “taking away” the debt, has the same effect as if the bank were removing some of your debt from its records and essentially giving you some money back. Subtracting a debt of 10 rubles from the amount of your account is exactly the same as depositing 10 rubles of your money into this account: while the amount of the account increases for 10 rubles. The combined effect of both in these circumstances tends to bring your bank balance back to zero. It follows that −6 × −5 has the same effect on your account as subtracting (removing) a debt of 5 rubles six times, which means it should increase your bank balance by 30 rubles.

One cat has one tail. Zero cats have eight tails. (Another reading is “There are no cats with eight tails.”) So we get: One cat has nine tails. - Note ed.

Stewart deserves the highest praise for his story about how great, amazing and useful the role of everyone in the global numbers community is. Kirkus Reviews Stewart does a brilliant job of explaining complex issues. New Scientist Britain's most brilliant and prolific popularizer of mathematics. Alex Bellos What is the book about? Essentially, mathematics is numbers, our main tool for understanding the world. In his book

Stewart deserves the highest praise for his story about how great, amazing and useful the role of everyone in the global numbers community is. Kirkus Reviews Stewart does a brilliant job of explaining complex issues. New Scientist Britain's most brilliant and prolific popularizer of mathematics. Alex Bellos What is the book about? Essentially, mathematics is numbers, our main tool for understanding the world. In his book, the most famous British popularizer of mathematics, Professor Ian Stewart, offers a delightful introduction to the numbers that surround us, from familiar combinations of symbols to the more exotic ones - factorials, fractals or the Apéry constant. On this path, the author tells us about prime numbers, cubic equations, the concept of zero, possible versions of the Rubik's cube, the role of numbers in the history of mankind and the relevance of their study in our time. With his characteristic wit and erudition, Stewart reveals to the reader the fascinating world of mathematics. Why the book is worth reading The most interesting thing about the most incredible numbers in the story of the best popularizer of mathematics from Britain, winner of the 2015 Lewis Thomas Prize. Ian Stewart examines the amazing properties of numbers from zero to infinity - natural, complex, irrational, positive, negative, prime, composite - and shows their history from the amazing discoveries of ancient mathematicians to the modern state of mathematical science. Under the experienced guidance of the professor, you will learn the secrets of mathematical codes and Sudoku, Rubik's cube and musical scales, see how one infinity can be larger than another, and also discover that you live in eleven-dimensional space. This book will delight those who love numbers and those who still think they don't love them. About the authorProfessor Ian Stewart is a world-famous popularizer of mathematics and the author of many fascinating books, and has been awarded a number of the highest international academic awards. In 2001 he became a member of the Royal Society of London. Emeritus Professor at the University of Warwick, he researches the dynamics of nonlinear systems and advances mathematical knowledge. Author of the best-selling book "The Greatest Mathematical Problems", published by the publishing house "Alpina Non-Fiction" in 2015. Key conceptsMathematics, numbers, numbers, riddles, higher mathematics, mathematical problems, mathematical research, history of mathematics, science, science.

Emeritus Professor of Mathematics at the University of Warwick, famous popularizer of science Ian Stewart, dedicated to the role of numbers in the history of mankind and the relevance of their study in our time.

Pythagorean hypotenuse

Pythagorean triangles have right angles and integer sides. The simplest of them has a longest side of length 5, the others - 3 and 4. There are 5 regular polyhedra in total. A fifth degree equation cannot be solved using fifth roots - or any other roots. Lattices on a plane and in three-dimensional space do not have five-lobed rotational symmetry, so such symmetries are absent in crystals. However, they can be found in lattices in four dimensions and in interesting structures known as quasicrystals.

Hypotenuse of the smallest Pythagorean triple

The Pythagorean theorem states that the longest side of a right triangle (the notorious hypotenuse) is related to the other two sides of this triangle in a very simple and beautiful way: the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Traditionally, we call this theorem by the name of Pythagoras, but in fact its history is quite vague. Clay tablets suggest that the ancient Babylonians knew the Pythagorean theorem long before Pythagoras himself; The fame of the discoverer was brought to him by the mathematical cult of the Pythagoreans, whose supporters believed that the Universe was based on numerical laws. Ancient authors attributed a variety of mathematical theorems to the Pythagoreans - and therefore to Pythagoras, but in fact we have no idea what kind of mathematics Pythagoras himself was involved in. We don't even know if the Pythagoreans could prove the Pythagorean Theorem or if they simply believed it to be true. Or, most likely, they had convincing evidence of its truth, which nevertheless would not be enough for what we consider evidence today.

Proofs of Pythagoras

The first known proof of the Pythagorean theorem is found in Euclid's Elements. This is a fairly complex proof using a drawing that Victorian schoolchildren would immediately recognize as “Pythagorean trousers”; The drawing really does resemble underpants drying on a line. There are literally hundreds of other proofs, most of which make the assertion more obvious.

Perigal's dissection is another puzzle proof.

There is also a proof of the theorem using arranging squares on a plane. Perhaps this is how the Pythagoreans or their unknown predecessors discovered this theorem. If you look at how the skew square overlaps two other squares, you can see how to cut a large square into pieces and then put them together into two smaller squares. You can also see right triangles, the sides of which give the dimensions of the three squares involved.

There are interesting proofs using similar triangles in trigonometry. At least fifty different proofs are known.

Pythagorean triples

In number theory, the Pythagorean theorem became the source of a fruitful idea: finding integer solutions to algebraic equations. A Pythagorean triple is a set of integers a, b and c such that

a 2 + b 2 = c 2 .

Geometrically, such a triple defines a right triangle with integer sides.

The smallest hypotenuse of a Pythagorean triple is 5.

The other two sides of this triangle are 3 and 4. Here

3 2 + 4 2 = 9 + 16 = 25 = 5 2 .

The next largest hypotenuse is 10 because

6 2 + 8 2 = 36 + 64 = 100 = 10 2 .

However, this is essentially the same triangle with double sides. The next largest and truly different hypotenuse is 13, for which

5 2 + 12 2 = 25 + 144 = 169 = 13 2 .

Euclid knew that there were an infinite number of different variations of Pythagorean triplets, and he gave what might be called a formula for finding them all. Later, Diophantus of Alexandria proposed a simple recipe, basically identical to Euclidean.

Take any two natural numbers and calculate:

their double product;

the difference of their squares;

the sum of their squares.

The three resulting numbers will be the sides of the Pythagorean triangle.

Let's take, for example, the numbers 2 and 1. Let's calculate:

double product: 2 × 2 × 1 = 4;

difference of squares: 2 2 – 1 2 = 3;

sum of squares: 2 2 + 1 2 = 5,

and we got the famous 3-4-5 triangle. If we take the numbers 3 and 2 instead, we get:

double product: 2 × 3 × 2 = 12;

difference of squares: 3 2 – 2 2 = 5;

sum of squares: 3 2 + 2 2 = 13,

and we get the next most famous triangle 5 – 12 – 13. Let’s try to take the numbers 42 and 23 and get:

double product: 2 × 42 × 23 = 1932;

difference of squares: 42 2 – 23 2 = 1235;

sum of squares: 42 2 + 23 2 = 2293,

no one has ever heard of the triangle 1235–1932–2293.

But these numbers work too:

1235 2 + 1932 2 = 1525225 + 3732624 = 5257849 = 2293 2 .

There is another feature of the Diophantine rule that has already been hinted at: given three numbers, we can take another arbitrary number and multiply them all by it. Thus, a 3–4–5 triangle can be turned into a 6–8–10 triangle by multiplying all sides by 2, or into a 15–20–25 triangle by multiplying all by 5.

If we switch to the language of algebra, the rule takes on the following form: let u, v and k be natural numbers. Then a right triangle with sides

2kuv and k (u 2 – v 2) has a hypotenuse

There are other ways of presenting the main idea, but they all boil down to the one described above. This method allows you to obtain all Pythagorean triples.

Regular polyhedra

There are exactly five regular polyhedra. A regular polyhedron (or polyhedron) is a three-dimensional figure with a finite number of flat faces. The faces meet each other on lines called edges; the edges meet at points called vertices.

The culmination of Euclidean's Principia is the proof that there can be only five regular polyhedra, that is, polyhedra in which each face is a regular polygon (equal sides, equal angles), all faces are identical, and all vertices are surrounded by an equal number of equally spaced faces. Here are five regular polyhedra:

tetrahedron with four triangular faces, four vertices and six edges;

cube, or hexahedron, with 6 square faces, 8 vertices and 12 edges;

octahedron with 8 triangular faces, 6 vertices and 12 edges;

dodecahedron with 12 pentagonal faces, 20 vertices and 30 edges;

An icosahedron with 20 triangular faces, 12 vertices and 30 edges.

Regular polyhedra can also be found in nature. In 1904, Ernst Haeckel published drawings of tiny organisms known as radiolarians; many of them are shaped like those same five regular polyhedra. Perhaps, however, he slightly corrected nature, and the drawings do not fully reflect the shape of specific living beings. The first three structures are also observed in crystals. You will not find dodecahedrons and icosahedrons in crystals, although irregular dodecahedrons and icosahedrons are sometimes found there. True dodecahedrons can occur as quasicrystals, which are similar to crystals in every way except that their atoms do not form a periodic lattice.

It can be interesting to make models of regular polyhedra from paper by first cutting out a set of interconnected faces - this is called developing a polyhedron; the development is folded along the edges and the corresponding edges are glued together. It is useful to add an additional glue pad to one of the ribs of each such pair, as shown in Fig. 39. If there is no such platform, you can use adhesive tape.

Fifth degree equation

There is no algebraic formula for solving 5th degree equations.

In general, a fifth degree equation looks like this:

ax 5 + bx 4 + cx 3 + dx 2 + ex + f = 0.

The problem is to find a formula for solutions to such an equation (it can have up to five solutions). Experience with quadratic and cubic equations, as well as equations of the fourth degree, suggests that such a formula should also exist for equations of the fifth degree, and, in theory, roots of the fifth, third and second degrees should appear in it. Again, we can safely assume that such a formula, if it exists, will be very, very complex.

This assumption ultimately turned out to be wrong. In fact, no such formula exists; at least there is no formula consisting of the coefficients a, b, c, d, e and f, made using addition, subtraction, multiplication and division, and taking roots. So there is something very special about the number 5. The reasons for this unusual behavior of the five are very deep, and it took a lot of time to understand them.

The first sign of trouble was that no matter how hard mathematicians tried to find such a formula, no matter how smart they were, they invariably failed. For some time, everyone believed that the reasons lay in the incredible complexity of the formula. It was believed that no one simply could understand this algebra properly. However, over time, some mathematicians began to doubt that such a formula even existed, and in 1823 Niels Hendrik Abel was able to prove the opposite. There is no such formula. Shortly thereafter, Évariste Galois found a way to determine whether an equation of one degree or another—5th, 6th, 7th, any kind—was solvable using this kind of formula.

The conclusion from all this is simple: the number 5 is special. You can solve algebraic equations (using nth roots for different values ​​of n) for powers 1, 2, 3, and 4, but not for powers 5. This is where the obvious pattern ends.

No one is surprised that equations of degrees greater than 5 behave even worse; in particular, the same difficulty is associated with them: there are no general formulas for solving them. This does not mean that the equations have no solutions; This also does not mean that it is impossible to find very precise numerical values ​​for these solutions. It's all about the limitations of traditional algebra tools. This is reminiscent of the impossibility of trisection of an angle using a ruler and compass. The answer exists, but the methods listed are insufficient and do not allow us to determine what it is.

Crystallographic limitation

Crystals in two and three dimensions do not have 5-ray rotational symmetry.

Atoms in a crystal form a lattice, that is, a structure that periodically repeats itself in several independent directions. For example, the pattern on wallpaper is repeated along the length of the roll; in addition, it is usually repeated in the horizontal direction, sometimes with a shift from one piece of wallpaper to the next. Essentially, wallpaper is a two-dimensional crystal.

There are 17 varieties of wallpaper patterns on a plane (see Chapter 17). They differ in types of symmetry, that is, in ways to rigidly move the pattern so that it lies exactly on itself in its original position. Types of symmetry include, in particular, various variants of rotational symmetry, where the pattern should be rotated by a certain angle around a certain point - the center of symmetry.

The order of rotational symmetry is how many times the body can be rotated in a full circle so that all the details of the pattern return to their original positions. For example, a 90° rotation is 4th order rotation symmetry*. The list of possible types of rotational symmetry in a crystal lattice again points to the unusualness of the number 5: it is not there. There are options with 2nd, 3rd, 4th and 6th order rotation symmetry, but none of the wallpaper designs have 5th order rotation symmetry. Rotation symmetry of order greater than 6 also does not exist in crystals, but the first violation of the sequence still occurs at number 5.

The same thing happens with crystallographic systems in three-dimensional space. Here the lattice repeats itself in three independent directions. There are 219 different types of symmetry, or 230 if we count the mirror image of a design as a separate variant - despite the fact that in this case there is no mirror symmetry. Again, rotational symmetries of orders 2, 3, 4, and 6 are observed, but not 5. This fact is called crystallographic confinement.

In four-dimensional space, lattices with 5th order symmetry exist; In general, for lattices of sufficiently high dimension, any predetermined order of rotational symmetry is possible.

Quasicrystals

Although 5th order rotational symmetry is not possible in 2D or 3D lattices, it can exist in slightly less regular structures known as quasicrystals. Using Kepler's sketches, Roger Penrose discovered planar systems with a more general type of fivefold symmetry. They are called quasicrystals.

Quasicrystals exist in nature. In 1984, Daniel Shechtman discovered that an alloy of aluminum and manganese could form quasicrystals; Initially, crystallographers greeted his report with some skepticism, but the discovery was later confirmed, and in 2011 Shechtman was awarded the Nobel Prize in Chemistry. In 2009, a team of scientists led by Luca Bindi discovered quasicrystals in a mineral from the Russian Koryak Highlands - a compound of aluminum, copper and iron. Today this mineral is called icosahedrite. By measuring the content of different oxygen isotopes in the mineral using a mass spectrometer, scientists showed that this mineral did not originate on Earth. It formed about 4.5 billion years ago, at a time when the solar system was just emerging, and spent most of its time in the asteroid belt, orbiting the Sun, until some disturbance changed its orbit and eventually brought it to Earth.