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Carl Sagan
Billions and Billions: Reflections on Life and Death at the Turn of the Millennium

Translator Natalia Kiechenko

Editor Vyacheslav Ionov

Scientific editor Vladimir Surdin, Ph.D. physical – mat. sciences

Project Manager I. Seregina

Proofreaders M. Milovidova, E. Aksenova

Computer layout A. Fominov

Cover designer Yu. Buga

© 1997 by The Estate of Carl Sagan with permission from Democritus Properties, LLC.

© Publication in Russian, translation, design. Alpina Non-Fiction LLC, 2017

All rights reserved. The work is intended exclusively for private use. No part of the electronic copy of this book may be reproduced in any form or by any means, including posting on the Internet or corporate networks, for public or collective use without the written permission of the copyright owner. For violation of copyright, the law provides for payment of compensation to the copyright holder in the amount of up to 5 million rubles (Article 49 of the Code of Administrative Offenses), as well as criminal liability in the form of imprisonment for up to 6 years (Article 146 of the Criminal Code of the Russian Federation).

* * *

To my sister Kari, one of six billion

Part I
The clarity and beauty of numbers

Chapter 1
Billions and Billions

There are people who think that the number of grains of sand is infinite. ...Others think that although this number is not infinite, it is impossible to imagine more. ...On the contrary, I will try to prove with geometric precision that will convince you that ... there are numbers greater than the number of grains of sand that can be contained not only in a space equal to the volume of the earth ... but also the whole world.

– Archimedes (c. 287–212 BC). Calculus of grains of sand 1
Quote by: Archimedes. Counting grains of sand (Psammit). – M.–L.: GTTI, 1932.

I swear, I didn’t say the phrase “billions and billions”! I could say, for example, “100 billion galaxies and 10 billion trillion stars.” It is impossible to describe space without resorting to large numbers. I repeatedly uttered the word “billion” in the programs of the television series “Cosmos”, which were watched by a great many viewers. But “billions and billions” – never. If only because it is too vague. “Billions and billions” – how much is that? Two or three billion? Twenty? One hundred? The spread is too wide. When preparing the new edition of the TV series, I carefully reviewed everything and made sure that I didn’t say anything like that.

This phrase was said by Johnny Carson, whose guest I have been on “The Tonight Show” at least three dozen times. He dresses up in a corduroy jacket and turtleneck, tousles his hair and, parodying me, a kind of double, launches something like “billions and billions” on the evening television. And I'm starting to get tired of the fact that this parody takes on a life of its own, uttering maxims with which friends and colleagues stun me the next morning. (Otherwise, I admit that the serious amateur astronomer Carson most often speaks in strict scientific language.)

Alas, “billions and billions” stuck. People like the sound of it. Every now and then people call me on the street, on a plane, at a party and, with slight embarrassment, ask me to do the courtesy of saying: “Billions and billions.”

“You see, these are not my words,” I explain.

“Well, okay,” they answer me. - Tell me anyway.

It turns out that Sherlock Holmes never said “elementary, Watson” (at least in the books of Arthur Conan Doyle), Jimmy Cagney 2
James Cagney (1899–1986) – American actor and dancer. – Note ed.

Humphrey Bogart’s character didn’t say “you’re a dirty rat” 3
Humphrey Bogart (1899–1957) – American actor. – Note ed.

Didn't say "play it again, Sam." But these phrases are so ingrained in popular culture that they are perceived as actually being said.

So the not-so-successful expression about billions is still attributed to me in computer magazines (“Carl Sagan would say that it takes billions and billions of bytes”), in economic reviews on the pages of newspapers, in stories about the salaries of sports stars, etc. d.

In the past, I would never have repeated this, either verbally or in writing – out of harm’s way. But now I've outgrown it. And if this is necessary for the story, please:

– Billions and billions!

Why has this phrase become so popular? Once upon a time, the symbol for large numbers was “million”. The super rich were millionaires. The population of the Earth at the time of Jesus was about 250 million people. The Convention of 1787 gave the Constitution to four million Americans; by the beginning of the Second World War there were already 132 million of us. There are 150 million km from the Earth to the Sun. About 40 million people died in World War I, and 60 million in World War II. There are 31.7 million seconds in a year (you can check it). The cumulative power of nuclear arsenals accumulated by the end of the 1980s would be enough to destroy a million Hiroshimas. For a long time, in most cases, the word “million” essentially meant “incredibly many.”

But times have changed. Nowadays there is a layer in the world billionaires, and not at all because the money has depreciated. The age of the Earth is generally accepted to be 4.6 billion years. The population has long exceeded 6 billion. Your two birthdays are separated by one year and a billion kilometers (the Earth moves around the Sun much faster than the Voyagers launched into space move away from it). Four B-2 bombers cost a billion dollars (according to other estimates, two or even four billion). The annual US defense budget, taking into account all hidden costs, exceeds $300 billion. In the event of a full-scale nuclear war between the US and Russia, about a billion people would immediately die. A few centimeters of matter is a chain of a billion atoms. Stars and galaxies also number in the many billions.

In 1980, when the television series “Cosmos” began airing, people were ready to count in billions. Millions were no longer enough; they did not strike the imagination. Meanwhile, these two words sound similar, it is easy to confuse them. Therefore, on the air of “Cosmos” I pronounced “billion” with such emphasized articulation that many viewers considered it an accent or a speech impediment.

I remember an old joke. A planetarium lecturer tells visitors that in 5 billion years, the Sun will become a red giant and engulf Mercury and Venus, and eventually, perhaps, the Earth. After the lecture, an alarmed listener grabs onto him:

- Sorry, what did you say? Will the sun burn the Earth in five billion years?

- Yes, approximately.

- God bless! I thought I heard “five million.”

Whether five million or five billion years, the future demise of the Earth is of purely theoretical interest to us. But when it comes to government budgets, the world's population, or the number of victims of a nuclear war, the difference between these values ​​is very important.

The phrase “billions and billions” has not yet lost its popularity, but it also lacks scope. A new standard of large numbers is just around the corner - trillion.

Global military spending already reaches $1 trillion a year. The total debt of developing countries to Western banks is approaching $2 trillion (compared to $60 billion in 1970). The US government's annual budget is also close to $2 trillion. The national debt is about $5 trillion. The cost of the technically dubious Reagan-era undertaking, the Strategic Defense Initiative, was estimated at $1 trillion to $2 trillion. All the plants on Earth weigh a trillion tons. On the scale of space, literally everything is measured in trillions. From the Solar System to the nearest star, Alpha Centauri, is about 40 trillion km.

In everyday life, people traditionally confuse millions, billions and trillions, and hardly a week goes by without a similar error in television news (most often a million and a billion “suffer”). Therefore, let me remind you once again: a million (million in English) is a thousand thousand, or one followed by six zeros; billion (billion) – a thousand million, one followed by nine zeros; trillion (trillion) – a thousand billion (or, this is the same thing, a million millions), which is written as a unit followed by 12 zeros.

This is how it is done in America. For a long time in British English, the word billion denoted the number that in America is called a trillion, and the American billion, quite rightly, was called by the British “a thousand million.” In Europe, a billion was denoted by the word milliard. I have been collecting postage stamps since childhood, and I have an uncanceled stamp issued in Germany in 1923, at the height of inflation, with the inscription “50 billion.” It cost 50 trillion German marks to send the letter. (In those days, you went to a bakery or grocery store with a wheelbarrow full of cash.) Nowadays, due to the great influence of the United States on the world, the word milliard has fallen out of use in many countries.

The most reliable way to understand what number we are talking about is very simple - count the zeros after the one. True, if there are a lot of zeros, this is a bummer. Therefore, groups of three zeros are separated by commas or spaces when writing. For example, a trillion looks like 1,000,000,000,000 or 1,000,000,000,000. (In Europe, dots are used instead of commas.) When faced with numbers larger than a trillion, you should count each time how many times they contain three zeros. It would be much more convenient, when naming a number, to immediately say how many zeros there are after the one.

Scientists and mathematicians, practical people, do just that - they use the so-called exponential notation. The number ten is written, to which an indicator is added in small print at the top right - a number corresponding to the number of characters after one. Thus, 10 6 = 1,000,000, 10 9 = 1,000,000,000, 10 12 = 1,000,000,000,000, etc. This exponent is called the exponent, power or order of a number. For example, 10 9 is read as “ten to the ninth power” (the exceptions are 10 2 and 10 3, which are usually called “ten squared” and “ten cubed”). The concept of power or order—along with some other terms from science and mathematics, such as “parameter”—is making its way into everyday language, but its meaning is increasingly blurred.

In addition to its clarity, exponential notation has the wonderful added benefit of being able to multiply any two numbers by simply adding their powers. Let's say 1000 × 1,000,000,000 = 10 3 × 10 9 = 10 12. Or let's take larger numbers: in the average galaxy there are 10 11 stars, the galaxies themselves are also 10 11, therefore, there are about 10 22 stars in space.

Nevertheless, exponential notation is met with hostility by people who are not good at mathematics (although it, on the contrary, is easier to understand), and typesetters who don’t feed them bread - let them type 109 instead of 10 9 (employees of the Random House publishing house, as you can see, are a happy exception).

The first six large numbers, which have names, are given in the box below. Each number is 1000 times larger than the previous one. The names of numbers greater than a trillion are practically not used. If you count around the clock without stopping, adding one per second, it will take more than a week to count to a million. A billion will take you half your life. You will not reach the quintillion, even if you live as long as the Universe exists.

Once you have mastered exponential notation, you can easily cope with incomprehensibly large numbers, such as the approximate number of microbes in a teaspoon of soil (10 8), grains of sand on all the beaches of the earth (about 10 20), living creatures on our planet (10 29), all life on Earth (10 41), atomic nuclei in the Sun (10 57) or elementary particles (electrons, protons, neutrons) throughout space (10 80). You still won't be able to introduce a billion or quintillion objects - and no one can. But thanks to exponential notation, we are able operate similar values ​​and use them in calculations. Not bad for self-taught people who came into this world with nothing and counted their fellow tribesmen on their fingers and toes!

The names of even larger numbers are sextillion (10 21), septillion (10 24), octillion (10 27), nonillion (10 30) and decillion (10 33). The mass of the Earth is 6 octillion grams.

In addition to the exponential notation accepted in science, each number can also be expressed in words using prefixes. For example, an electron has one femtometer (10–15 m) across, the wavelength of yellow light is half a micrometer (0.5 μm), the human eye can distinguish an insect the size of one tenth of a millimeter (10–4 m), the radius of the Earth is 6300 km ( 6.3 megameters), the weight of an average mountain is 100 petagrams (10 17 g). Here are all the possible prefixes and their meanings:

Truly large numbers are the life and blood of modern science, but one should not think that this is a modern invention.

In India, arithmetic has long mastered huge numbers. In Indian newspapers one often reads about expenses in lakhs or crores of rupees. The system is as follows: Das - 10, San - 100, Hazar - 1000, Lakh - 10 5, Crore - 10 7, Arahb - 10 9, Karab - 10 11, Nie - 10 13, Padham - 10 15 and Sankh - 10 17. The Mayan Indians who lived on the territory of modern Mexico, whose civilization was destroyed by newcomers from Europe, compiled a calendar, the length of which pales in comparison to the measly few thousand years that, according to Europeans, had passed since the creation of the world. Among the ruins of the city of Coba in the Mexican state of Quintana Roo, inscriptions were discovered according to which the Mayans estimated the age of the Universe at about 10 29 years. Hindus believed that the current incarnation of the Universe is 8.6 × 10 9 years old - and they were almost right. And the mathematician Archimedes, who lived in Sicily in the 3rd century. BC e., in his book “Calculus of Grains of Sand” calculated that 10 63 grains of sand are needed to fill the entire space. Even in those days, billions and billions were clearly not enough to solve truly large-scale problems.

Chapter 2
Persian chess

There can be no language more comprehensive than analytical equations, and simpler, free from errors and ambiguities, that is, more worthy of expressing the immutable relationships of the real world... Mathematical analysis, being a faculty of the human mind, makes up for the brevity of our lives and the imperfection of our feelings.

– Jean Baptiste Joseph Fourier. Analytical theory of heat (1822) 4
Quote from: Life of Science. Anthology of introductions to the classics of natural science. – M.: Nauka, 1973.

In the version I know, this story took place in Ancient Persia, although it could just as well have happened in India and even China. In any case, it was a long time ago. The Grand Vizier, the chief adviser to the ruler, invented a new game in which it was necessary to move pieces on a square board lined with 64 squares of red and black. The most important figure was the ruler, the next most important was the vizier, as one would expect, given the personality of the inventor. The player's goal was to destroy the enemy's main piece, and according to the corresponding words in the Persian language (check - ruler, checkmate - death), the game was called "chess". Literally, “death of the ruler.” In Russian, this game is still called that way, which apparently reflects the special revolutionary spirit of the Russian people. Time passed, the pieces, their moves and the rules of the game changed. Thus, the place of the vizier is now taken by the queen, who has incomparably greater capabilities.

How the Shah could have liked the game “Kill the Ruler” is a mystery. However, the legend says, the Shah was so delighted with the new entertainment that he invited the Grand Vizier to appoint himself a reward. The proposal did not take him by surprise. The vizier replied that he was a modest man and his request would be the most modest. Here is a game board laid out in eight columns and eight rows. Let only one grain of wheat be placed on the first cell, twice as much on the second, and twice as much on the third. more twice as many and so on until all the cells are filled. The Shah protested. Such a small price to pay for such a wonderful invention! He offered jewelry, beauties, palaces. But the sage, humbly lowering his head, rejected any gifts. All he needs is a little wheat. And the ruler, secretly complaining about the unpretentiousness and stubbornness of his adviser, agreed.

However, when the keeper of the royal granary began to count out the grain, an unpleasant surprise revealed itself. It all started small: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024... But the further, the more monstrous, unimaginably huge the numbers became. The last, 64th cell corresponds to almost 18.5 quintillion (see sidebar below). The Grand Vizier was probably on a high fiber diet.

How much do 18.5 quintillion grains of wheat weigh? If we take the size of each grain to be equal to a millimeter, then their total weight will be about 75 billion tons - much more than the reserves of any Shah. Strictly speaking, this is the harvest for 150 years at modern production volumes. What follows is lost in the darkness of time. Whether the Shah ceded his power to the sage, reproaching himself for neglecting arithmetic, or preferred to play the new game “Vizimat”, remained unknown.

Perhaps the story of the invention of Persian chess is just a fairy tale. But the ancient Persians and Indians actually made many brilliant discoveries in mathematics and were well aware of the results that successive doubling gave. If chess games were played on a board with a hundred squares (10 × 10), the checker would owe the vizier wheat with a total weight equal to the weight of the Earth. A numerical sequence in which each subsequent number is the result of multiplying the previous one by a fixed value is called a geometric progression, and the corresponding process of increasing the total is called exponential growth.

Geometric progressions are found in all important areas of life, ordinary and exotic. Let's take compound interest as an example. If your ancestor 200 years ago, shortly after the Revolutionary War, deposited $10 into the bank at 5% interest, the account would already have $10 x 1.05,200, or $172,925.81. (To find out how much 1.05 200 is, you simply need to multiply the number 1.05 by itself 200 times.) It is a pity that few ancestors care so much about the well-being of distant descendants, and $10 - a substantial sum in those days - would not have been found everyone has it. If your generous ancestor placed a deposit at 6% per annum, you would inherit more than a million, at 7% - more than $7.5 million. A fantastic 10% per annum would bring you a tidy sum - $1.9 billion.

Inflation works in a similar way. At an inflation rate of 5% per year, a dollar in a year will cost $0.95, in two years (0.95)² = $0.91, in 10 years it will “lose weight” to $0.61, in 20 – to $0.37, etc. d. This most directly concerns pensioners, whose annual payments are fixed and are not indexed for inflation.

Repeated doubling and, therefore, exponential growth is a characteristic feature of the reproduction of biological organisms. Let's start with a simple example - a bacterium that reproduces by division. After the required time, each of the two daughter bacteria also divides into two. With a sufficient amount of food and the absence of poisons in the habitat, the colony of bacteria grows exponentially. Under the most favorable conditions, their numbers double approximately every 15 minutes: four doublings per hour, 96 per day. One bacterium weighs about one trillionth of a gram, but in just a day of unbridled reproduction its descendants will be equal in weight to a mountain, in about a day and a half - to the Earth, and in two days they will outweigh the Sun... A little more time will pass, and the entire Universe will be filled with bacteria. The prospects are not very rosy, but, fortunately, unrealistic. Why? Because such exponential growth inevitably runs into one or another natural barrier. Bugs, for example, eat all the food, poison each other, or stop mating when there are too many of them. Exponential growth cannot continue forever, otherwise it would consume everything, but long before that, a limiting factor comes into play. The exponential curve reaches a plateau (see figure).

This feature is very important in light of the AIDS epidemic. Nowadays, in many countries, the number of people with symptoms of this disease is growing exponentially, doubling every year. This means that every year there are twice as many carriers of the AIDS virus than there were last year. AIDS has already caused enormous loss of life. If the exponential spread of the epidemic continued, humanity would face an unprecedented catastrophe. In 10 years, the number of cases would increase a thousand times, and in 20 years – a million. Multiplying the current number of patients by a million, we get a result that far exceeds the population of the Earth. If the annual doubling of the number of AIDS patients had no natural limits, and the disease itself inevitably led to a fatal outcome (and was absolutely incurable), then all earthlings would die of AIDS, and very quickly.

However, some people have innate immunity. Additionally, according to the U.S. Department of Health's Center for Infectious Diseases, the exponential rise in AIDS incidence initially occurred almost exclusively in high-risk groups—primarily homosexual men, hemophiliacs, and intravenous drug users—who have virtually no sexual contact. with the rest of the population. If a cure for AIDS is not found, heroin addicts who share needles will die. Not all thanks to innate immunity, but almost all. The same fate awaits homosexuals who have promiscuous and unprotected relationships, but it will not happen to those who never neglect the means of protection, remain faithful to a permanent partner, and are also among the lucky few who are immune to this disease. 100% heterosexual, since the early 1980s. those who do not cheat on each other (or practice safe sex), provided that they do not share syringes - and the majority of such people - are generally insured against AIDS. After the incidence curves of the most at-risk groups reach a plateau, it will be the turn of the less vulnerable groups. Currently in America, these are apparently heterosexual youth who do not know how to curb their passions and are carried away by unsafe experiments. Many of them will die, some will be saved by luck, innate immunity or abstinence, and the peak of the increase in incidence will move to the next riskiest group - perhaps the next generation of homosexual men. As a result, the exponential curve for all of humanity will reach a plateau, and AIDS will kill far fewer people than the entire population of the Earth (which, however, is unlikely to console many of its victims and their loved ones).

* * *

The Earth's overpopulation crisis is also associated with the phenomenon of exponential growth. For most of the existence of mankind, its population was practically constant - birth rates and deaths balanced each other. This state is called dynamic equilibrium. With the development of agriculture, including methods of cultivating the very wheat that the Grand Vizier craved, the human population began to increase and entered a phase of exponential growth. And this is anything but balance. Now the population of our planet is doubling in 40 years. Every 40 years our number doubles. The English priest Thomas Malthus noted back in 1798: a population growing exponentially - in Malthus's terminology, in geometric progression - will starve with any increase in food production. No green revolution, no hydroponics, no development of deserts will compensate for the exponential growth in the number of eaters.

There is also no solution to this problem outside of Earth. There are now about 240,000 more people born than there are deaths every day. Our capabilities, to put it mildly, are insufficient to send 240,000 settlers into space per day. No bases in Earth orbit, on the Moon or on Mars will accommodate any significant part of such a rapidly growing population. Even if we could transfer everyone to other star systems on ships with superluminal speeds, it would not help. All habitable planets in the Milky Way would be overpopulated within a millennium. Everything is useless until we reduce the rate of reproduction. Exponential growth is serious business.

The figure below shows the Earth's population change curve. Now we are just in a phase of sharp exponential growth (or we are about to exit it). But many countries, such as the United States, Russia and China, have already reached or will soon reach a state in which population growth slows down and a dynamic equilibrium sets in. This is the so-called zero population growth. However, the growth in geometric progression is so large-scale that the overall situation will still not change if at least a small part of earthlings continues to reproduce exponentially. The total human population will also grow exponentially.

There is a convincingly proven connection between poverty and high birth rates. In countries large and small, capitalist and communist, Catholic and Muslim, in the West and in the East - almost everywhere, exponential population growth slows down or stops when poverty is overcome. A so-called demographic shift is taking place. The most urgent strategic need of the human species is to realize this shift in every corner of our planet. Not only out of moral obligation, but also from the point of view of direct benefit, rich countries must help poor countries achieve economic stability. One of the main reasons for the global demographic crisis is poverty.

There are interesting exceptions to the demographic shift. There are countries where, despite a high per capita income, the birth rate is still significant. These are states where the population has virtually no access to contraception and/or women have virtually no political influence. The connection between these factors is obvious.

At the time of writing this book, there were about 6 billion people living on Earth. If current growth rates continue, in 40 years there will be 12 billion of us, in 80 years - 24 billion, in 120 years - 48 billion... The general consensus is that the Earth is not capable of feeding so many people. There are already so many of us that eradicating poverty on a global scale seems to be the cheapest and, of course, the most humane way of overcoming the crisis that humanity will have for many decades to come. Our task is to ensure a widespread demographic shift and bring the population growth curve to a plateau. We must lift people in all countries out of poverty, provide them with safe and effective means of birth control, and make women a real force in society (by allowing them into executive, legislative, judicial, law enforcement agencies and institutions that influence public opinion). Otherwise, other processes over which we have no control are activated.

* * *

By the way...

The idea of ​​fission of the atomic nucleus first came to the mind of the Hungarian emigre physicist Leo Szilard in London in September 1933. He was interested in the question of whether man could release the colossal energy contained in the nucleus of an atom. What happens if you send a neutron into a nucleus? (Due to the lack of electric charge, the neutron will not be repelled by positively charged protons and will collide with the nucleus.) While the scientist was waiting for the traffic light to turn green at the Southampton Row intersection, it dawned on him that there might be some substance, some chemical element, the atom of which collision with one neutron emits two! Each of these two neutrons can knock out a couple more neutrons from other atoms... And before Szilard’s mind’s eye a picture of a nuclear chain reaction appeared - an avalanche-like process of exponential multiplication of neutrons and the decay of atomic nuclei. That evening, in a small room at the Strand Palace Hotel, he calculated that just a few kilograms of the substance, if it could cause a controlled chain reaction in it, could supply energy to a small city for a whole year... Or instantly destroy this city if all this energy were released at once. Subsequently, Szilard emigrated to the United States and began methodically sorting through chemical elements in search of one whose atoms, when bombarded with neutrons, emit more neutrons than they collide with. Uranus seemed promising to him. Szilard persuaded Albert Einstein to write a now famous letter to President Roosevelt calling for the creation of an atomic bomb in the United States as soon as possible. Szilard led the first successful experiment on the fission of uranium nuclei, carried out in Chicago in 1942, which became the first step towards the creation of nuclear weapons, and for the rest of his life he tried to open mankind’s eyes to the danger of the monstrous force that he himself had unleashed. This is how Leo Szilard discovered the power of exponential growth.

* * *

Each of us has two parents, four grandparents, eight great-grandparents, 16 great-great-grandparents, etc. With each transition back one generation, the number of ancestors in the direct line doubles. The analogies with the legend of Persian chess are obvious. Let's say generations change every 25 years. Then 64 generations fit into 64 × 25 = 1600 years (how many years ago Ancient Rome approached its collapse). It turns out that everyone living today should have had in 400 AD. e. about 18.5 quintillion ancestors (see sidebar)? And without taking into account relatives along the lateral lines. But this is much more than the entire population of the Earth, then and now, much more than the total number of people who have ever lived. Where is the mistake? On the premise that all our direct ancestors are different people. Of course, this is not true. Many lines of inheritance go back to the same ancestor. We are connected to each of us by many intersecting family ties. As we move deeper into time, the number of intersections becomes enormous.

This applies to all humanity. If you go deep enough into the past, you can find the common ancestor of any two people on Earth. Every time a new US president is elected, someone - usually in England - is sure to find his family ties to the British queen. This is believed to bring English speakers together. For two representatives of the same people or ethnic group, or natives of a fairly isolated region, who have a documented genealogy, finding a common ancestor is not a problem. But even if traces of it are lost in the distance of time, it is definitely there. We are all relatives - absolutely everyone living on Earth.

CALCULATION OF THE REMUNERATION THAT THE SHAH SHOULD HAVE PAID TO THE VAZIR

Let's calculate how many grains of wheat needed to be placed on the chessboard from the Persian legend. Don't be scared, it's not difficult!

The following elegant move will allow us to get an almost exact result.

The exponent tells you how many times we multiplied the number 2 by itself. 2 2 = 4, 2 4 = 16, 2 10 = 1024, etc. Let S– the total number of grains on the board, from one grain on the first cell to 263 on the 64th. Then obviously:

S = 1 + 2 + 2 2 + 2 3 +… + 2 62 + 2 63 .

Simply multiplying both sides of the equation by two, we bring it to the form:

2 S = 2 + 2 2 + 2 3 + 2 4 +… + 2 63 + 2 64 .

Subtract the first equation from the second:

2 S – S = S = 2 64 – 1.

This is the answer.

To represent the size of this number, let's move on to regular decimal notation. 2 10 is approximately 1000, or 10 3 (with a difference of 2.4%). Thus, 2 20 = 2 (10 × 2) = (2 10) 2 = approximately (10 3) 2 = 10 6, which is 10 taken six times, i.e. a million. Likewise, 2 60 = (2 10) 6 = approximately (10 3) 6 = 10 18. Then 2 64 = 2 4 × 2 60 = about 16 × 10 18, or 16 followed by eighteen zeros - 16 quintillion grains. An accurate calculation gives 18.6 quintillion.

* * *

Another striking example of geometric progression is associated with the concept of the so-called half-life. A radioactive “parent” element, say plutonium, decays into a daughter element, usually less dangerous. This is not an instant process; it occurs over time according to a certain law. After a certain period - the half-life - half of the atoms will decay. The remaining ones will continue to divide, after another similar time interval, half of them will also decay, etc. For example, with a half-life equal to a year, half of the available number of atoms will decay in a year, half of the remaining half, or a quarter of the original number, in two years, one eighth of the initial number - after three years, one thousandth - after 10 years, etc. Each element has its own half-life. This indicator must be taken into account when solving the problem of storing spent fuel from a nuclear power plant or when calculating the radioactive contamination of an area during a nuclear war. This is an example of exponential decrease, the opposite of the exponential growth represented in the Persian legend.

Translator Natalia Kiechenko

Editor Vyacheslav Ionov

Scientific editor Vladimir Surdin, Ph.D. physical – mat. sciences

Project Manager I. Seregina

Proofreaders M. Milovidova, E. Aksenova

Computer layout A. Fominov

Cover designer Yu. Buga

© 1997 by The Estate of Carl Sagan with permission from Democritus Properties, LLC.

© Publication in Russian, translation, design. Alpina Non-Fiction LLC, 2017

All rights reserved. The work is intended exclusively for private use. No part of the electronic copy of this book may be reproduced in any form or by any means, including posting on the Internet or corporate networks, for public or collective use without the written permission of the copyright owner. For violation of copyright, the law provides for payment of compensation to the copyright holder in the amount of up to 5 million rubles (Article 49 of the Code of Administrative Offenses), as well as criminal liability in the form of imprisonment for up to 6 years (Article 146 of the Criminal Code of the Russian Federation).

To my sister Kari, one of six billion

There are people who think that the number of grains of sand is infinite. ...Others think that although this number is not infinite, it is impossible to imagine more. ...On the contrary, I will try to prove with geometric precision that will convince you that ... there are numbers greater than the number of grains of sand that can be contained not only in a space equal to the volume of the earth ... but also the whole world.

I swear, I didn’t say the phrase “billions and billions”! I could say, for example, “100 billion galaxies and 10 billion trillion stars.” It is impossible to describe space without resorting to large numbers. I repeatedly uttered the word “billion” in the programs of the television series “Cosmos”, which were watched by a great many viewers. But “billions and billions” – never. If only because it is too vague. “Billions and billions” – how much is that? Two or three billion? Twenty? One hundred? The spread is too wide. When preparing the new edition of the TV series, I carefully reviewed everything and made sure that I didn’t say anything like that.

This phrase was said by Johnny Carson, whose guest I have been on “The Tonight Show” at least three dozen times. He dresses up in a corduroy jacket and turtleneck, tousles his hair and, parodying me, a kind of double, launches something like “billions and billions” on the evening television. And I'm starting to get tired of the fact that this parody takes on a life of its own, uttering maxims with which friends and colleagues stun me the next morning. (Otherwise, I admit that the serious amateur astronomer Carson most often speaks in strict scientific language.)

Alas, “billions and billions” stuck. People like the sound of it. Every now and then people call me on the street, on a plane, at a party and, with slight embarrassment, ask me to do the courtesy of saying: “Billions and billions.”

“You see, these are not my words,” I explain.

“Well, okay,” they answer me. - Tell me anyway.

It turns out that Sherlock Holmes never said “basically, Watson” (at least in Arthur Conan Doyle’s books), Jimmy Cagney never said “you dirty rat,” and Humphrey Bogart’s character never said “play that again, Sam.” But these phrases are so ingrained in popular culture that they are perceived as actually being said.

So the not-so-successful expression about billions is still attributed to me in computer magazines (“Carl Sagan would say that it takes billions and billions of bytes”), in economic reviews on the pages of newspapers, in stories about the salaries of sports stars, etc. d.

In the past, I would never have repeated this, either verbally or in writing – out of harm’s way. But now I've outgrown it. And if this is necessary for the story, please:

– Billions and billions!

Why has this phrase become so popular? Once upon a time, the symbol for large numbers was “million”. The super rich were millionaires. The population of the Earth at the time of Jesus was about 250 million people. The Convention of 1787 gave the Constitution to four million Americans; by the beginning of the Second World War there were already 132 million of us. There are 150 million km from the Earth to the Sun. About 40 million people died in World War I, and 60 million in World War II. There are 31.7 million seconds in a year (you can check it). The cumulative power of nuclear arsenals accumulated by the end of the 1980s would be enough to destroy a million Hiroshimas. For a long time, in most cases, the word “million” essentially meant “incredibly many.”

But times have changed. Nowadays there is a layer in the world billionaires, and not at all because the money has depreciated. The age of the Earth is generally accepted to be 4.6 billion years. The population has long exceeded 6 billion. Your two birthdays are separated by one year and a billion kilometers (the Earth moves around the Sun much faster than the Voyagers launched into space move away from it). Four B-2 bombers cost a billion dollars (according to other estimates, two or even four billion). The annual US defense budget, taking into account all hidden costs, exceeds $300 billion. In the event of a full-scale nuclear war between the US and Russia, about a billion people would immediately die. A few centimeters of matter is a chain of a billion atoms. Stars and galaxies also number in the many billions.

In 1980, when the television series “Cosmos” began airing, people were ready to count in billions. Millions were no longer enough; they did not strike the imagination. Meanwhile, these two words sound similar, it is easy to confuse them. Therefore, on the air of “Cosmos” I pronounced “billion” with such emphasized articulation that many viewers considered it an accent or a speech impediment.

I remember an old joke. A planetarium lecturer tells visitors that in 5 billion years, the Sun will become a red giant and engulf Mercury and Venus, and eventually, perhaps, the Earth. After the lecture, an alarmed listener grabs onto him:

- Sorry, what did you say? Will the sun burn the Earth in five billion years?

- Yes, approximately.

- God bless! I thought I heard “five million.”

Whether five million or five billion years, the future demise of the Earth is of purely theoretical interest to us. But when it comes to government budgets, the world's population, or the number of victims of a nuclear war, the difference between these values ​​is very important.

The phrase “billions and billions” has not yet lost its popularity, but it also lacks scope. A new standard of large numbers is just around the corner - trillion.

Global military spending already reaches $1 trillion a year. The total debt of developing countries to Western banks is approaching $2 trillion (compared to $60 billion in 1970). The US government's annual budget is also close to $2 trillion. The national debt is about $5 trillion. The cost of the technically dubious Reagan-era undertaking, the Strategic Defense Initiative, was estimated at $1 trillion to $2 trillion. All the plants on Earth weigh a trillion tons. On the scale of space, literally everything is measured in trillions. From the Solar System to the nearest star, Alpha Centauri, is about 40 trillion km.

Translator Natalia Kiechenko

Editor Vyacheslav Ionov

Scientific editor Vladimir Surdin, Ph.D. physical – mat. sciences

Project Manager I. Seregina

Proofreaders M. Milovidova, E. Aksenova

Computer layout A. Fominov

Cover designer Yu. Buga

© 1997 by The Estate of Carl Sagan with permission from Democritus Properties, LLC.

© Publication in Russian, translation, design. Alpina Non-Fiction LLC, 2017

All rights reserved. The work is intended exclusively for private use. No part of the electronic copy of this book may be reproduced in any form or by any means, including posting on the Internet or corporate networks, for public or collective use without the written permission of the copyright owner. For violation of copyright, the law provides for payment of compensation to the copyright holder in the amount of up to 5 million rubles (Article 49 of the Code of Administrative Offenses), as well as criminal liability in the form of imprisonment for up to 6 years (Article 146 of the Criminal Code of the Russian Federation).

* * *

To my sister Kari, one of six billion

Part I
The clarity and beauty of numbers

Chapter 1
Billions and Billions

There are people who think that the number of grains of sand is infinite. ...Others think that although this number is not infinite, it is impossible to imagine more. ...On the contrary, I will try to prove with geometric precision that will convince you that ... there are numbers greater than the number of grains of sand that can be contained not only in a space equal to the volume of the earth ... but also the whole world.

– Archimedes (c. 287–212 BC). Calculus of grains of sand

I swear, I didn’t say the phrase “billions and billions”! I could say, for example, “100 billion galaxies and 10 billion trillion stars.” It is impossible to describe space without resorting to large numbers. I repeatedly uttered the word “billion” in the programs of the television series “Cosmos”, which were watched by a great many viewers. But “billions and billions” – never. If only because it is too vague. “Billions and billions” – how much is that? Two or three billion? Twenty? One hundred? The spread is too wide. When preparing the new edition of the TV series, I carefully reviewed everything and made sure that I didn’t say anything like that.

This phrase was said by Johnny Carson, whose guest I have been on “The Tonight Show” at least three dozen times. He dresses up in a corduroy jacket and turtleneck, tousles his hair and, parodying me, a kind of double, launches something like “billions and billions” on the evening television. And I'm starting to get tired of the fact that this parody takes on a life of its own, uttering maxims with which friends and colleagues stun me the next morning. (Otherwise, I admit that the serious amateur astronomer Carson most often speaks in strict scientific language.)

Alas, “billions and billions” stuck. People like the sound of it. Every now and then people call me on the street, on a plane, at a party and, with slight embarrassment, ask me to do the courtesy of saying: “Billions and billions.”

“You see, these are not my words,” I explain.

“Well, okay,” they answer me. - Tell me anyway.

It turns out that Sherlock Holmes never said “basically, Watson” (at least in Arthur Conan Doyle’s books), Jimmy Cagney never said “you dirty rat,” and Humphrey Bogart’s character never said “play that again, Sam.” But these phrases are so ingrained in popular culture that they are perceived as actually being said.

So the not-so-successful expression about billions is still attributed to me in computer magazines (“Carl Sagan would say that it takes billions and billions of bytes”), in economic reviews on the pages of newspapers, in stories about the salaries of sports stars, etc. d.

In the past, I would never have repeated this, either verbally or in writing – out of harm’s way. But now I've outgrown it. And if this is necessary for the story, please:

– Billions and billions!

Why has this phrase become so popular? Once upon a time, the symbol for large numbers was “million”. The super rich were millionaires. The population of the Earth at the time of Jesus was about 250 million people. The Convention of 1787 gave the Constitution to four million Americans; by the beginning of the Second World War there were already 132 million of us. There are 150 million km from the Earth to the Sun. About 40 million people died in World War I, and 60 million in World War II. There are 31.7 million seconds in a year (you can check it). The cumulative power of nuclear arsenals accumulated by the end of the 1980s would be enough to destroy a million Hiroshimas. For a long time, in most cases, the word “million” essentially meant “incredibly many.”

But times have changed. Nowadays there is a layer in the world billionaires, and not at all because the money has depreciated. The age of the Earth is generally accepted to be 4.6 billion years. The population has long exceeded 6 billion. Your two birthdays are separated by one year and a billion kilometers (the Earth moves around the Sun much faster than the Voyagers launched into space move away from it). Four B-2 bombers cost a billion dollars (according to other estimates, two or even four billion). The annual US defense budget, taking into account all hidden costs, exceeds $300 billion. In the event of a full-scale nuclear war between the US and Russia, about a billion people would immediately die. A few centimeters of matter is a chain of a billion atoms. Stars and galaxies also number in the many billions.

In 1980, when the television series “Cosmos” began airing, people were ready to count in billions. Millions were no longer enough; they did not strike the imagination. Meanwhile, these two words sound similar, it is easy to confuse them. Therefore, on the air of “Cosmos” I pronounced “billion” with such emphasized articulation that many viewers considered it an accent or a speech impediment.

I remember an old joke. A planetarium lecturer tells visitors that in 5 billion years, the Sun will become a red giant and engulf Mercury and Venus, and eventually, perhaps, the Earth. After the lecture, an alarmed listener grabs onto him:

- Sorry, what did you say? Will the sun burn the Earth in five billion years?

- Yes, approximately.

- God bless! I thought I heard “five million.”

Whether five million or five billion years, the future demise of the Earth is of purely theoretical interest to us. But when it comes to government budgets, the world's population, or the number of victims of a nuclear war, the difference between these values ​​is very important.

The phrase “billions and billions” has not yet lost its popularity, but it also lacks scope. A new standard of large numbers is just around the corner - trillion.

Global military spending already reaches $1 trillion a year. The total debt of developing countries to Western banks is approaching $2 trillion (compared to $60 billion in 1970). The US government's annual budget is also close to $2 trillion. The national debt is about $5 trillion. The cost of the technically dubious Reagan-era undertaking, the Strategic Defense Initiative, was estimated at $1 trillion to $2 trillion. All the plants on Earth weigh a trillion tons. On the scale of space, literally everything is measured in trillions. From the Solar System to the nearest star, Alpha Centauri, is about 40 trillion km.

In everyday life, people traditionally confuse millions, billions and trillions, and hardly a week goes by without a similar error in television news (most often a million and a billion “suffer”). Therefore, let me remind you once again: a million (million in English) is a thousand thousand, or one followed by six zeros; billion (billion) – a thousand million, one followed by nine zeros; trillion (trillion) – a thousand billion (or, this is the same thing, a million millions), which is written as a unit followed by 12 zeros.

This is how it is done in America. For a long time in British English, the word billion denoted the number that in America is called a trillion, and the American billion, quite rightly, was called by the British “a thousand million.” In Europe, a billion was denoted by the word milliard. I have been collecting postage stamps since childhood, and I have an uncanceled stamp issued in Germany in 1923, at the height of inflation, with the inscription “50 billion.” It cost 50 trillion German marks to send the letter. (In those days, you went to a bakery or grocery store with a wheelbarrow full of cash.) Nowadays, due to the great influence of the United States on the world, the word milliard has fallen out of use in many countries.

The most reliable way to understand what number we are talking about is very simple - count the zeros after the one. True, if there are a lot of zeros, this is a bummer. Therefore, groups of three zeros are separated by commas or spaces when writing. For example, a trillion looks like 1,000,000,000,000 or 1,000,000,000,000. (In Europe, dots are used instead of commas.) When faced with numbers larger than a trillion, you should count each time how many times they contain three zeros. It would be much more convenient, when naming a number, to immediately say how many zeros there are after the one.

Scientists and mathematicians, practical people, do just that - they use the so-called exponential notation. The number ten is written, to which an indicator is added in small print at the top right - a number corresponding to the number of characters after one. Thus, 10 6 = 1,000,000, 10 9 = 1,000,000,000, 10 12 = 1,000,000,000,000, etc. This exponent is called the exponent, power or order of a number. For example, 10 9 is read as “ten to the ninth power” (the exceptions are 10 2 and 10 3, which are usually called “ten squared” and “ten cubed”). The concept of power or order—along with some other terms from science and mathematics, such as “parameter”—is making its way into everyday language, but its meaning is increasingly blurred.

In addition to its clarity, exponential notation has the wonderful added benefit of being able to multiply any two numbers by simply adding their powers. Let's say 1000 × 1,000,000,000 = 10 3 × 10 9 = 10 12. Or let's take larger numbers: in the average galaxy there are 10 11 stars, the galaxies themselves are also 10 11, therefore, there are about 10 22 stars in space.

Nevertheless, exponential notation is met with hostility by people who are not good at mathematics (although it, on the contrary, is easier to understand), and typesetters who don’t feed them bread - let them type 109 instead of 10 9 (employees of the Random House publishing house, as you can see, are a happy exception).

The first six large numbers, which have names, are given in the box below. Each number is 1000 times larger than the previous one. The names of numbers greater than a trillion are practically not used. If you count around the clock without stopping, adding one per second, it will take more than a week to count to a million. A billion will take you half your life. You will not reach the quintillion, even if you live as long as the Universe exists.

Once you have mastered exponential notation, you can easily cope with incomprehensibly large numbers, such as the approximate number of microbes in a teaspoon of soil (10 8), grains of sand on all the beaches of the earth (about 10 20), living creatures on our planet (10 29), all life on Earth (10 41), atomic nuclei in the Sun (10 57) or elementary particles (electrons, protons, neutrons) throughout space (10 80). You still won't be able to introduce a billion or quintillion objects - and no one can. But thanks to exponential notation, we are able operate similar values ​​and use them in calculations. Not bad for self-taught people who came into this world with nothing and counted their fellow tribesmen on their fingers and toes!

The names of even larger numbers are sextillion (10 21), septillion (10 24), octillion (10 27), nonillion (10 30) and decillion (10 33). The mass of the Earth is 6 octillion grams.

In addition to the exponential notation accepted in science, each number can also be expressed in words using prefixes. For example, an electron has one femtometer (10–15 m) across, the wavelength of yellow light is half a micrometer (0.5 μm), the human eye can distinguish an insect the size of one tenth of a millimeter (10–4 m), the radius of the Earth is 6300 km ( 6.3 megameters), the weight of an average mountain is 100 petagrams (10 17 g). Here are all the possible prefixes and their meanings:

Truly large numbers are the life and blood of modern science, but one should not think that this is a modern invention.

In India, arithmetic has long mastered huge numbers. In Indian newspapers one often reads about expenses in lakhs or crores of rupees. The system is as follows: Das - 10, San - 100, Hazar - 1000, Lakh - 10 5, Crore - 10 7, Arahb - 10 9, Karab - 10 11, Nie - 10 13, Padham - 10 15 and Sankh - 10 17. The Mayan Indians who lived on the territory of modern Mexico, whose civilization was destroyed by newcomers from Europe, compiled a calendar, the length of which pales in comparison to the measly few thousand years that, according to Europeans, had passed since the creation of the world. Among the ruins of the city of Coba in the Mexican state of Quintana Roo, inscriptions were discovered according to which the Mayans estimated the age of the Universe at about 10 29 years. Hindus believed that the current incarnation of the Universe is 8.6 × 10 9 years old - and they were almost right. And the mathematician Archimedes, who lived in Sicily in the 3rd century. BC e., in his book “Calculus of Grains of Sand” calculated that 10 63 grains of sand are needed to fill the entire space. Even in those days, billions and billions were clearly not enough to solve truly large-scale problems.

Chapter 2
Persian chess

There can be no language more comprehensive than analytical equations, and simpler, free from errors and ambiguities, that is, more worthy of expressing the immutable relationships of the real world... Mathematical analysis, being a faculty of the human mind, makes up for the brevity of our lives and the imperfection of our feelings.

– Jean Baptiste Joseph Fourier. Analytical theory of heat (1822)

In the version I know, this story took place in Ancient Persia, although it could just as well have happened in India and even China. In any case, it was a long time ago. The Grand Vizier, the chief adviser to the ruler, invented a new game in which it was necessary to move pieces on a square board lined with 64 squares of red and black. The most important figure was the ruler, the next most important was the vizier, as one would expect, given the personality of the inventor. The player's goal was to destroy the enemy's main piece, and according to the corresponding words in the Persian language (check - ruler, checkmate - death), the game was called "chess". Literally, “death of the ruler.” In Russian, this game is still called that way, which apparently reflects the special revolutionary spirit of the Russian people. Time passed, the pieces, their moves and the rules of the game changed. Thus, the place of the vizier is now taken by the queen, who has incomparably greater capabilities.

How the Shah could have liked the game “Kill the Ruler” is a mystery. However, the legend says, the Shah was so delighted with the new entertainment that he invited the Grand Vizier to appoint himself a reward. The proposal did not take him by surprise. The vizier replied that he was a modest man and his request would be the most modest. Here is a game board laid out in eight columns and eight rows. Let only one grain of wheat be placed on the first cell, twice as much on the second, and twice as much on the third. more twice as many and so on until all the cells are filled. The Shah protested. Such a small price to pay for such a wonderful invention! He offered jewelry, beauties, palaces. But the sage, humbly lowering his head, rejected any gifts. All he needs is a little wheat. And the ruler, secretly complaining about the unpretentiousness and stubbornness of his adviser, agreed.

However, when the keeper of the royal granary began to count out the grain, an unpleasant surprise revealed itself. It all started small: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024... But the further, the more monstrous, unimaginably huge the numbers became. The last, 64th cell corresponds to almost 18.5 quintillion (see sidebar below). The Grand Vizier was probably on a high fiber diet.

How much do 18.5 quintillion grains of wheat weigh? If we take the size of each grain to be equal to a millimeter, then their total weight will be about 75 billion tons - much more than the reserves of any Shah. Strictly speaking, this is the harvest for 150 years at modern production volumes. What follows is lost in the darkness of time. Whether the Shah ceded his power to the sage, reproaching himself for neglecting arithmetic, or preferred to play the new game “Vizimat”, remained unknown.

Perhaps the story of the invention of Persian chess is just a fairy tale. But the ancient Persians and Indians actually made many brilliant discoveries in mathematics and were well aware of the results that successive doubling gave. If chess games were played on a board with a hundred squares (10 × 10), the checker would owe the vizier wheat with a total weight equal to the weight of the Earth. A numerical sequence in which each subsequent number is the result of multiplying the previous one by a fixed value is called a geometric progression, and the corresponding process of increasing the total is called exponential growth.

Geometric progressions are found in all important areas of life, ordinary and exotic. Let's take compound interest as an example. If your ancestor 200 years ago, shortly after the Revolutionary War, deposited $10 into the bank at 5% interest, the account would already have $10 x 1.05,200, or $172,925.81. (To find out how much 1.05 200 is, you simply need to multiply the number 1.05 by itself 200 times.) It is a pity that few ancestors care so much about the well-being of distant descendants, and $10 - a substantial sum in those days - would not have been found everyone has it. If your generous ancestor placed a deposit at 6% per annum, you would inherit more than a million, at 7% - more than $7.5 million. A fantastic 10% per annum would bring you a tidy sum - $1.9 billion.

Inflation works in a similar way. At an inflation rate of 5% per year, a dollar in a year will cost $0.95, in two years (0.95)² = $0.91, in 10 years it will “lose weight” to $0.61, in 20 – to $0.37, etc. d. This most directly concerns pensioners, whose annual payments are fixed and are not indexed for inflation.

Repeated doubling and, therefore, exponential growth is a characteristic feature of the reproduction of biological organisms. Let's start with a simple example - a bacterium that reproduces by division. After the required time, each of the two daughter bacteria also divides into two. With a sufficient amount of food and the absence of poisons in the habitat, the colony of bacteria grows exponentially. Under the most favorable conditions, their numbers double approximately every 15 minutes: four doublings per hour, 96 per day. One bacterium weighs about one trillionth of a gram, but in just a day of unbridled reproduction its descendants will be equal in weight to a mountain, in about a day and a half - to the Earth, and in two days they will outweigh the Sun... A little more time will pass, and the entire Universe will be filled with bacteria. The prospects are not very rosy, but, fortunately, unrealistic. Why? Because such exponential growth inevitably runs into one or another natural barrier. Bugs, for example, eat all the food, poison each other, or stop mating when there are too many of them. Exponential growth cannot continue forever, otherwise it would consume everything, but long before that, a limiting factor comes into play. The exponential curve reaches a plateau (see figure).

This feature is very important in light of the AIDS epidemic. Nowadays, in many countries, the number of people with symptoms of this disease is growing exponentially, doubling every year. This means that every year there are twice as many carriers of the AIDS virus than there were last year. AIDS has already caused enormous loss of life. If the exponential spread of the epidemic continued, humanity would face an unprecedented catastrophe. In 10 years, the number of cases would increase a thousand times, and in 20 years – a million. Multiplying the current number of patients by a million, we get a result that far exceeds the population of the Earth. If the annual doubling of the number of AIDS patients had no natural limits, and the disease itself inevitably led to a fatal outcome (and was absolutely incurable), then all earthlings would die of AIDS, and very quickly.

However, some people have innate immunity. Additionally, according to the U.S. Department of Health's Center for Infectious Diseases, the exponential rise in AIDS incidence initially occurred almost exclusively in high-risk groups—primarily homosexual men, hemophiliacs, and intravenous drug users—who have virtually no sexual contact. with the rest of the population. If a cure for AIDS is not found, heroin addicts who share needles will die. Not all thanks to innate immunity, but almost all. The same fate awaits homosexuals who have promiscuous and unprotected relationships, but it will not happen to those who never neglect the means of protection, remain faithful to a permanent partner, and are also among the lucky few who are immune to this disease. 100% heterosexual, since the early 1980s. those who do not cheat on each other (or practice safe sex), provided that they do not share syringes - and the majority of such people - are generally insured against AIDS. After the incidence curves of the most at-risk groups reach a plateau, it will be the turn of the less vulnerable groups. Currently in America, these are apparently heterosexual youth who do not know how to curb their passions and are carried away by unsafe experiments. Many of them will die, some will be saved by luck, innate immunity or abstinence, and the peak of the increase in incidence will move to the next riskiest group - perhaps the next generation of homosexual men. As a result, the exponential curve for all of humanity will reach a plateau, and AIDS will kill far fewer people than the entire population of the Earth (which, however, is unlikely to console many of its victims and their loved ones).

* * *

The Earth's overpopulation crisis is also associated with the phenomenon of exponential growth. For most of the existence of mankind, its population was practically constant - birth rates and deaths balanced each other. This state is called dynamic equilibrium. With the development of agriculture, including methods of cultivating the very wheat that the Grand Vizier craved, the human population began to increase and entered a phase of exponential growth. And this is anything but balance. Now the population of our planet is doubling in 40 years. Every 40 years our number doubles. The English priest Thomas Malthus noted back in 1798: a population growing exponentially - in Malthus's terminology, in geometric progression - will starve with any increase in food production. No green revolution, no hydroponics, no development of deserts will compensate for the exponential growth in the number of eaters.

There is also no solution to this problem outside of Earth. There are now about 240,000 more people born than there are deaths every day. Our capabilities, to put it mildly, are insufficient to send 240,000 settlers into space per day. No bases in Earth orbit, on the Moon or on Mars will accommodate any significant part of such a rapidly growing population. Even if we could transfer everyone to other star systems on ships with superluminal speeds, it would not help. All habitable planets in the Milky Way would be overpopulated within a millennium. Everything is useless until we reduce the rate of reproduction. Exponential growth is serious business.

The figure below shows the Earth's population change curve. Now we are just in a phase of sharp exponential growth (or we are about to exit it). But many countries, such as the United States, Russia and China, have already reached or will soon reach a state in which population growth slows down and a dynamic equilibrium sets in. This is the so-called zero population growth. However, the growth in geometric progression is so large-scale that the overall situation will still not change if at least a small part of earthlings continues to reproduce exponentially. The total human population will also grow exponentially.

There is a convincingly proven connection between poverty and high birth rates. In countries large and small, capitalist and communist, Catholic and Muslim, in the West and in the East - almost everywhere, exponential population growth slows down or stops when poverty is overcome. A so-called demographic shift is taking place. The most urgent strategic need of the human species is to realize this shift in every corner of our planet. Not only out of moral obligation, but also from the point of view of direct benefit, rich countries must help poor countries achieve economic stability. One of the main reasons for the global demographic crisis is poverty.

There are interesting exceptions to the demographic shift. There are countries where, despite a high per capita income, the birth rate is still significant. These are states where the population has virtually no access to contraception and/or women have virtually no political influence. The connection between these factors is obvious.

At the time of writing this book, there were about 6 billion people living on Earth. If current growth rates continue, in 40 years there will be 12 billion of us, in 80 years - 24 billion, in 120 years - 48 billion... The general consensus is that the Earth is not capable of feeding so many people. There are already so many of us that eradicating poverty on a global scale seems to be the cheapest and, of course, the most humane way of overcoming the crisis that humanity will have for many decades to come. Our task is to ensure a widespread demographic shift and bring the population growth curve to a plateau. We must lift people in all countries out of poverty, provide them with safe and effective means of birth control, and make women a real force in society (by allowing them into executive, legislative, judicial, law enforcement agencies and institutions that influence public opinion). Otherwise, other processes over which we have no control are activated.

* * *

By the way...

The idea of ​​fission of the atomic nucleus first came to the mind of the Hungarian emigre physicist Leo Szilard in London in September 1933. He was interested in the question of whether man could release the colossal energy contained in the nucleus of an atom. What happens if you send a neutron into a nucleus? (Due to the lack of electric charge, the neutron will not be repelled by positively charged protons and will collide with the nucleus.) While the scientist was waiting for the traffic light to turn green at the Southampton Row intersection, it dawned on him that there might be some substance, some chemical element, the atom of which collision with one neutron emits two! Each of these two neutrons can knock out a couple more neutrons from other atoms... And before Szilard’s mind’s eye a picture of a nuclear chain reaction appeared - an avalanche-like process of exponential multiplication of neutrons and the decay of atomic nuclei. That evening, in a small room at the Strand Palace Hotel, he calculated that just a few kilograms of the substance, if it could cause a controlled chain reaction in it, could supply energy to a small city for a whole year... Or instantly destroy this city if all this energy were released at once. Subsequently, Szilard emigrated to the United States and began methodically sorting through chemical elements in search of one whose atoms, when bombarded with neutrons, emit more neutrons than they collide with. Uranus seemed promising to him. Szilard persuaded Albert Einstein to write a now famous letter to President Roosevelt calling for the creation of an atomic bomb in the United States as soon as possible. Szilard led the first successful experiment on the fission of uranium nuclei, carried out in Chicago in 1942, which became the first step towards the creation of nuclear weapons, and for the rest of his life he tried to open mankind’s eyes to the danger of the monstrous force that he himself had unleashed. This is how Leo Szilard discovered the power of exponential growth.

* * *

Each of us has two parents, four grandparents, eight great-grandparents, 16 great-great-grandparents, etc. With each transition back one generation, the number of ancestors in the direct line doubles. The analogies with the legend of Persian chess are obvious. Let's say generations change every 25 years. Then 64 generations fit into 64 × 25 = 1600 years (how many years ago Ancient Rome approached its collapse). It turns out that everyone living today should have had in 400 AD. e. about 18.5 quintillion ancestors (see sidebar)? And without taking into account relatives along the lateral lines. But this is much more than the entire population of the Earth, then and now, much more than the total number of people who have ever lived. Where is the mistake? On the premise that all our direct ancestors are different people. Of course, this is not true. Many lines of inheritance go back to the same ancestor. We are connected to each of us by many intersecting family ties. As we move deeper into time, the number of intersections becomes enormous.

This applies to all humanity. If you go deep enough into the past, you can find the common ancestor of any two people on Earth. Every time a new US president is elected, someone - usually in England - is sure to find his family ties to the British queen. This is believed to bring English speakers together. For two representatives of the same people or ethnic group, or natives of a fairly isolated region, who have a documented genealogy, finding a common ancestor is not a problem. But even if traces of it are lost in the distance of time, it is definitely there. We are all relatives - absolutely everyone living on Earth.

CALCULATION OF THE REMUNERATION THAT THE SHAH SHOULD HAVE PAID TO THE VAZIR

Let's calculate how many grains of wheat needed to be placed on the chessboard from the Persian legend. Don't be scared, it's not difficult!

The following elegant move will allow us to get an almost exact result.

The exponent tells you how many times we multiplied the number 2 by itself. 2 2 = 4, 2 4 = 16, 2 10 = 1024, etc. Let S– the total number of grains on the board, from one grain on the first cell to 263 on the 64th. Then obviously:

S = 1 + 2 + 2 2 + 2 3 +… + 2 62 + 2 63 .

Simply multiplying both sides of the equation by two, we bring it to the form:

2 S = 2 + 2 2 + 2 3 + 2 4 +… + 2 63 + 2 64 .

Subtract the first equation from the second:

2 S – S = S = 2 64 – 1.

This is the answer.

To represent the size of this number, let's move on to regular decimal notation. 2 10 is approximately 1000, or 10 3 (with a difference of 2.4%). Thus, 2 20 = 2 (10 × 2) = (2 10) 2 = approximately (10 3) 2 = 10 6, which is 10 taken six times, i.e. a million. Likewise, 2 60 = (2 10) 6 = approximately (10 3) 6 = 10 18. Then 2 64 = 2 4 × 2 60 = about 16 × 10 18, or 16 followed by eighteen zeros - 16 quintillion grains. An accurate calculation gives 18.6 quintillion.

* * *

Another striking example of geometric progression is associated with the concept of the so-called half-life. A radioactive “parent” element, say plutonium, decays into a daughter element, usually less dangerous. This is not an instant process; it occurs over time according to a certain law. After a certain period - the half-life - half of the atoms will decay. The remaining ones will continue to divide, after another similar time interval, half of them will also decay, etc. For example, with a half-life equal to a year, half of the available number of atoms will decay in a year, half of the remaining half, or a quarter of the original number, in two years, one eighth of the initial number - after three years, one thousandth - after 10 years, etc. Each element has its own half-life. This indicator must be taken into account when solving the problem of storing spent fuel from a nuclear power plant or when calculating the radioactive contamination of an area during a nuclear war. This is an example of exponential decrease, the opposite of the exponential growth represented in the Persian legend.

Carl Sagan's latest work is now available in Russian. “The Futurist” talks about the saddest and most long-awaited book on the birthday of the famous astronomer and popularizer of science.

“I swear, I didn’t say the phrase “billions and billions”!” the famous scientist and Pulitzer Prize winner begins the book with this. - I could say, for example, “100 billion galaxies and 10 billion trillion stars.” It is impossible to describe space without resorting to large numbers. I repeatedly uttered the word “billion” in the programs of the television series “Cosmos”, which were watched by a great many viewers. But “billions and billions” - never. If only because it’s too vague.”

In fact, this famous line was said by famous TV host Johnny Carson on the Tonights Show. He parodied the pronunciation of Sagan, who constantly emphasized the b in the word billions so that listeners would not confuse millions and billions. The scientist was not offended by the comedian and so titled his latest book - a collection of essays devoted to the most important problems of the environment and the future of humanity.

In the context of this book, “billions and billions” is not just a playful reference to a legend that has become popular. It is a metaphor for exponential growth, population explosion and nuclear decay, illustrating the scale of the problems facing the planet and humanity, billions and billions of people.

Hope for tomorrow

The book contains a lot of mathematics, which permeates all areas of our lives. Using the exponential curve, the author explains things of a very different order: the nuclear chain reaction, the AIDS epidemic and the problem of overpopulation.

“There is a convincingly proven connection between poverty and high birth rates. In countries large and small, capitalist and communist, Catholic and Muslim, in the West and in the East - almost everywhere, exponential population growth slows down or stops when poverty is overcome.<...>If current growth rates are maintained, in 40 years there will be 12 billion of us, in 80 years - 24 billion, in 120 years - 48 billion... There are already so many of us that eradicating poverty on a global scale seems to be the cheapest and, of course, the most humane way to overcome the crisis ", writes Sagan.

The scientist easily moves between topics: from the origins of human love for sports to foreign policy, from the history of chess to the problems of poverty, from the physics of light and sound to the search for alien life. Carl Sagan pays special attention to the study of the satellites of the planets of the solar system and the search for life on Mars.

The second part of the book is devoted to the problems of interaction between scientists and society. Carl Sagan covers the history of the fight to ban CFCs, the chemical compounds used in refrigerators and air conditioners. Widespread use of these substances depletes the ozone layer, which protects the planet from cosmic radiation. This is one example of the successful struggle of scientists against the conservatism of authorities and public organizations. The story of global warming is far from over. Carl Sagan analyzes research on the greenhouse effect and suggests ways to avoid catastrophic climate change. In the third, “When the Heart and the Mind Are at War,” Sagan demonstrates his views on political issues, in particular, relations between the United States and Russia, the problem of abortion and much more. This is one of Sagan's most political and incisive books, demonstrating that a scientific approach can be applied to any field.

In the two decades that have passed since the scientist’s death, the world has not changed much. Some of the forecasts did not come true: Sweden did not have time to switch to alternative energy until 2010. Until now, relations between Russia and the United States remain contradictory, the activities of many enterprises still threaten the environment, and faith conflicts with scientific rationalism. ​​And yet scientific progress cannot be stopped: the search for exoplanets continues, and the spacecraft of the ExoMars mission are preparing to look for life in the Martian soil. Sagan believed in future generations. This book is a message to those who will live on Earth in the coming centuries, something like the Voyager golden record, in the creation of which the astrophysicist was directly involved.

Roulette of Death

Like many of Sagan's works, Billions and Billions is about the future. But this book is read with particular bitterness. In the last chapter, Carl Sagan talks about his struggle with myelodysplastic syndrome, a blood disease that later ended his life. He undergoes a bone marrow transplant with a 30% chance of cure - "like playing Russian roulette with four bullets in the drum instead of one." The donor was his younger sister Kari. This book is dedicated to her.

“I would like to believe that even after death I will not lose life, that some part of me - thinking, feeling, storing memories - will continue to exist. But neither the attractiveness of this belief, nor the antiquity and ubiquity of cultural traditions of honoring the afterlife does not replace for me the absence of any facts indicating that these are not just dreams,” this is the essence of a real scientist, courageously looking into the face of death.

Thousands of Christians, Hindus, Jews, and Muslims prayed for the scientist when it became clear that Sagan was losing the fight against the disease. He was sincerely grateful to these people.

And yet, it was important for him to maintain the ability to think critically until the last day - this helped Sagan feel alive. People from all over the world mourned the scientist and thanked him for opening their eyes and inspiring them to work in the name of science. Until now, many scientists and popularizers call Carl Sagan their teacher. And, as his widow Ann Druyan notes in the epilogue, this is what makes her feel that Karl is still with us.