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Dies ist neben einigen engen Kurven Hauptursache für das zum Teil als unerträglich empfundene Geräuschniveau einer solchen Strecke. Finally he came to Gauss's slate, on which was written a single number, 5,, with no supporting arithmetic.

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Eventually he entered the arithmetic class, in which most pupils remained until their confirmation, that is, until about their fifteenth year. Here an event occurred which is worthy of notice because it was of some influence on Gauss' later life, and he often told it in old age with great joy and animation. Büttner once gave the class the exercise of writing down all the numbers from 1 to and adding them. The pupil who finished an exercise first always laid his tablet in the middle of a big table; the second laid his on top of this, and so forth.

The problem had scarcely been given when Gauss threw his tablet on the table and said in Brunswick low dialect: While the other pupils were figuring, multiplying, and adding, Büttner went back and forth, conscious of his dignity; he cast a sarcastic glance at his quick pupil and showed a little scorn.

In the end, however, he found on Gauss' tablet only one number, the answer, and it was correct. But the young boy was in a position to explain to the teacher how he arrived at this result.

Many of the other answers were wrong and were at once "rectified" by the whip. The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. The first sequence of numbers above [1, 3, 6, 10, 15, The tenth number on the list is the number of beans required to build a triangle with ten rows, starting with one bean in the first row and ending with ten beans in the last row.

So the N th triangular number is got by simply adding the first N numbers: If you want to find the th triangular number, there is a long laborious method in which you attack the problem head on and add up the first numbers.

Indeed, Gauss's schoolteacher liked to set this problem for his class, knowing that it always took his students so long that he could take 40 winks. As each student finished the task they were expected to come and place their slate tablets with their answer written on it in a pile in front of the teacher. While the other students began laboring away, within seconds the ten-year-old Gauss had laid his tablet on the table.

Furious, the teacher thought that the young Gauss was being cheeky. But when he looked at Gauss's slate, there was the answer — 5, — with no steps in the calculation. Rather than tackling the problem head on, Gauss had thought laterally. He argued that the best way to discover how many beans there were in a triangle with rows was to take a second similar triangle of beans which could be placed upside down on top of the first triangle. Now Gauss had a rectangle with rows each containing beans.

Calculating the total number of beans in this rectangle built from the two triangles was easy: There is nothing special here about Fericite erau clipele pe care talentatul Carl Friedrich le petrecea cu manualul de la Hamburg.

Practical Analysis in One Variable. We motivate the need for induction using a story about the mathematician Gauss when he was His old-fashioned arithmetic teacher liked to show off to his students by asking them to add a large number of sequential numbers by hand, which the teacher knew from a book could be done quickly using the formula.

By the way, long sums of numbers arise in integration and in models such as computing compound interest on a savings account or adding up populations of animals. Addition formulas like 3. How did young Gauss manage to compute the sum so quickly? He did not know the formula 3. In Pädagogitches Archive , Vol. The anecdote appears on p.

A Selection of Mathematical Stories and Anecdotes. Gauss very early in life exhibited a remarkable cleverness with numbers, becoming a "wonder child" at the age of two.

There are a couple of oft-told stories illustrating the boy's unusual ability. One of the stories tells how on a Saturday evening Gauss's father was making out the weekly payroll for the laborers of the small bricklaying business that he operated in the summer. The father was quite unaware that his young three-year-old son Carl was following the calculations with critical attention, as so was surprised at the end of the computation to hear the little boy announce that the reckoning was wrong and that it should be so and so instead.

A check of the figures showed that the boy was correct, and on subsequent Saturday evenings the youngster was propped up on a high stool so that he could assist with the accounts.

Gauss enjoyed telling this story later in life, and used to joke that he could figure before he could talk. The other story dates from Gauss's schooldays, when he was about ten years old. At the first meeting of the arithmetic class, Master Büttner asked the pupils to write down the numbers from 1 through and add them.

It was the custom that the pupils lay their slates, with their answers thereon, on the master's desk upon completion of the problem. Master Büttner had scarcely finished stating the exercise when young Gauss flung his slate on the desk. The other pupils toiled on for the rest of the hour while Carl sat with folded hands under the scornful and sarcastic gaze of the master. At the conclusion of the period, Master Büttner looked over the slates and discovered that Carl alone had the correct answer, and upon inquiry Carl was able to explain how he had arrived at his result.

Fadiman, Clifton, and Andre Bernard, general editors. Bartlett's Book of Anecdotes. Little, Brown and Company. At school, Gauss showed little of his precocious talent until the age of nine, when he was admitted to the arithmetic class.

The master had set what appeared to be a complicated problem involving the addition of a series of numbers in arithmetical progression.

Although he had never been taught the simple formula for solving such problems, Gauss handed in his slate within seconds. For the next hour the boy sat idly while his classmates labored.

At the end of the lesson there was a pile of slates on top of Gauss's, all with incorrect answers. The master was stunned to find at the bottom the slate from the youngest member of the class bearing the single correct number.

He was so impressed that he bought the best available arithmetic textbook for Gauss and thereafter did what he could to advance his progress. Gauss was born in Brunswick, Germany as the only son of poor peasants living in miserable conditions. He exhibited such early genius that his family and neighbors called him the "wonder child". When he was two years old, he gradually got his parents to tell him how to pronounce all the letters of the alphabet.

Then, by sounding out combinations of letters, he learned on his own to read aloud. He also picked up the meanings of the number symbols and learned to do arithmetical calculations. The story as told by Eric T. Coming to the end of his long computations, Gerhardt was startled to hear the little boy pipe up, 'Father, the reckoning is wrong, it should be When Gauss was ten years old he was allowed to attend an arithmetic class taught by a man Buttner who had a reputation for being cynical and having little respect for the peasant children he was teaching.

The teacher had given the class a difficult summation problem in order to keep them busy and so that they might appreciate the "shortcut" formula he was preparing to teach them. Gauss took one look at the problem, invented the shortcut formula on the spot, and immediately wrote down the correct answer. This act was apparently so astonishing that Herr Buttner was transformed into a champion for this young boy. The boy flashed through the book. Buttner, realizing that he could teach this young genius no more, recommend him to the Duke of Brunswick, who granted him financial assistance to continue his education into secondary school and finally into the University of Gottingen.

A Modeling Approach Using Technology: Integrated Mathematics, Level 2. According to mathematical lore, one day his teacher asked the class to add all the natural numbers from 1 to Students were instructed to place their slates on the table when finished.

To the surprise of the teacher, young Gauss placed his slate on the table after only a few moments. To find the sum of the first natural numbers, Gauss used a method involving a finite series.

For example, the sum of the first numbers can be written as the arithmetic series S The story of and other weird math facts.

Curiousmath Web site posting dated Monday, February 10, , 4: About years ago, a young boy who grew up to be a great mathematician by the name of Gauss pronounced "Gowss" was at school when the class got in trouble for being too loud and misbehaving.

Their teacher, looking for something to keep them quiet for a while, told her students that she wanted them to "add up all of the numbers from 1 to and put the answer on her desk.

About 30 seconds later, the year-old Gauss tossed his slate small chalkboard onto the teacher's desk with the answer "" written on it and said to her in a snotty tone, "There it is. That's 50 pairs of So he just multiplied by 50 to get A Language for Learning. Therefore, let us consider a numerical problem, which has its roots in the following story about the young Carl Frederick Gauss, who was to become one of the greatest mathematicians of all time.

The story is probably apocryphal, but is still a good story. This version follows Polya , who also uses it to introduce recursion in mathematical problems. When Gauss was in primary school, the teacher, hoping to keep his students occupied while he attended to other matters, gave a tough problem: While the other children were just getting started, young Gauss walked to the teacher's desk and handed in his slate.

The teacher, thinking this an act of impudance, did not even bother to look at Gauss's work until all the other children had handed in theirs. When he did look at it, he was surprised to find that it contained just a single number, the right answer. In Matierialien für eine wissenschaftliche Biographie von Gauss , compiled by F. Link to PDF file pp.

Den Schülern der under des Lehrers Büttner Leitung stehenden Rechenklasse der Katharineenschule in Braunschweig wurde die Aufgabe vorgelegt, die Summe einer Reihe auf einander folgende Zahlen zu bilden. Jeder, der die Rechnung beendet haben würde, sollte die Tafel auf einen Sammeltisch legen.

Der alte Büttner musterte den schnellfertigen Knaben mit spöttischem Mitleid, während die andern Schüler die Stunde hindurch weiter Rechneten. Er hatte das Summationsprinzip für die arithmetischen Reihen auf den ersten Blick herausgefunden. Red Orbit Breaking News, 29 September It wasn't just that they solved it in record time, it's that they figured out a whole new way of doing it.

Desiree Martinez and Amber Lopez, both freshman algebra students at Pojoaque Valley High School, figured out the answer to a math problem made famous by 18th-century mathematician Johann Carl Friedrich Gauss in 6 and 11 minutes, respectively. Those are the two best times their teacher, Lanse Carter, has seen in his four years of teaching. The problem is to add all integers from one to Carter said he gives his students one hint before they start, which is to look for a pattern.

Go ahead, play at home—the answer and method will be near the end of this article. Don't forget to show your work. In the meantime, here's a history lesson: Gauss' math teacher, J. Buttner, reputed to be a rather surly man, assigned the problem when he wanted to occupy his students for up to an hour and was dismayed when Gauss, then 13 some histories peg his age at 10, others as young as 7 , solved it in about a minute, flinging his slate upon the table barely after Buttner finished stating the problem and saying "Ligget se," Brunswick German for "there 'tis.

Martinez and Lopez "through their own ingenuity, found a pattern I wasn't looking for," Carter said, adding that pattern recognition is a key concept in mathematics. Now for the answer. Realizing he had 50 pairs of numbers that, when added, equal , he multiplied 50 times and came up with the answer, 5, Martinez and Lopez also found a pattern, but it was different from Gauss'.

A third attempt, adding all the numbers between 21 and 30, resulted with a total of It gave me a wonderful feeling as well," Carter said.

How Do You See It? Discovering Mathematical Patterns and Sequences. When Karl Gauss, a brilliant German mathematician, was 10 years old in the late 18th century he was presented a very difficult problem.

His teacher, a stern and lazy man, wrote on the board the task that these young men had to perform. The problem was to add up all the numbers from 1 to Knowing this would take his students time, the arrogant teacher began to go back to his seat and prepare himself for a long quiet day.

As soon as he sat down, Gauss approached him and put the slate, a small board that these students used to do their work on, face down on his teacher's desk. All the students were shocked at how fast and seemingly effortlessly Gauss completed the problem. The teacher just glared at him. By the end of the school day, the last of the boys set his slate down. The teacher had a feeling that no person came up with the right answer.

He began turning over each of the slates, each revealing a wrong answer. Finally he came to Gauss' slate. All the kids snickered as the teacher slowly turned his slate over. The teacher's face was pale and stunned. Gauss had the correct answer! What's even more surprising is that he had written very little besides the answer. How did young Gauss come up with the answer? Oddly, he did not have an equation like we have now. He did it purely on observation. Look at the series of numbers Take the first and last term of the series, 1 and Gauss says that this combination when adding together equals Looking at another combination, 2 and 99 is also and so forth.

Thinking that this pattern repeated for all the others, he knew he had a certain number of 's. The question was how many 's were there? Gauss was brilliant for his age and became ever more brilliant as his life went on.

Gauss' method is wonderful to look at but there still must be an easier way to figure out the sum of a finite arithmetic series. We can solve this problem using the equation: Carl Friedrich Gauss — — sein Leben. April in Braunschweig als Sohn eines Gassenschlächters geboren, verblüffte Carl Friedrich Gauss — der von sich selbst sagte, er habe eher rechnen als sprechen gelernt — schon als Kind seine Lehrer.

In der mit Schülern überfüllten Schulstube erteilte der Lehrer die Aufgabe, alle Zahlen von 1 bis zu addieren. Lange vor seinen Mitschülern hatte der kleine Carl Friedrich das richtige Ergebnis parat. Geschwinde, Ewald, and Hans-Jürgen Schönig. At the age of seven, Carl Friedrich Gauss started elementary school, and his potential was noticed almost immediately. His teacher, Büttner, and his assistant, Martin Bartels, were amazed when Gauss summed the integers from 1 to instantly by spotting that the sum was 50 pairs of numbers each pair summing to Gauss's Formula for the sum of integers was born.

Physics for Scientists and Engineers with Modern Physics. Upper Saddle River, N. Gauss was born on April 30th, in the Duchy of Brunswick, now a part of Germany. He was a child prodigy, and many stories are told of his early mathematical prowess. It is well-documented that he corrected an error in his father's payroll calculations at the age of three, and as an adult he explained that by his recollection he could count before he could talk. Probably the most famous story about young Gauss occured in , when he was nine years old.

Büttner, assigned his class the task of adding all of the numbers from 1 to Gauss turned in his slate after only a few seconds, with only the final answer written down. Büttner studiously ignored him until the class had finished, and was astonished to find that Gauss's answer was correct. He asked Gauss how he had arrived at his answer, and Gauss explained: After Gauss' death in February of , a medal was struck in his honor The history of every real prince begins with a childhood surrounded by legends.

Gauss was not an exception At the age of seven, Carl Friedrich entered Catherine's School. In that school students were not taught how to count until the third grade, so for the first two years nobody paid attention to little Carl. The children usually got to the third grade at the age of 10 and stayed in that grade until confirmation at the age of The teacher Büttner had to devote himself simultaneously to children of different ages and knowledge.

For this reason, he often gave some of the students long exercises in calculation in order to be able to talk to other students. Once, he asked a group of students, among them Gauss, to sum up all natural numbers from 1 to As a student finished the calculations, he would place his slate on the teacher's desk.

The order of the slates was taken into account when giving marks. Ten-year-old Gauss turned in his slate as soon as Büttner had finished assigning the task.

To everybody's surprise, only Gauss' answer turned out to be correct. The explanation was simple: The fame of the infant prodigy spread all over Brunswick. String Crossings Andrew Glassner's Notebook. Passage appears on p. The story goes that when Gauss was a child, his math teacher came to class unprepared one day. The teacher decided to fill the class time by instructing the students to add up all the numbers from 1 to Most of the students started writing down all the numbers in a big column in preparation for adding them up.

But in only a few seconds Gauss announced to the teacher that the answer was 5, The teacher assumed that Gauss had simply learned this result as a piece of trivia.

So she set him about the more time-consuming task of adding the numbers from 1 to After only a moment's paperwork, Gauss announced the answer was , Monkeyshines Explores Math, Money, and Banking.

Most of us imagine mathematicians to be old people with beards and thick glasses, yet many of the important mathematical discoveries have been made by fairly young people. One of the youngest and most famous mathematicians in all of history was Carl Friedrich Gauss who was born in Germany in and died in He came from a poor family. His father was a gardener and his mother a housekeeper. Young Gauss showed his mathematical ability at a very early age.

When he was three years old he watched his father add up a long column of numbers. Gauss pointed out an error and gave his father the correct answer. When the father checked the addition, he found his son was indeed correct.

When Gauss was ten years old he began his first lessons in arithmetic. The teacher gave the class a long and difficult problem so they would have to spend hours to find the answer. The problem involved adding up a sequence of numbers like: There is a trick to solve problems like this but it was unknown to the young students.

However, Gauss discovered the trick for himself and quickly solved the problem while all the other students worked for hours and all the answers were wrong except for Gauss's. Recognizing that Gauss was special, his teacher helped him to advance in his studies. Goldstein, Martin, and Inge F.

The Experience of Science: Most mathematicians are no less bored by adding up long columns of figures than the rest of us. They do not consider it their job, and are usually annoyed when nonmathematicians assume that it is. The point may be illustrated by two episodes in the life of Karl Gauss — , one of the greatest of mathematicians. Gauss was born a poor boy, the son of a bricklayer, in Braunschweig, Germany. The schoolmaster in the local school Gauss attended, a certain Herr Büttner, was a hard taskmaster who gave his classes practice in arithmetic by asking them to add up long sequences of large numbers.

For his own convenience, so that he would not have to do the tedious arithmetic involved to check his pupils' almost invariably erroneous answers, the sequences of numbers he assigned his classes to add were chosen to form what it called an arithmetic series—the successive numbers in the long list differed by a constant amount. For example, the series 11, 14, 17, 20, 23, 26 is such a series, in which each term increases by 3.

Büttner then makes use of a well-known formula for the sum of such a series: For the series given above, the sum is. In any event, Büttner wrote on the blackboard a list of large numbers forming such a series, and after finishing turned around to face the class, expecting as usual to have a free hour or so while his pupils sweated and struggled, to find little Gauss handing in his slate with the correct sum written out.

Gauss had recognized the numbers as forming an arithmetic series, figured out on the spot the formula for the sum, and calculated it. Büttner, to his everlasting credit, though no mathematician himself, knew one when he saw one. With his own money he bought Gauss the best textbook on arithmetic then available and brought the boy's abilities to the attention of people who could help him in his career.

Gauss et le GAUS. La Revue de Physique. An Invitation to Mathematics. Karl Friedrich Gauss — was one of the finest mathematicians of all time. The son of a bricklayer, it is said that he spotted formulae for certain arithmetic sums for himself at the age of His teacher had a habit of setting the class long strings of numbers to add up to keep them occupied, all the time knowing a formula for the answer.

Gauss outwitted him and all his teacher could do was to buy him a text book and announce that the boy was beyond him. Knuth and Oren Patashnik. To evaluate S n [the sum of the first n positive integers] we can use a trick that Gauss reportedly came up with in , when he was nine years old: Gauss's trick in chapter 1 can be viewed as an application of these three basic laws [i.

These two equations can be added by using the associative law: And we can now apply the distributive law and evaluate a trivial sum: Blog posting, Jeudi 14 juillet Translated by Albert Froderberg. Biographies about or by great men generally contain more or less noteworthy anecdotes, intended to illustrate the budding genius.

It is a field in which memory gladly accommodates itself to a fixed path and where imagination easily overtakes the uncertain facts. The situation is especially pernicious in the case of child prodigies, who are often encountered in mathematics, music, and chess.

Myths appear with treacherous ease. Gauss was a mathematical prodigy—it is certain that he was one of the most outstanding examples of this genre, but basically this is unimportant. First-hand accounts of this come from Gauss himself, who in his old age liked to talk of his childhood. From a critical viewpoint they are naturally suspect, but his stories have been confirmed by other persons, and in any case they have anecdotal interest.

During the summers Gebhard Gauss was foreman for a masonry firm, and on Saturdays he used to pay the week's wages to his workers. One time, just as Gebhard was about to pay a sum, Carl Friedrich rose up and said, "Papa, you have made a mistake," and then he named another figure.

The three-year-old child had followed the calculation from the floor, and to the open-mouthed surprise of those standing around, a check showed that Carl Friedrich was correct. Gauss used to say laughingly that he could reckon before he could talk. He asked the adults how to pronounce the letters of the alphabet and learned to read by himself.

When Carl Friedrich was seven years old he enrolled in St. His teacher was J. The large classroom had a low ceiling, and the schoolmaster walked about on the uneven floor, cane in hand, among his approximately pupils. Caning was the foremost pedagogical aid both for learning and discipline, and Büttner is thought to have used it constantly, either as a consequence of necessity or because of his temperament.

Gauss stayed in these surroundings for two years without any ill consequences. It is the traditional picture of that period's public education, when the caning pedagogy was generally accepted—by the adults of course—but we shall soon see that Büttner was more likely above than below average among his colleagues.

When Gauss was about ten years old and was attending the arithmetic class, Büttner asked the following twister of his pupils: The problem is not difficult for a person familiar with arithmetic progressions, but the boys were still at the beginner's level, and Büttner certainly thought that he would be able to take it easy for a good while. But he thought wrong. In a few seconds Gauss laid his slate on the table, and at the same time he said in his Braunschweig dialect: While the other pupils added until their brows began to sweat, Gauss sat calm and still, undisturbed by Büttner's scornful or suspicious glances.

At the end of the period the results were examined. Most of them were wrong and were corrected with the rattan cane. On Gauss's slate, which lay on the bottom, there was only one number: It seems unnecessary to point out that this was correct. Now Gauss had to explain to the amazed Büttner how he had found his result: This is a total of 50 pairs of numbers, each of which adds up to Thus Gauss had found the symmetry property of arithmetic progressions by pairing together the terms as one does when deriving the summation formula for an arbitrary arithmetic progression—a formula which Gauss probably discovered on his own.

What this actually entails is that one writes the series both "forward" and "backward"; that is. The event is symbolic. For the rest of his life Gauss was to present his results in the same calm, matter-of-fact way, fully conscious of their correctness.

The evidence of his struggles would be wiped away from the completed work in the same way. And, like Büttner, many learned persons would wish to be given a detailed explanation, but here a difference would appear, for Gauss would not feel compelled to give one. Hannoversch Münden web site. Der Fürst der Mathematiker konnte früher rechnen als sprechen Den Anekdoten nach war der am Zwolf Kapitel aus Seinem Leben.

Auf dieser Seite also glauben die verborgenen Quelladern des Genius riefeln zu hören. Und doch, wie hoch man die Gunst dieser Einflüsse auch anschlagen mag, ein Wunder bleibt es, mit welcher Macht er in diesem Erdenfinde hervorbrach.

Aus sich selbst, mit gelegentlicher Nachfrage bei seiner Umgebung, lernt er lesen; am erstaunlichsten aber zeigt sich von frühester Kindheit an die intuitive Kraft seiner Auffassung von Zahlenverhältnissen: Als es ans Unszahlen geht, zirpt er warnend dazwischen, und siehe da, der Alte hat sich verrechnet und was der Kleine angiebt ist das Richtige.

Katharine, der er seit angehört, eine arithmetische Reihe summirt werden soll. Der alte Büttner mustert den schnellfertigen kleinsten seiner Unglückswürmer mit spöttischem Mitleid: In der Schule hatte der Lehrer die Aufgabe gestellt, alle Zahlen von 1 bis zusammenzuzählen. Auf seiner Tafel steht die richtige Zahl , und viele andere sind falsch oder noch nicht fertig.

Er hatte den geometrischen Aufbau der Zahlen sofort vor Augen gehabt und erkannt: God Created the Integers: Gauss's talents were obvious as soon as he stepped into a classroom at the age of seven. When the class began to be unruly, the teacher, J. Büttner, assigned them the task of adding up all of the integers from 1 to As his classmates struggled to fit their calculations on their individual slates, Gauss wrote down the answer immediately: As soon as the problem was stated, Gauss recognized that the set of integers from 1 to was identical to 50 pairs of integers each adding up to Herr Büttner approached Gauss's parents to persuade them to let their son stay after school for special math instruction.

Gauss's parents were at first skeptical. They had recognized their son's calculating ability when, at the age of three, he corrected a mistake his father made in paying out wages to men who worked [for] him Discrete Structures, Logic, and Computability. When Gauss—mathematician Karl Friedrich Gauss — —was a year-old boy, his schoolmaster, Buttner, gave the class an arithmetic progression of numbers to add up to keep them busy. We should recall that an arithmetic progression is a sequence of numbers where each number differs from its successor by the same constant.

Gauss wrote down the answer just after Buttner finished writing the problem. Although the formula was known to Buttner, no child of 10 had ever discovered it. For example, suppose we want to add up the seven numbers in the following arithmetic progression:.

The example illustrates a use of the following formula for the sum of an arithmetic progression of n numbers a 1 , a 2 , Insights and connections—that's what mathematicians look for. Carl Friedrich Gauss, who was born in in Braunschweig, Germany, the son of a masonry foreman, was a master of exposing unsuspected connections.

Like Erdös, Gauss was a mathematical prodigy, and in his old age he liked to tell stories of his childhood triumphs. Like the time, at the age of three, he spotted an error in his father's ledger and stopped him just as he was about to overpay his laborers.

Like the fact that he could calculate before he could read. And he certainly could calculate. At the age of ten, he was a show-off in arithmetic class at St. Catherine elementary school, "a squalid relic of the Middle Ages The student who solved the problem first was supposed to go and lay his slate on Büttner's desk; the next to solve it would lay his slate on top of the first slate, and so on.

Büttner thought the problem would preoccupy the class, but after a few seconds Gauss rushed up, tossed his slate on the desk, and returned to his seat.

Büttner eyed him scornfully, as Gauss sat there quietly for the next hour while his classmates completed their calculations. As Büttner turned over the slates, he saw one wrong answer after another, and his cane grew warm from constant use. Finally he came to Gauss's slate, on which was written a single number, 5,, with no supporting arithmetic. Astonished, Büttner asked Gauss how he did it, "and when Gauss explained it to him," said Erdös, "the teacher realized that this was the most important event in his life and from then on worked with Gauss always," plying him with textbooks, for which "Gauss was grateful all his life.

What was Gauss's trick? In his mind he apparently pictured writing the summation sequence twice, forward and backward, one sequence above the other:. Gauss realized that you could add the numbers vertically instead of horizontally. There are vertical pairs, each summing to So the answer is times divided by 2, since each number is counted twice. Gauss easily did the arithmetic in his head.

Gauss found a very nice way of showing that if you add all the numbers from one up through any number n , the answer is n times n plus one, all divided by two. This method of summing such a series is really straight from the Book. Bulletin Institute of Mathematics and its Applications 13 3—4: Reprinted in Makers of Mathematics , , London: Gauss' precocity is legendary.

At the age of 3 he was correcting his father's weekly wage calculations. When he was 7 he entered his first school, a squalid prison run by one Büttner, a brutal taskmaster. Two years later Gauss was admitted to the arithmetic class. Büttner had the endearing habit of giving out long problems of the kind, such as summing progressions, where the answer could readily be obtained from a formula—a formula known of course to the teacher, but not to the pupils.

Each boy, on completing his task, had to place his slate on the master's desk. On one occasion no sooner had Büttner dictated the last number than his youngest pupil flung his slate on the desk and waited for an hour while the other boys toiled. When Büttner looked at Gauss' slate, he found there a single number—no calculation at all.

Gauss liked to recall this incident in his later years, and to point out that his was the only correct answer. Ponder this, July Although it is contended that the solution for finding the sum of consecutive integers has ancient roots, perhaps stretching back to Pythagorus, it is the story of Gauss's school age experience that has become legend.

As the story goes, Gauss's teacher tried to occupy the class during an unsupervised absence by proposing a simple problem: Find the sum of all integers from 1 to As his classmates laboriously -- and quietly, one hopes -- proceeded to work the solution by rote addition, Gauss reasoned the problem as follows:.

He imagined adding, not the consecutive integers, but two series of addition, the integers progressing forward in one series and in reverse in the other. He concluded that the sum of the two series was the product of the largest integer in the series and that integer plus 1: The reaction of Gauss's classmates -- and his teacher -- to his shortcut remains a mystery.

Fortunately, his result has been preserved. We shall start with an arithmetic progression whose first term and common difference are 1. This is the progression. According to the tradition in the schools at that time, when a mathematics problem was given to a class, the pupil who finished first placed his slate board down in the middle of a large table, and then the next to finish put his slate down on top of it.

One day, when young Carl was a pupil in Mr. Büttner's arithmetic class, Mr. Büttner posed the problem of adding an arithmetic progression.

He had barely finished describing this task when Gauss threw his slate board on the table saying, in low Brunswick dialect, "Ligget se" "there she lies".

While the other pupils continued to work on this problem, Mr. Büttner, conscious of his dignity, walked up and down the room, and occasionally threw a contemptuous and caustic glance at the smallest of his pupils, who had finished the task too quickly. At last the other slates began to come in; and when the slates were turned over, Mr.

Büttner found that Gauss' solution was correct even though many of the others were wrong and were corrected with a slapping. Therefore the number that Gauss wrote on his slate should have been The method we have just described for summing an arithmetic progression is both fast and simple, and because it is simple, it is not prone to computational errors.

We shall now repeat the method to obtain the more general sum. Janzen, Beau director and animator. Video on YouTube or Vimeo. Usenet posting in news group alt. This sounds like the story recounted by Eric Temple Bell about K. Gauss at the age of about 8 years, except that probably nobody considered Gauss to be "dull", just not yet at that age a great mathematician.

I don't know the starting number nor the increment, but they formed an arithmetic progression, the kids were probably supposed to derive each term before adding it, and the teacher had a secret formula for determining the answer.

My guess is that Gauss figured out that the teacher had access to something he wasn't sharing and independently derived a slick way to find the sum, by rearranging the order of summing. Maybe it wasn't exactly divine inspiration, but it still took a pretty impressive mind to come up with that technique at that age.

Gauss just wrote the answer on his slate no calculations , and he and Büttner sort of glared at each other for an hour while the other boys slaved away. Gauss later said that his answer was the only correct one turned in that day. The story has a happy ending -- the teacher, recognizing that there wasn't much more that he could teach this unusual student, arranged for a tutor to take charge of Gauss's education, and the tutor and Gauss became lifelong friends and collaborators.

Kaplan, Robert, and Ellen Kaplan. The Art of the Infinite: The Pleasures of Mathematics. In order to savor once more this all too fugitive experience, here is a very different way of seeing that. Some people relish the geometric approach, some of the symbolic.

This tells you at once that personality plays as central a role in mathematics as in any of the arts. Proofs—those minimalist structures that end up on display in glass cases—come from people mulling things over in strikingly different ways, with different leapings and lingerings.

But is it always from the same premises that we explore? Is there some sort of common sense that is to reason what Jung's collective unconscious used to be to the psyche? One of these approaches, or some third, must have been in the mind of the ten-year-old Gauss—the Mozart of mathematics—when, in his first arithmetic class, he so startled his teacher. It was and Herr Büttner was in the habit of handing out brutally long sums like these, which the children had to labor over.

When each one finished, he added his slate to the pile growing on the teacher's desk. The morning might well be over before all had finished. But Gauss no sooner heard the problem than he wrote a single number on his slate and banged it down.

A History of Mathematics: Gauss was born into a family that, like many others of the time, had recently moved into town, hoping to improve its lot from that of impoverished farm workers. One of the benefits of living in Brunswick was that the young Carl could attend school. There are many stories told about Gauss's early-developing genius, one of which comes from his mathematics class when he was 9. At the beginning of the year, to keep his pupils occupied, the teacher, J.

Büttner, assigned them the task of summing the first integers. He had barely finished explaining the assignment when Gauss wrote the single number on his slate and deposited it on the teacher's desk.

Gauss had noticed that the sum in question was simply 50 times the sum of the various pairs 1 and , 2 and 99, 3 and 98, Die Vermessung der Welt. Büttner hatte ihnen aufgetragen, alle Zahlen von eins bis hundert zuzammenzuzählen. Das würde Studen dauern, und es war beim besten Willen nicht zu schaffen, ohne irgendwann einen Additionsfehler zu machen, für den man bestraft werden konnte.

Na los, hatte Büttner gerufen, keine Maulaffen feilhalten, aufangen, los! Jedenfalls hatte er sich nicht unter Kontrolle gehabt und stand nach drei Minuten mit seiner Schiefertafel, auf die nur eine einzige Zeile geschrieben war, vor dem Lehrerpult. So, sagte Büttner und griff nach dem Stock.

Sein Blick fiel auf des Ergebnis, und seine Hand erstarrte. Er fragte, was das solle. Darum sei es doch gegangen, eine Addition aller Zahlen von eins bis hundert. Hundert und eins ergebe hunderteins. Neunundneunzig und zwei ergebe hunderteins. Achtundneunzig und drei ergebe hunderteins. Das könne man fünfzigmal machen. Also fünfzig mal hunderteins. Helping Children Learn Mathematics. The Art of Mathematics. The great mathematicians feel mathematics in a way the rest of us do not.

And their genius for mathematics is immediately recognizable. When Gauss was eight years old, he and his classmates were asked by their teacher to find the sum of the integers from 1 to The children began laboriously to calculate on their slates. Gauss noticed that the integers 1, 2, 3, There are exactly 50 such pairs and the sum of the integers in each pair is Hence, the desired sum is the same as 50 times , which is Gauss wrote this number on his slate and handed it to the teacher.

The whole process took him only seconds. Gaea, Natur, und Leben. Im siebenten Jahre lam der Knabe in die Katharinenschule und wurde zunächst während zweier Jahre im Lesen und Schreiben unterrichtet, ohne sich irgendwie vor seinen Mitschülern auszuzeichnen. Eduard Heinrich Mayer Verlagsbuchhandlung. The anecdote appears on pp. The Pleasures of Counting. There is a well known story, repeated, with his usual trimmings, by Bell in his Men of Mathematics , that when Gauss was ten his teacher, Bütner, seeking an hour's repose, set his pupils the term sum.

A more restrained account of Gauss's early life, and a more sympathetic estimate of Bütner will be found in Bühler's excellent biography. More Stories and Anecdoetes of Mathematicians and the Mathematical. Mathematical Associaiton of America. What Gauss did was to observe that the sum of an arithmetic series is the number of terms multiplied times the average of the first and last term. The story has, however, been transmogrified with time. It is thought that the actual sum that Gauss was asked to calculate was.

John Wiley and Sons. The number of risk parameters in a portfolio equals the sum of the number of assets it includes. There is an amusing and perhaps apocryphal story about this result and the famous mathematician Carl Friedrich Gauss, who was born in in Braunschweig, Germany.

When Gauss was a child at St. Catherine elementary school, his teacher who was named Büttner asked the students in his class to sum the numbers from one to Büttner's intent was to distract the students for a while so that he could tend to other business. To Büttner's surprise and annoyance, however, Gauss, after a few seconds, raised his hand and gave the answer—5, Büttner was obviously shocked at how quickly Gauss could add, but Gauss confessed that he had found a short cut.

He described how he began by adding one plus two plus three but became bored and started adding backward from He then noticed that one plus equals , as does two plus 99 and three plus He immediately realized that if he multiplied by and divided by 2, so as not to double count, he would arrive at the answer.

Dare i numeri fa bene. Il professore ha fama di essere assai burbero e dai modi scostanti. Inoltre, pieno di pregiudizi fino al midollo, non ama gli allievi che provengono da famiglie povere, convinto che siano costituzionalmente inadeguati ad affrontare programmi culturali complessi e di un certo spessore. Un episodio in particolare viene ricordato nelle storie della matematica.

Proprio mentre comincia a gongolarsi al pensiero di quanto un suo trucchetto avrebbe lasciato a bocca aperta gli alunni, viene interrotto da Gauss che, in modo fulmineo afferma: Engines of Our Ingenuity, No. Podcast and Web site. Link to Web page Viewed and [Based in part on material presented here.

You may've heard the story about the great mathematician Carl Friedrich Gauss. He was a young schoolboy in the s. To keep his class quiet, the teacher told them to sum all the numbers from one to a hundred. Gauss immediately turned his slate over on the teacher's desk. After an hour, the teacher had all the slates, and he found the right answer on that bottom one.

The other boys made errors in the tedious arithmetic, while Gauss saw a shortcut. He saw that he could add one and a hundred, two and 99, on down to fifty. Now Brian Hayes goes looking for the truth of the story, and finds that it's been retold in many dozens of versions. He traces them from to the present, and what he learns is quite amazing. In some, Gauss is the youngest student.

In some he's the only one to get it right. Some versions have the teacher knowing the answer when he makes the assignment, others don't. There's a variant on how Gauss might've worked the problem. Instead of doing it as I've just described, he might've added zero and a hundred, one and 99, until he reached 49 and Then he'd have a hundred, fifty times, with the middle fifty to tack on. Same answer, slightly different tactics. Riesen und Zwerge im Zahlenreich: Und doch ist das Ergebnis richtig: Leipzig, Teubner, ; von ihnen will ich wenigstens eine hier wiedergeben, die ebenso im Rechenunterricht der Kleinen wie bei der Behandlung der arithmetischen Reihen in Obersekunda oder Prima ihre Stelle haben kann.

Die Aufgabe lautete diesmal: Der Lehrer freut sich schon, den allzu fixen Buben ertappt zu haben. Doch nach her ergibt sich: How to be a little Gauss. There is a story about Carl Friedrich Gauss. The teacher wanted to get some work done, or get some sleep, or whatever. Anyway, to the teacher's annoyance, little Gauss [Here the lecturer holds his hand out to show that little Gauss was about 2 feet tall, to the amusement of the audience] To the teacher's annoyance, little Gauss came up to the teacher with the answer, right away.

The teacher probably had to spend the rest of the class time verifying little Gauss's [2 feet tall] result. Some people find that story hard to believe, even impossible.

I think that the story has the ring of truth to it. I believe that the story is true, or close to it. There are versions of the story, in which the numbers are one to a thousand [murmur in the audience].

I think that you people can duplicate little Gauss's [2 feet tall] trick [doubt in the audience]. I'm going to give you two very small hints. But, that's all you will need, to be just like little Gauss [2 feet tall].

Nobody use your calculators, or even paper and pencil for a while. You are going to be slower than little Gauss [Lecturer hesitates, then shows "2 feet tall"]. But, you're going to be just as smart.

We want to find X. Well, it's going to take 99 additions to solve this. It's going to take a while. There's got to be an easier way. Do we get the same answer? It was algebra, right? It doesn't matter what order you add things up, you get the same answer. So "yes" we get the same answer [Lecturer writes "X" to the right of the equal sign].

That's going to take just as long, isn't it? There are 99 additions there, too. What if we add up the even numbers that's 49 additions , then add up the odd numbers that's 49 additions , and then add up the two totals? That's, uh, 99 additions. Darn, that's no better. When we finally total them up, we get the same answer, right? Does that look helpful? This is the second hint, by the way [points at those numbers]. Do you see something magical about that? Do you all see it? How many s do we have?

Lozansky, Edward, and Cecil Rousseau. It was known in antiquity that if a 1 , a 2 , However it is one thing for a formula to be known by practicing mathematicians and quite another for it to be deduced in an instant by a ten-year-old boy. Dazu kamen noch logistische Probleme. In den er Jahren trug zudem die starke wirtschaftliche Erholung der Stadt zur Verbesserung der Finanzsituation bei.

Zwischen und wurden über U-Bahn-Wagen generalüberholt und mit Klimaanlagen ausgestattet. Auf diesem Wege sollten Komfort, Zuverlässigkeit und Lebensdauer erhöht werden, um Neuanschaffungen hinauszögern zu können. Ersetzt werden sollten nur die jeweils ältesten Wagen jeder Division, so dass trotz des eigentlich überalterten Wagenparks lediglich neue Fahrzeuge beschafft werden mussten. Dazu verbesserten Wartungspläne den Erhaltungszustand der Wagen deutlich.

Parallel dazu begann eine umfangreiche Sanierung der Strecken. Innerhalb von zehn Jahren wurden die Gleise dabei fast auf kompletter Länge erneuert. Die Williamsburg Bridge und die Manhattan Bridge, die starke Korrosionsschäden aufwiesen, wurden über Jahre hinweg generalsaniert. Gleichzeitig bemühte sich die MTA, den Service zu verbessern, der ebenso lange vernachlässigt worden war. Dies reichte von neuen Uniformen über Schulungen für das Personal bis hin zu korrekten Zielschildern an den Fahrzeugen.

Einige Linienwege wurden überdies dem veränderten Bedarf der Kundschaft angepasst. Ein weiteres erklärtes Ziel war die Senkung der Kriminalität oder zumindest eine Verbesserung des subjektiven Sicherheitsgefühls.

Tatsächlich dauerte es noch bis in die er Jahre, bis die Kriminalität in der Stadt und damit auch in der U-Bahn deutlich zurückging. Dennoch blieb der Ruf als langsames, verfallenes, dreckiges und unsicheres Fortbewegungsmittel weiterhin an der Subway haften. In den er Jahren begannen die Renovierungsarbeiten an den Stationen.

Ab Ende der er Jahre wurden wieder Neubaustrecken geplant. Anfang hat der neue Kopfbahnhof South Ferry die alte Wendeschleife ersetzt. Diese Station wurde aber aus Kostengründen gestrichen. Auch die seit geplante Second Avenue Line soll bis in vier Bauabschnitten endlich errichtet werden. Allerdings stehen viele New Yorker diesem Projekt skeptisch gegenüber, weil die U-Bahn unter der Second Avenue bereits im Rahmen des Second Independent System geplant worden war und sämtliche bisherigen Anläufe am Geldmangel scheiterten.

Auch das Aktionsprogramm von brachte nur einige kurze Tunnelabschnitte und Bauvorleistungen hervor. Bauten und Technik blieben seither beinahe unverändert. Daher repräsentiert fast das gesamte Netz den Stand jener Zeit, so dass die Subway bisweilen als altertümlich und marode angesehen wird. Es handelt sich hierbei zumeist um kurze Verbindungsstücke zwischen bestehenden Strecken oder um kurze Verlängerungsstücke im Bereich der Endstationen.

Deren obere Enden verbinden Querträger , die wiederum zwischen den einzelnen Gleisen abgestützt sind. Erst später kam es zur Verwendung von Mauerwerk aus Stahlbeton , was die notwendige Anzahl der Pfeiler stark reduzierte. Die aufgeständerten Streckenabschnitte stammen mehrheitlich aus der Zeit der Doppelverträge und sind meist als vollwandige Stahl- Tragwerkskonstruktionen ausgeführt und überwiegend blaugrün gestrichen. Die Schienen sind auf Holzschwellen montiert, die im Grunde direkt auf dieser Tragwerkskonstruktion ruhen.

Dies ist neben einigen engen Kurven Hauptursache für das zum Teil als unerträglich empfundene Geräuschniveau einer solchen Strecke. Sie haben insbesondere die Eigenschaft, dass sie nicht ohne weiteres von U-Bahn-Wagen befahren werden können. Dies liegt neben den engen Kurven und den geringeren Gleismittenabständen insbesondere an ihrer Tragwerkkonstruktion, die für die schweren U-Bahn-Wagen aus Stahl zu schwach ausgelegt ist. Das rührt daher, dass die damals privaten Bauherren und Betreiber mangels staatlicher Beteiligung an diesen Projekten Geld sparen mussten.

Von den Old Els, also den alten Hochbahnen ohne verstärktes Tragwerk, sind heute keine mehr in Betrieb. Sie waren wegen ihrer geringen Verkehrsleistung unwirtschaftlich geworden und wurden nach der Vereinigung von der Stadt sukzessive stillgelegt. An einigen Stellen waren jedoch auch politische Gründe für den Abbruch der Hochbahnen ausschlaggebend. Ein kleinerer Teil der Strecken verläuft ebenerdig, auf einem Bahndamm oder im Einschnitt. Die Trassen ähneln durch ihren Oberbau aus eingeschotterten Schwellengleisen denen konventioneller Eisenbahnstrecken und sind meist aus solchen hervorgegangen.

Stellenweise ragt üppiger baumartiger Bewuchs bis tief in die Trasse hinein. Eine weitere New Yorker Eigenheit stellen Strecken mit mehr als zwei Betriebsgleisen dar, die sich vor allem auf den stark befahrenen Strecken im Stadtzentrum finden. Bei dreigleisigen Abschnitten wird das mittlere Gleis, das center track , je nach Tageszeit nur in der jeweiligen Hauptlastrichtung für diesen Zweck verwendet.

Viergleisige Strecken sind teilweise in zwei Stockwerken mit je zwei Gleisen angelegt, teilweise mit vier Gleisen nebeneinander.

Auch die Bahnhöfe sind auf dieses Betriebsmuster ausgerichtet. Während die local tracks an jeder Station Bahnsteige haben, verfügen die Expressgleise nur an ausgewählten Bahnhöfen, den Expressbahnhöfen express stations , über eine Zustiegsmöglichkeit. Meist ist etwa jede vierte bis sechste Station entlang einer mehrgleisigen Strecke ein solcher Expressbahnhof, im Bereich von Midtown Manhattan etwa Alle Stationen, die nicht von Expresszügen bedient werden, sind entsprechend local stations.

Stationen mit dreigleisigem Expresshalt weisen je einen Mittelbahnsteig zwischen dem Expressgleis und dem jeweiligen Lokalgleis der zugehörigen Fahrtrichtung auf.

Dabei kann wiederum zwischen Tunnellage, aufgeständerten und ebenerdigen Anlagen unterschieden werden. Bei Strecken mit mehr als zwei Gleisen bedienen diese immer nur das Bummler-Gleispaar. Expressbahnhöfe besitzen dagegen in der Regel zwei Mittelbahnsteige , die richtungsgleiches Umsteigen ermöglichen.

Bei dreigleisigen Abschnitten wird das mittlere Gleis von beiden Bahnsteigen bedient, wobei die Zugtüren dort immer nur in Fahrtrichtung rechts geöffnet werden. An manchen dieser Abgänge ist nicht einmal ein Fahrkartenautomat vorhanden. Speziell bei älteren Bahnhöfen setzt sich die Architektur der Tunnel nahtlos fort.

Die Raumhöhe geht zudem in vielen Fällen nicht über die der angrenzenden Tunnel hinaus. Über den Bahnsteigen ist es wegen abgehängter Lampen und Versorgungsleitungen mitunter noch etwas niedriger. Die Gestaltung der unterirdischen Bahnhöfe zeigt auch signifikante Unterschiede zwischen den zumeist zwischen und eröffneten Bahnhöfen der IRT und den zumeist zwischen und eröffneten Bahnhöfen der IND.

Durch den Fortschritt der Technik war es auch möglich, die Raumhöhe auf dem Bahnsteigniveau etwas höher und damit freundlicher zu gestalten und die Stützsäulen auf den Bahnsteigen schlanker auszuführen. Nach und nach wurden zahllose Treppenaufgänge und Zwischengeschosse ganz oder teilweise geschlossen, besonders in den er Jahren, um Kriminalität in den weitläufigen und oft menschenleeren Zwischengeschossen vorzubeugen.

In den Zwischengeschossen ist häufig durch unterschiedliche Wandfliesen zu erkennen, dass ursprünglich durchgehende Zwischengeschosse auf die Eingangsbereiche an den Bahnsteigenden zurückgebaut wurden. Die dazwischen liegenden Bereiche werden häufig als Lagerräume genutzt. Die Bahnsteige sind normalerweise mit Holzplatten beplankt und auf zwei stählernen Längsträgern montiert.

Dazu kommt in der Regel auf mindestens halber Länge eine Überdachung aus Blech. Die Namen der Bahnhöfe sind nicht immer eindeutig.

Die meisten Bahnhöfe sind nicht behindertengerecht. Auch ist der nachträgliche Einbau von Rolltreppen und Aufzügen aufgrund der vielfach beengten Platzverhältnisse nur schwer möglich und daher teuer, so dass bisher nur stark frequentierte oder günstig gelegene Stationen damit ausgerüstet wurden.

Eine Strecke line ist zunächst gleichbedeutend zu einer Eisenbahnstrecke , also einer Verbindung mit einem Schienenweg. Sie hat im Regelfall eine eigene Kilometrierung und existiert unabhängig vom tatsächlich stattfindenden Verkehr. Allerdings sind die Abgrenzungen teilweise mehr dem Volksmund denn der tatsächlichen Streckenführung zuzuordnen. Sämtliche Streckenabschnitte der Subway haben eigene Namen. Dazu hat es sich eingebürgert, das Kürzel der jeweiligen Betreibergesellschaft davor zu setzen, weil die Namen in der Vergangenheit nicht immer eindeutig waren.

Zu den Bemühungen um die Konsolidierung der U-Bahn nach der Vereinigung gehörte auch, jene Präfixe offiziell nicht mehr zu verwenden. Doch die Bevölkerung hat diese Praxis bis heute kaum übernommen. Die Strecken der IRT weisen dabei ein schmaleres Lichtraumprofil auf, so dass sie nicht von den Fahrzeugen der anderen beiden Abteilungen befahren werden können. Dieser Umstand rührt daher, dass die Interborough ihr U-Bahn-Netz nach den Standards der alten Hochbahnen baute, die aus Platz- und Gewichtsgründen nur schmalere Fahrzeuge einsetzen konnten.

Die technische Ausstattung ist bei Schmal- und Breitprofilstrecken [27] im Prinzip identisch, so dass ein gemeinschaftlicher Betrieb auf der gleichen Strecke grundsätzlich möglich ist.

Langfristig ist geplant, alle schmalprofiligen Strecken auf Breitprofil umzubauen. Die Strecken an der Oberfläche verursachen dagegen weniger Probleme; dort müssen im Regelfall nur die Bahnsteigkanten gekappt werden.

Heute bezieht sich die Bezeichnung Division aber auf Profil und Rollmaterial. Somit gehörten auch diejenigen aufgeständerten Trassen zur U-Bahn, die von diesen Zügen befahren wurden und in ihre Tunnel mündeten. Die Einteilung geschah unabhängig davon, dass einige Abschnitte schon auf U-Bahn-Standard umgebaut waren.

Der letzte Hochbahnabschnitt auf der Insel Manhattan wurde am August stillgelegt, und am Historisch gesehen handelt es sich dabei um Ausflugsbahnen nach Coney Island und die daran angeschlossenen U-Bahn-Tunnel, die im Zuge der Doppelverträge entstanden. Die dadurch definierten netzinternen Grenzen spiegelten sich sowohl bei Wartung und eingesetztem Wagenpark als auch im Regelbetrieb wider.

Trotz einiger Gleisverbindungen gab es kaum Verkehr über beide Divisionen hinweg, was sich bis heute im Grunde nicht geändert hat. In Queens gab es noch eine Besonderheit: Die Canarsie Line in Bushwick. Das Netz der Independent Subway und späteren IND Division wurde in den er Jahren als Einheit geplant und gebaut, so dass es hier keine historisch begründeten Streckeneinteilungen gab.

Nach der Vereinigung der U-Bahn kamen noch einige weitere Streckenabschnitte hinzu. Darüber hinaus werden einige Neubaustrecken aus der Zeit nach zu dieser Streckengruppe gezählt. Dazu entstanden im Laufe der Zeit einige Bauvorleistungen für geplante, aber letztlich nie realisierte Strecken.

Die Old Els sind meist nur noch dort zu sehen, wo sie einst mit anderen, noch existierenden Strecken verbunden waren. Die Gleise und Bahnsteige an der Ninth Avenue sind noch erhalten und werden gelegentlich für Filmaufnahmen verwendet. Dabei wurden an einigen Stellen Kreuzungs- und Abzweigbahnhöfe gebaut, die heute jedoch nur teilweise in Betrieb sind. Allerdings ist dieser Umstand meist nicht sofort zu erkennen, weil die zusätzlich errichteten Betriebsanlagen entweder nicht zugänglich sind oder anderweitig genutzt werden.

Dort bilden die beiden mittleren Bahnsteiggleise den Streckenstumpf einer nie gebauten Verbindung hinüber nach Williamsburg. Als in den er Jahren die Bahnsteige für Zehn-Wagen-Züge verlängert wurden, sparte man einige Stationen aus, um sie stillzulegen und dadurch die Reisegeschwindigkeit zu erhöhen. Dort sind die Bahnsteige zwar noch erhalten, werden seitdem jedoch ohne Halt durchfahren.

Besonders in Manhattan fallen in vielen Stationen zudem ungenutzte und für den Betrieb scheinbar überflüssige Bahnsteige auf. Dadurch gingen die Fahrgastzahlen an bestimmten Bahnhöfen zurück, so dass der Zusatzaufwand nicht mehr gerechtfertigt war.

An anderen Stellen wiederum gab es Bahnsteige, die ein Betriebsgleis von beiden Seiten erschlossen, die sich als unpraktisch erwiesen und daher aufgegeben wurden. Normalerweise werden bei auf gleicher Strecke fahrenden Linien die Umsteigemöglichkeiten zu diesen Linien nur angesagt, wenn gleichzeitig auch Anschluss zu anderen, nicht auf gleicher Strecke fahrenden, Linien besteht.

Die Linien routes oder auch services bezeichnen im Unterschied zu den Strecken den tatsächlich stattfindenden Verkehr als eine Gruppe von Zügen , die eine bestimmte Haltestellenfolge bedienen.

Ihre Nummerierung und Verläufe haben sich über die Zeit mehrfach tiefgreifend geändert. Der Unterschied zu den Strecken wird im Zusammenhang deutlich: Nachdem er eingestiegen ist, würde er aber sagen: Die Nummerierung geschieht nach einem bestimmten Muster. Linien, die die schmalprofiligen Strecken der Division A befahren, sind mit Zahlen von 1 an aufwärts durchnummeriert. Die heutige Systematik gilt prinzipiell seit dem 5. Die Farbe gibt dagegen schon seit Juni die Stammstrecke einer Linie wieder.

Der Unterschied liegt darin, ob ein Zug auf einer bestimmten Strecke das Bummler-Gleispaar local tracks benutzt und entsprechend an jedem Bahnhof hält, oder auf den Express-Gleisen fährt und dabei eben nicht überall hält. Das Verhalten lässt sich hier anhand der Form des Liniensymbols erkennen: Locals fahren mit rundem, ein Express mit rautenförmigem Logo als so genannter diamond service.

Dabei halten die Züge auf bestimmten Streckenabschnitten abwechselnd an jeder zweiten Station. August und dem Die Linie 1 bedient aktuell die historische Station South Ferry, sie wurde im April wieder eröffnet, nachdem die Neubaustation durch den Hurrikan Sandy stark beschädigt wurde.

Die Fahrzeuge werden elektrisch betrieben. Bei beiden Divisions werden im Regelfall Züge zu je zehn Wagen gebildet, die dementsprechend unterschiedliche Längen und Beförderungskapazitäten haben. Die derzeit eingesetzten Fahrzeuge stammen aus den Jahren bis und sind der zweiten, dritten und vierten Generation U-Bahn-Wagen zuzurechnen. Die Wagenkästen bestehen allesamt aus rostfreiem Stahl und sind, bis auf rote Zierelemente an den Fronten der R und R, nicht farbig lackiert.

Der Ersatz der R42 hat hier allerdings Priorität, da diese sich — obwohl jünger — in einem bedeutend schlechteren Zustand befinden. Yards sind dabei mit Betriebswerkstätten vergleichbare Anlagen. In zwei Betriebshöfen sind zudem Bautrupps stationiert. Er wurde von der BMT in Betrieb genommen. Die Anlage ist auf diesem Satellitenbild deutlich als graubrauner Fleck zu erkennen.

Die Hauptwerkstatt arbeitet sieben Tage die Woche rund um die Uhr. Daneben werden hier die vielen historischen Wagen restauriert und untergestellt. In den letzten Jahren sind einige Anlagen ausgebaut worden, um die wachsende Fahrzeugflotte bedienen zu können. Historisch gesehen wurden die Betriebshöfe ebenso wie das Netz an sich von den drei Gesellschaften getrennt errichtet und betrieben.

Bis auf zwei liegen alle in der Bronx. Daneben unterhielt die Interborough fünf weitere Betriebshöfe und Werkstätten für die Manhattaner Hochbahnen. Hallen und Abstellgleise waren dabei manchmal ebenerdig angeordnet, so dass die Wagen den Höhenunterschied einzeln mittels Lastenaufzug überwinden mussten. Heute befindet sich dort ein Gewerbegebiet. Genauso errichtete die Independent ihre eigenen Betriebshöfe. Als die Rockaway Line in Betrieb ging, waren zusätzliche Abstellkapazitäten notwendig.

Der Linden Yard ist der einzige Betriebshof, der nicht mit elektrischen Triebwagen erreichbar ist. Ungenutzte Expressgleise werden heute im Gegensatz zu früher nicht mehr zum Abstellen von Zügen verwendet, um Graffiti und Vandalismus vorzubeugen. Es handelt sich dabei um eine elektronische Geldbörse in Form einer dünnen Plastikkarte mit Magnetstreifen. Allerdings ist nicht zu allen diesen Verkehrsmitteln ein kostenloser Übergang möglich, teilweise wird beim Umsteigen zwischen zwei Verkehrsmittel eine neue Fahrt zum jeweiligen Tarif abgerechnet.

Es gibt sie als Einzel- und Mehrfahrtenkarte sowie als Zeitkarte. Juni erhöhte die MTA die Fahrpreise: Ebenso wurde am 3. Das Aufladen der Metrocard ist weiterhin kostenlos. Bis wurden bei den U-Bahnen ganz normale Fahrscheine zum damaligen Einheitstarif von fünf Cent verkauft.

Mai eingestellt, und die Fahrgäste warfen die Fünf-Cent-Münzen von diesem Zeitpunkt an direkt in die neuen, automatischen Drehkreuze ein. Die New York Subway war damit zu einem so genannten geschlossenen System geworden. Dieses System überstand so jahrzehntelang alle weiteren Tariferhöhungen, bei denen dann entweder neue Tokens ausgegeben oder nur die Verkaufspreise geändert wurden.

Ab wurde schrittweise die MetroCard eingeführt. Sie sollte einerseits das mühselige Entleeren der über Die Drehkreuze akzeptierten aber auch weiterhin die alten Tokens. Erst seit dem April kann man in der Subway nur noch mit der MetroCard fahren. Ab Ende soll das neue System installiert und bis Ende fertig gestellt werden. Das neue System soll die MetroCard ab ersetzen. Dabei ist der Fahrplan abgesehen von den Wochenenden für die tageszeitlichen Schwankungen der Verkehrsnachfrage grob in vier Zeiträume eingeteilt, in denen jeweils andere Taktfolgen herrschen.

Die Rush Hours bezeichnen dabei die Hauptverkehrszeiten und gehen von Einige Linien weichen von dieser Einteilung jedoch ab. Der Fahrplan ist so ausgelegt, dass zur Hauptverkehrszeit auf den am stärksten befahrenen Strecken pro Richtung etwa alle drei bis fünf Minuten ein Zug kommt. Gibt es dort auch Expressbetrieb, so fahren diese Züge meist ebenso oft.

Auf einigen Streckenabschnitten verhält es sich jedoch genau umgekehrt, so dass die Expresslinien stattdessen an jeder Station halten.

Bei der New Yorker U-Bahn werden die Zugfahrten durch Stellwerke , Signaltechnik , streckenseitige Zugbeeinflussung und streckenseitige Geschwindigkeitsüberwachung gesichert. Elektronische Stellwerke entsprechen der jüngsten Generation mit zusätzlichen Funktionen. Unabhängig von der Stellwerkstechnik bleibt die Logik der Stellwerke gleich: Übliche Home-Signale bestehen aus einem Signalschirm mit drei Lampen, um folgende grundlegende Fahrbegriffe zu zeigen:.

Vor Verzweigungsweichen werden sowohl Geschwindigkeits- als auch Richtungssignalisierung benutzt; dabei kommen ferner zwei unterschiedliche Signalisierungs-Philosophien zur Anwendung:. Auch wenn dies eine einfache Zugbeeinflussung ist, muss berücksichtigt werden, dass die Fahrsperre erst dann wieder wirksam geschaltet werden darf, wenn sichergestellt ist, dass der Zug vollständig das zugehörige Signal passiert hat.

Ein Zeitschalter Timer wird gestartet, sobald ein Zug einen bestimmten Punkt passiert, und nach einer vordefinierten Zeit das vorausliegende Hauptsignal auf Fahrt gestellt; die Timer-Zeit wird anhand der Geschwindigkeitsbeschränkung und der Entfernung vom Einschaltpunkt zum Hauptsignal ermittelt. Linien L und 7 [37] die Signalisierung , Zugbeeinflussung und Geschwindigkeitsüberwachung modernisiert.

Im Laufe der über jährigen Geschichte der New Yorker U-Bahn haben sich derart viele Unfälle ereignet, dass der Ruf dieses Verkehrsmittels bis heute darunter leidet. Vor allem in den er und er Jahren kam es zu zahlreichen Vorfällen wegen des schlechten Erhaltungszustands von Fahrzeugen und Betriebsanlagen. Allein im Jahr zählte man 30 solcher Entgleisungen.

Unfälle wie jene vom Mai oder vom Juli sind auf schlechte Ausbildung der Mitarbeiter und Mängel in den Dienstvorschriften zurückzuführen. Die Terroranschläge am Allerdings hatte auch hier die Tragwerkkonstruktion gelitten, und die Ausgänge waren verschüttet. Als die Sperrzone am September auf den Bereich südwestlich Broadway und Canal Street beschränkt wurde, konnten fast alle Strecken wieder normal betrieben werden. Aus wirtschaftspolitischen Erwägungen setzten die Stadt und mit ihr die MTA alles daran, die beschädigten Tunnelabschnitte so schnell wie möglich wieder in Betrieb zu nehmen.

Januar konnte man dort auch wieder einsteigen. Oktober war die BMT Broadway Line wieder durchgehend befahrbar, nachdem man einige provisorische Pfeiler eingezogen hatte; der Bahnhof Cortlandt Street blieb jedoch noch bis zum Die Züge fuhren dort bis dahin mit stark verminderter Geschwindigkeit durch. Der Bahnhof Cortlandt Street an dieser Strecke bleibt jedoch geschlossen, weil er noch mitten in der Baugrube liegt.

Die beiden Seitenbahnsteige wurden solange provisorisch abgemauert. Obwohl New York in der zweiten Hälfte des Jahrhunderts zu den am schnellsten wachsenden Städten der Erde zählte, wurde durch die herrschende wirtschaftliche und politische Elite der Bau eines solchen Nahverkehrsnetzes zunächst verhindert.

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