PASCAL

Pascal was born in France/Clermont on June 19, 1623. Bright child began learning from the age of seven. Two sisters, like himself, had a very beautiful and cultured. Jak Qualine especially, has played an important role in Pascal's life. Sisters, this effect sometimes good, but most have been in a bad direction.           . Such as Descartes and Fermat to be contemporary with the great mathematician somewhere for him was a misfortune. Therefore, you can create a stand-alone discovery of probability theory Fermat ‘a shared. Issues related to religion and philosophy was more in mathematics, taking little time left. Her sister was dominant in this regard. In contrast, gave much more than it can do.           Pascal, who was a child very early development. However body’s was very weak. Conversely, the head was too shiny. Although very young age, showed interest in mathematics was very striking. In fact, mathematical problems dealing with the day and night began. The deterioration of health will suspect's father, a gap was a barrier to his mathematical work, though, it's more of this behavior led to Pascal's mathematical work. Euclide ' s "Elements" in a short time swallowing his geometry book read like a novel.           Without any assistance and no geometry without reading, at a very young age of the sum of a triangle inside a 180-degree angle, that has proved to be two right angles. Pascal avoids sister told by Gilbert; Euclides'in Pascal to recommend the first thirty-two in his book Elements found in the order. Thirty-second proposition is related to the sum of angles inside a triangle is the proof. Pascal himself was a geometry process. Father Pascal family's plight to wrestle with the government authorities dropped. Beautiful and bright sister, Jacqueline, tax dispute with his father to entertain the Cardinal de Richelieu'yu in front of a game being played out in games without introducing himself. The artist who is a fan of self-learning Cardinal, and then the whole family donations to Pascal's father gives an officer.           Pascal, sixteen years ago, in 1639, the most beautiful of the geometry theorem proved. Nineteenth century the famous British mathematician Sylvester, Pascal's theorem to this big "cat’s cradle" name was given. When it comes to the age of eighteen, now a museum that is stored in the calculator industry in Paris has found. In physics, the weight of air, fluid and pressure on the balance of state laws found Pascal. Apollonius and others that accept work one results revealed four hundred to one proposalPascal's this piece a geometric metric is not a projection geometry. Aristo, the multitude of mathematical knowledge he was the. However, there is no plurality in Pascal's geometry.            Twenty-three years old, sisters, and the pressure and influence within the Christian religion to some sect input. This topic also spent a lot of shock. But still he prevailed mathematics. Pascal, like palm trees began to dry hills. The same year broke digestive organs.

Mean while, had a temporary paralysis. This gave him much pain. In spite of everything, the idea of the work was going on and head.         In 1648 Torricelli’s (1608 -1647) by reviewing their work, it moved to the front. Change of pressure with height has detected. Descartes, Pascal, and especially to talk with the various issues came to learn about barometers. These two mental states of knowledge creation and many did not match. Descartes, on conical written by a boy of sixteen years of work is written expressly refused to believe. Even further, the idea of Pascal's barometer experiment, Messene’s stole from work has become suspect. Descartes will hold Jesuits, Pascal Jansen defended the sect. Pascal's outspoken sister Jacqueline's it turns out that these two were even jealous of each other quite. Therefore, the above-mentioned meeting was a cold call and visit. Some advice to the young man had Descartes. Pascal did not take him seriously. In the year 1658 a night, insomnia and toothache of the twisted Pascal, nippers at a time when the dominant, in order to forget the terrible pain, the most famous mathematicians of the deal was lost in a graceful curve Cycloid’s. All the pain he saw the latest or, so dived on Cycloids that was all forgotten their pain and suffering. He worked on the full eight days Cycloids geometry. Are associated with this curve was able to solve various problems. This meeting with some of the signature nickname Detonville Amos, the French and British mathematician place to read, to have been printed. In 1658 the patient felt very self. The exception of short intervals, the drowsiness, severe headaches and relieve him never was much persecution. Full four years was convulsed with pain. Year in June of 1662 at the age of thirty-nine died. After his death in the autopsy, the cause of the pain from a serious brain disease was found forward.                This is probably because the output of the theory, gambler Chevalier de Mere to Pascal was to recommend it. The most important task of the fifty-two card games was played. Meanwhile the backgammon dice, with the same shape of different colored ball is of great importance. Accordingly, the famous Pascal triangle was born. This is Pascal's triangle, was used in later years too. In particular, this method of expansion and binomial series expansions are easily found.  Pascal’s Triangle, binomial expansion coefficients to find that the benefits of. This is Pascal's triangle, probability theory is also used expertly. This triangle, biology applications, mathematics, statistics and many application areas of modern physics is found. Christian religion

Pascal Triangle

Eugène Charles Catalan defined the numbers, called today the Catalan numbers, while considering the solution of the problem of dissecting a polygon into triangles by means of non-intersecting diagonals. Catalan was not the first to solve the problem, however, since Segner had solved it in the 18 th century, although his solution was not as elegant as Catalan's. Euler had worked on simplifying the solution to the problem as did Binet at almost the same time as

Catalan, around 1838. The Catalan Numbers are a sequence of numbers which show up in many contexts. They were discovered by Leonhard Euler when he was attempting to find a general formula to express the number of ways to divide a polygon with N sides into triangles using non-intersecting diagonals . The Catalan Numbers' correspondence to the division of polygons is shown below:

Number of sides 3 4 5 6 7 8 9

Number of way to partition it into triangles 1 2 5 14 42 132 429

 

 

The first few Catalan numbers for n = 1, 2, ... are 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ... (Sloane's A000108).

An explicit formula for is given by

(1)

where denotes a central binomial coefficient and is the usual factorial. A recurrence

relation for is obtained from

Add the indeed: (2)

(3)

There are at least two ways they can be found in Pascal's triangle, one in the middle column going down the center, subtracting the element immediately adjacent (see numbers in red),

1 2 6 20 70 252 ............- 1 4 15 56 210 ............1 1 2 5 14 42 ............

and another one row above, taking the Nth term over and subtracting the term immediately to the right (numbers in blue).

1 3 10 35 126 462 ............- 1 5 21 84 330 ............

1 2 5 14 42 132 ...........

See the next examples:

1 2 6 20 70 252 ............- 1 6 28 120 ............1 2 5 14 42 132 ............

and:

1 = 11 = 2-12 = 6-3-15 = 20-10-1-414=70-35-15-5-142=252-126-56-21-6-1

1 = 12 = 3-15 =10-4-114=35-15-5-142=126-56-21-6-1

You can see in next Pascal Triangle that each Catalan number is the sum of specific Pascal numbers.(© Dirk Laureyssens, 2004)

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The lines containing the three points of the intersection of the three pairs of opposite sides of a (not necessarily regular) hexagon.

There are 6! (i.e., 6 factorial) possible ways of taking all polygon vertices in any order, but among these are six equivalent cyclic permutations and two possible orderings, so the total

number of different hexagons (not all simple) is

There are therefore a total of 60 Pascal lines created by connecting polygon vertices in any order.

The 60 Pascal lines form a very complicated pattern which can be visualized most easily in the degenerate case of a regular hexagon inscribed in a circle, as illustrated above for magnifications ranging over five powers of 2. Only 45 lines are visible in this figure since each of the three thick lines (located at angles to each other) represents a degenerate group of four Pascal lines, and six of the Pascal lines are lines at infinity (Wells 1991).

The pattern for a general ellipse and hexagon (illustrated above) is much more complicated, and is difficult to distinguish

WHAT İS THE PASCAL?

Pascal’s Triangle, binomial expansion coefficients to find that the benefits of. This is Pascal's triangle; probability theory is also used expertly. This triangle, biology applications, mathematics, statistics and many application areas of modern physics is found. Christian Religion. (According to some sources, ancient Chinese had recognized the triangle some Pascal triangle Khayyam triangle is actually a he had said.) Probability theory out the cause to Pascal by gambler Chevalier de mere was to propose. The most important task of the fifty-two card games was played. Meanwhile the backgammon dice, with the same shape of different colored ball is of great importance. Accordingly, the famous

Pascal triangle was born. This is Pascal's triangle, was used in later years too. In particular, this method of expansion and binomial series expansions are easily found.

Formula ,

example:

Indicates the number of subsets of a set ’PASCAL’ let’s make a triangle. Number of elements of the cluster: s(A)=0............................................ ...............1s(A)=1............................................ ............1.....1s(A)=2............................................ .......1.....2.....1s(A)=3............................................ ..1.....3.....3.....1s(A)=4..........................................1. ....4.....6.....4.....1s(A)=5......................................1..... 5.....10....10.....5....1 ...

BINOM EXPANSION (A+B).n = expansion in the numbers in the Pascal triangle coefficients of the term. Forces from 0 up to n a’s decrease, the b force increased from 0 up to n to be written.(A+B).5=?coefficients =1 5 10 10 5 1A’s forces: a5 a4 a3 a2 a 1B ‘s forces: 1 b b2 b3 b4 b6

(a+b)5=1a5+5a4b+10a3b2+10a2b3+5ab4+1b5

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KAYNAKÇA2)www2.lv.psu.edu/.../fa-cb-bc-kf/1200

3)www.bydigi.net/geometri-matematik

3) mathworld.wolfram.com/PascalLines.html

4) http://www.gap-system.org/~history/Biographies/Pascal.html

5)http://www.macs.hw.ac.uk/~greg/calculators/pascal/Blaise_Pascal's_Gallery.htm

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