• Пожаловаться

John Paulos: INNUMERACY: Mathematical Illiteracy and Its Consequences

Здесь есть возможность читать онлайн «John Paulos: INNUMERACY: Mathematical Illiteracy and Its Consequences» весь текст электронной книги совершенно бесплатно (целиком полную версию). В некоторых случаях присутствует краткое содержание. категория: Математика / Математика / на английском языке. Описание произведения, (предисловие) а так же отзывы посетителей доступны на портале. Библиотека «Либ Кат» — LibCat.ru создана для любителей полистать хорошую книжку и предлагает широкий выбор жанров:

любовные романы фантастика и фэнтези приключения детективы и триллеры эротика документальные научные юмористические анекдоты о бизнесе проза детские сказки о религиии новинки православные старинные про компьютеры программирование на английском домоводство поэзия

Выбрав категорию по душе Вы сможете найти действительно стоящие книги и насладиться погружением в мир воображения, прочувствовать переживания героев или узнать для себя что-то новое, совершить внутреннее открытие. Подробная информация для ознакомления по текущему запросу представлена ниже:

John Paulos INNUMERACY: Mathematical Illiteracy and Its Consequences

INNUMERACY: Mathematical Illiteracy and Its Consequences: краткое содержание, описание и аннотация

Предлагаем к чтению аннотацию, описание, краткое содержание или предисловие (зависит от того, что написал сам автор книги «INNUMERACY: Mathematical Illiteracy and Its Consequences»). Если вы не нашли необходимую информацию о книге — напишите в комментариях, мы постараемся отыскать её.

Paulos speaks mainly of the dangers of mathematical innumeracy; that is, the common misconceptions of the layperson in regards to numbers, exploring the relationship between math and the human mind. Paulos discusses innumeracy with quirky anecdotes, scenarios and facts, encouraging readers in the end to look at their world in a more quantitative way. Topics include probability and coincidence, the birthday problem, innumeracy in pseudoscience, and statistics and trade-offs in society.

John Paulos: другие книги автора


Кто написал INNUMERACY: Mathematical Illiteracy and Its Consequences? Узнайте фамилию, как зовут автора книги и список всех его произведений по сериям.

INNUMERACY: Mathematical Illiteracy and Its Consequences — читать онлайн бесплатно полную книгу (весь текст) целиком

Ниже представлен текст книги, разбитый по страницам. Система сохранения места последней прочитанной страницы, позволяет с удобством читать онлайн бесплатно книгу «INNUMERACY: Mathematical Illiteracy and Its Consequences», без необходимости каждый раз заново искать на чём Вы остановились. Поставьте закладку, и сможете в любой момент перейти на страницу, на которой закончили чтение.

Шрифт:

Сбросить

Интервал:

Закладка:

Сделать

I once wrote to a significant minority of the Forbes 400, a list of the four hundred richest Americans, asking for $25,000 in support for a project I was working on at the time. Since the average wealth of the people I contacted was approximately $400 million (4 x 108, certainly a gargantuan number of dollars) and I was asking for only 1/16,000th of that wealth, I hoped that linear proportionality would hold, reasoning that if some stranger wrote me asking for support of a worthy project of his and asked me for $25, more than 1/16,000th of my own net worth, I would probably comply with his request. Alas, though I received a number of kind responses, I didn't receive any money.

ARCHIMEDES AND PRACTICALLY INFINITE NUMBERS

There is a fundamental property of numbers named after the Greek mathematician Archimedes which states that any number, no matter how huge, can be exceeded by adding together sufficiently many of any smaller number, no matter how tiny. Though obvious in principle, the consequences are sometimes resisted, as they were by the student of mine who maintained that human hair just didn't grow in miles per hour. Unfortunately, the nanoseconds used up in a simple computer operation do add up to lengthy bottlenecks on intractable problems, many of which would require millennia to solve in general. It takes some getting accustomed to the fact that the minuscule times and distances of microphysics as well as the vastness of astronomical phenomena share the dimensions of our human world.

It's clear how the above property of numbers led to Archimedes' famous pronouncement that given a fulcrum, a long enough lever, and a place to stand, he alone could physically lift the earth. An awareness of the additivity of small quantities is lacking in innumerates, who don't seem to believe that their little aerosol cans of hairspray could play any role in the depletion of the ozone layer of the atmosphere, or that their individual automobile contributes anything to the problem of acid rain.

The pyramids, impressive as they are, were built a stone at a time over a period very much shorter than the five thousand to ten thousand years required to move the 12,000-foot Mount Fuji by truck. A similar but more classic calculation of this type was made by Archimedes, who estimated the number of grains of sand needed to fill up the earth and heavens. Though he didn't have exponential notation, he invented something comparable, and his calculations were essentially equivalent to the following.

Interpreting "the earth and heavens" to be a sphere about the earth, we observe that the number of grains of sand needed to fill it depends on the radius of the sphere and the coarseness of the sand. Assuming there are fifteen grains per linear inch, there are 15 x 15 per planar inch and 153 grains per cubic inch. Since there are twelve inches per foot, there are 123 cubic inches per cubic foot and thus 153 x 123 grains per cubic foot. Similarly, there are 153 x 123 x 5,2803 grains per cubic mile. Since the formula for the volume of a sphere is 4/3 x pi x the cube of the radius, the number of grains of sand needed to fill a sphere of radius one trillion miles (approximately Archimedes' estimate) is 4/3 x pi x 1,000,000,000,0003 x 153 x 123 x 5,2803. This equals approximately 1054 grains of sand.

There is a sense of power connected with such calculations which is hard to explain but which somehow involves a mental encompassing of the world. A more modern version is the calculation of the approximate number of subatomic bits that would fill up the universe. This number plays the role of "practical infinity" for computer problems which are solvable but only theoretically.

The size of the universe is, to be a little generous, a sphere about 40 billion light-years in diameter. To be even more generous and also to simplify the rough calculation, assume it's a cube 40 billion light-years on a side. Protons and neutrons are about 10-12 centimeters in diameter. The Archimedean question computer scientist Donald Knuth poses is how many little cubes 10- 13 centimeters in diameter (1/10 the diameter of these nucleons) would fit into the universe. An easy calculation shows the number to be less than 10125. Thus, even if a computer the size of the universe had working parts that were smaller than nucleons, it would contain fewer than 10125 such parts, and thus computations on problems which require more parts wouldn't be possible. Perhaps surprisingly, there are many such problems, some of them quite ordinary and of practical importance.

A comparably tiny time unit is the amount of time required for light, which travels at 300,000 kilometers per second, to traverse the length of one of the above tiny cubes, whose edges are 10-13 centimeters. Taking the universe to be about 15 billion years old, we determine that fewer than 1042 such time units have passed since the beginning of time. Thus, any computer calculation which requires more than 1042 steps (each of which is certainly going to require more time than our unit of time) requires more time than the present history of this universe. Again, there are many such problems.

Taking a human being to be spherical and about a meter in diameter (assume a person is squatting), we end with some biologically revealing comparisons that are somewhat easier to visualize. The size of a human cell is to that of a person as a person's size is to that of Rhode Island. Likewise, a virus is to a person as a person is to the earth; an atom is to a person as a person is to the earth's orbit around the sun; and a proton is to a person as a person is to the distance to Alpha Centauri.

THE MULTIPLICATION PRINCIPLE AND MOZART'S WALTZES

Now is probably a good time to reiterate my earlier remark that an occasional difficult passage may be safely ignored by the innumerate reader. The next few sections in particular may contain several such passages. The occasional trivial passage likewise may be quite safely ignored by the numerate reader. (Indeed, the whole book may be safely ignored by all readers, but I'd prefer that, at most, only isolated paragraphs will be.)

The so-called multiplication principle is deceptively simple and very important. It states that if some choice can be made in M different ways and some subsequent choice can be made in N different ways, then there are M x N different ways these choices can be made in succession. Thus, if a woman has five blouses and three skirts, then she has 5 x 3 = 15 choices of outfit, since each of the five blouses (B1, B2, B3, B4, B5) can be worn with any of the three skirts (S1, S2, S3), to yield the following fifteen outfits: B1,S1; B1,S2; B1,S3; B2,S1; B2,S2; B2,S3; B3,S1; B3,S2; B3,S3; B4,S1; B4,S2; B4,S3; B5,S1; B5,S2; B5,S3. From a menu with four appetizers, seven entrees, and three desserts, a diner can design 4x7x3 = 84 different dinners, assuming he orders each course.

Likewise, the number of possible outcomes when rolling a pair of dice is 6 x 6 = 36; any of the six numbers on the first die can be combined with any of the six numbers on the second die. The number of possible outcomes where the second die differs from the first is 6 x 5 = 30; any of the six numbers of the first die can be combined with any of the remaining five numbers on the second die. The number of possible outcomes when rolling three dice is 6 x 6 x 6 = 216. The number of outcomes in which the numbers on the three dice differ is 6 x 5 x 4 = 120.

Читать дальше

Шрифт:

Сбросить

Интервал:

Закладка:

Сделать

Похожие книги на «INNUMERACY: Mathematical Illiteracy and Its Consequences»

Представляем Вашему вниманию похожие книги на «INNUMERACY: Mathematical Illiteracy and Its Consequences» списком для выбора. Мы отобрали схожую по названию и смыслу литературу в надежде предоставить читателям больше вариантов отыскать новые, интересные, ещё не прочитанные произведения.


Отзывы о книге «INNUMERACY: Mathematical Illiteracy and Its Consequences»

Обсуждение, отзывы о книге «INNUMERACY: Mathematical Illiteracy and Its Consequences» и просто собственные мнения читателей. Оставьте ваши комментарии, напишите, что Вы думаете о произведении, его смысле или главных героях. Укажите что конкретно понравилось, а что нет, и почему Вы так считаете.