When was the Pascal Triangle discovered?
In 1653 he wrote the Treatise on the Arithmetical Triangle which today is known as the Pascal Triangle. Although other mathematicians in Persia and China had independently discovered the triangle in the eleventh century, most of the properties and applications of the triangle were discovered by Pascal.
When was the triangle invented?
Gerolamo Cardano, also, published the triangle as well as the additive and multiplicative rules for constructing it in 1570. Pascal's Traité du triangle arithmétique (Treatise on Arithmetical Triangle) was published posthumously in 1665.
Which triangle is the same as Pascal's triangle?
In 13th century, Yang Hui (1238–1298) presented the arithmetic triangle that is the same as Pascal's triangle. Pascal's triangle is called Yang Hui's triangle in China. The "Yang Hui's triangle" was known in China in the early 11th century by the Chinese mathematician Jia Xian (1010–1070). Petrus Apianus (1495–1552) published the triangle on ...
What is Pascal's Traité du triangle arithmétique?
Pascal's Traité du triangle arithmétique (Treatise on Arithmetical Triangle) was published in 1655. In this, Pascal collected several results then known about the triangle, and employed them to solve problems in probability theory.
Who discovered the Pascal's triangle?
mathematician Jia XianIt is named for the 17th-century French mathematician Blaise Pascal, but it is far older. Chinese mathematician Jia Xian devised a triangular representation for the coefficients in the 11th century.
What is the history of the Pascal triangle?
Pascal's triangle is named after the French mathematician and philosopher Blaise Pascal (1623-1662), who was the first to write an organized work about the properties and applications of the triangle in his treatise, Traité du triangle arithmétique (Treatise on Arithmetical Triangle) in 1653.
Who invented Pascal's Triangle in India?
The Indian mathematician Bhattotpala (c. 1068) later gives rows 0-16 of the triangle. At around the same time, it was discussed in Persia by the mathematician Al-Karaji (953–1029) and the poet-astronomer-mathematician Omar Khayyám (1048-1131); thus the triangle is referred to as the "Khayyam triangle" in Iran.
Why is Pascal's triangle named after Pascal?
Pascal's Triangle is a special triangular arrangement of numbers used in many areas of mathematics. It is named after the famous 17 th century French mathematician Blaise Pascal because he developed so many of the triangle's properties.
How old is Pascal's triangle?
Pascal's triangle was known in China in the early 11th century through the work of the Chinese mathematician Jia Xian (1010–1070). In the 13th century, Yang Hui (1238–1298) presented the triangle and hence it is still called Yang Hui's triangle (杨辉三角; 楊輝三角) in China.
Why is Pascal's triangle so important?
Pascal's triangle is important because it contains numerous patterns that can be used to make complex calculations much easier.
Where is Pascal's triangle used in real life?
One real life situation that Pascal's Triangle is used for is Probability, and combinations. We have situations like this all of the time. For example, say you are at an ice cream shop and they have 5 different ice creams. You want to know how many different ways you can pick two of the ice creams and eat them.
Who is the father of triangles?
Many have speculated that the claims of the 5th-century BC Greek mathematician Pythagoras being the first to deduce facts about right-angled triangles just did not add up. Some believe later commentaries indicate it was the collaborative work of his followers, the Pythagoreans.
Is Pascal's Triangle infinite?
Pascal's triangle is a never-ending equilateral triangle of numbers that follow a rule of adding the two numbers above to get the number below. Two of the sides are “all 1's” and because the triangle is infinite, there is no “bottom side.”
What are 5 interesting things about Pascal's triangle?
7 Amazing Facts About Pascal's TriangleThe numbers on each row are binomial coefficients.The numbers on the second diagonal form counting numbers.The numbers on the third diagonal are triangular numbers.The sum of the numbers on each row are powers of 2.A series of diagonals form the Fibonacci Sequence.
How can I memorize Pascal's triangle?
0:396:09Pascal's Triangle and the Binomial Theorem | Don't Memorise - YouTubeYouTubeStart of suggested clipEnd of suggested clipIn the Pascal's triangle each number is the sum of the two numbers directly above it yes notice thatMoreIn the Pascal's triangle each number is the sum of the two numbers directly above it yes notice that 2 is the sum of 1 and 1 this 3 is the sum of 1 and 2 and this 3 is the sum of 2.
Is Pascal an old language?
Pascal, a computer programming language developed about 1970 by Niklaus Wirth of Switzerland to teach structured programming, which emphasizes the orderly use of conditional and loop control structures without GOTO statements.
How did Pascal make his triangle?
The triangle is constructed by starting with a 1 in the upper left corner and each number in the triangle is a result of adding the number to its left to the number above it. Pascal used this triangle to conclude many results.
What is Pascal and who invented it?
Blaise Pascal laid the foundation for the modern theory of probabilities, formulated what came to be known as Pascal's principle of pressure, and propagated a religious doctrine that taught the experience of God through the heart rather than through reason.
What are 5 interesting things about Pascal's triangle?
7 Amazing Facts About Pascal's TriangleThe numbers on each row are binomial coefficients.The numbers on the second diagonal form counting numbers.The numbers on the third diagonal are triangular numbers.The sum of the numbers on each row are powers of 2.A series of diagonals form the Fibonacci Sequence.
Who invented the triangle?
Chinese mathematician Jia Xian devised a triangular representation for the coefficients in the 11th century. His triangle was further studied and popularized by Chinese mathematician Yang Hui in the 13th century, for which reason in China it is often called the Yanghui triangle.
How to fill out a triangle?
Then the triangle can be filled out from the top by adding together the two numbers just above to the left and right of each position in the triangle. Thus, the third row, in Hindu-Arabic numerals, is 1 2 1, the fourth row is 1 4 6 4 1, the fifth row is 1 5 10 10 5 1, and so forth. The first row, or just 1, gives the coefficient for the expansion of (x + y)0 = 1; the second row, or 1 1, gives the coefficients for (x + y)1 = x + y; the third row, or 1 2 1, gives the coefficients for (x + y)2 = x2 + 2xy + y2; and so forth.
How to make a triangle in Chinese?
The triangle can be constructed by first placing a 1 (Chinese “—”) along the left and right edges. Then the triangle can be filled out from the top by adding together the two numbers just above to the left and right of each position in the triangle.
Who created the triangle of numbers?
Pascal’s triangle, in algebra, a triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression, such as (x + y)n. It is named for the 17th-century French mathematician Blaise Pascal, but it is far older. Chinese mathematician Jia Xiandevised a triangular representation for the coefficients in the 11th century. His triangle was further studied and popularized by Chinese mathematician Yang Huiin the 13th century, for which reason in Chinait is often called the Yanghui triangle. It was included as an illustration in Chinese mathematician Zhu Shijie’s Siyuan yujian(1303; “Precious Mirror of Four Elements”), where it was already called the “Old Method.” The remarkable pattern of coefficients was also studied in the 11th century by Persian poet and astronomer Omar Khayyam.
What is algebra in math?
algebra, branch of mathematics in which arithmetical operations and formal manipulations are applied to abstract symbols rather than specific numbers. The notion that there exists such a distinct subdiscipline of mathematics, as well as the term algebrato denote it, resulted from a slow historical development. This article presents that…
What is East Asian mathematics?
East Asian mathematics: Theory of root extraction and equations
Who created the fractal?
Polish mathematician Wacław Sierpiński described the fractal that bears his name in 1915, although the design as an art motif dates at least to 13th-century Italy. Begin with a solid equilateral triangle, and remove the triangle formed by connecting the midpoints of each side.
Introduction
Mathematics is a world of numbers. We have different sets of numbers that can be arranged in different sequences and patterns. One such arrangement of numbers is the Pascal’s triangle. What is it and who invented it? Let us find out.
Definition
Pascal’s Triangle is the triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression. The numbers are so arranged that they reflect as a triangle. We can also say that Pascal’s triangle is an arrangement of binomial coefficients in triangular form.
Key Facts and Summary
Pascal’s Triangle is the triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression. The numbers are so arranged that they reflect as a triangle.
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How to Use Pascal’s Triangle?
Pascal’s triangle can be used in various probability conditions. Suppose if we are tossing the coin one time, then there are only two possibilities of getting outcomes, either Head (H) or Tail (T).
How to find Fibonacci numbers on a Pascal triangle?
There are various ways to show the Fibonacci numbers on the Pascal triangle. R. Knott was able to find the Fibonacci appearing as sums of “rows” in the Pascal tri angle. He moved all the rows over by one place and here the sums of the columns would represent the Fibonacci numbers.
What are the properties of Pascal's triangle?
Properties of Pascal’s Triangle 1 Each number is the sum of the two numbers above it. 2 The outside numbers are all 1. 3 The triangle is symmetric. 4 The first diagonal shows the counting numbers. 5 The sums of the rows give the powers of 2. 6 Each row gives the digits of the powers of 11. 7 Each entry is an appropriate “choose number.” 8 And those are the “binomial coefficients.”
What was Pascal's contribution to mathematics?
This triangle was among many of Pascal’s contributions to mathematics. He also came up with significant theorems in geometry, discovered the foundations of probability and calculus and also invented the Pascaline-calculator. Still, he is best known for his contributions to the Pascal triangle.
How to construct a triangle with no numbers?
The easiest way to construct the triangle is to start at row zero and write only the number one. From there , to obtain the numbers in the following rows, add the number directly above and to the left of the number with the number above and to the right of it. If there are no numbers on the left or right side, replace a zero for that missing number and proceed with the addition. Here is an illustration of rows zero to five.
What are prime numbers in a triangle?
If a row starts with a prime number or is a prime numbered row, all the numbers that are in that row (not counting the 1’s) are di visible by that prime. If we look at row 5 (1 5 10 10 51), we can see that 5 and 10 are divisible by 5.
How to describe a triangle?
A different way to describe the triangle is to view the first line is an infinite sequence of zeros except for a single 1. To obtain successive lines, add every adjacent pair of numbers and write the sum between and below them. The non-zero part is Pascal’s triangle.
What is the name of the treatise Pascal wrote on the triangle?
Pascal's Traité du triangle arithmétique ( Treatise on Arithmetical Triangle) was published posthumously in 1665. In this, Pascal collected several results then known about the triangle, and employed them to solve problems in probability theory.
What is the set of numbers that form Pascal's triangle?
The set of numbers that form Pascal's triangle were known before Pascal. However, Pascal developed many uses of it and was the first one to organize all the information together in his treatise, Traité du triangle arithmétique (1653). The numbers originally arose from Hindu studies of combinatorics and binomial numbers and the Greeks' study ...
When was the triangle of binomial coefficients first used?
The earliest explicit depictions of a triangle of binomial coefficients occur in the 10th century in commentaries on the Chandas Shastra, an Ancient Indian book on Sanskrit prosody written by Pingala in or before the 2nd century BC.
Which theorem is related to the triangle?
Several theorems related to the triangle were known, including the binomial theorem. Khayyam used a method of finding n th roots based on the binomial expansion, and therefore on the binomial coefficients. In 13th century, Yang Hui (1238–1298) presented the arithmetic triangle that is the same as Pascal's triangle.
Who named the triangle?
The triangle was later named after Pascal by Pierre Raymond de Montmort (1708) who called it "Table de M. Pascal pour les combinaisons" (French: Table of Mr. Pascal for combinations) and Abraham de Moivre (1730) who called it "Triangulum Arithmeticum PASCALIANUM" (Latin: Pascal's Arithmetic Triangle), which became the modern Western name.
Who was the first person to describe the Khayyam-Pascal triangle?
It was later repeated by the Persian poet-astronomer-mathematician Omar Khayyám (1048–1131); thus the triangle is referred to as the Khayyam-Pascal triangle or simply the Khayyam triangle in Iran.
Who published the triangle?
Petrus Apianus (1495–1552) published the triangle on the frontispiece of his book on business calculations in the 16th century. This is the first record of the triangle in Europe.
What is Pascal's triangle?
Pascal ’ s triangle, in mathematics, is a geometric arrangement of the binomial coefficients. It is a well-known set of numbers aligned in the shape of a pyramid. The numbers represent the binomial coefficients, which are representations of the number of subsets of a given size. The numbers in Pascal ’ s triangle are also the coefficients of the expansion of (a + b) n , (a + b) raised to the n th power. So, for n equals to three, the expansion is (a + b) × (a + b) × (a + b), which equals (a 2 + 2ab+ b 2) × (a+ b), which equals (a 3 + 3ab 2 + 3ba 2 + b 3 ). The coefficients are 1, 3, 3, 1. These are listed in the third row of Pascal ’ s triangle.
How are figurate numbers formed?
Actually, figurate numbers can be formed from any polygon. Another set of figurate numbers could be formed using the pentagon, a polygon with five sides. The figurate numbers were studied heavily to learn about counting numbers and arrangements. For example, if a woman was asked to determine which of two sacks of gold coins was worth more, she would probably have to count the coins. To count the coins, the best approach would be to stack the coins into short stacks of a given number. Then, the number of stacks could be counted. Counting numbers, looking at the patterns, and studying the ways objects could be arranged led to the numbers in Pascal ’ s triangle. The study of combining or arranging objects by various rules to create new arrangements of objects is called combinatorics, an important branch of mathematics. Pascal ’ s triangle in its current form is shown below. It is the same as the above combinatorial triangle rotated 45 degrees clockwise.
What is a binomial number?
Binomial numbers or coefficients — Numbers that stand for the number of subsets of equal size within a larger set.
How to solve a new row in Pascal's triangle?
Each new row in Pascal ’ s triangle is solved by taking the top two numbers and adding them together to get the number below.
What number is the first to organize all the information together?
1 1 1 1 1 1 2 3 4 1 3 6 1 4 1. The combination of numbers that form Pascal ’ s triangle were well known before Pascal, but he was the first one to organize all the information together in his treatise “ The Arithmetical Triangle.
What are some examples of unsolved problems in Pascal's days?
A number of unsolved problems in Pascal ’ s days encouraged the formation of probability theory. The Gambler ’ s Ruin and the Problem of Point are two examples of such problems.
What is the problem of points?
The Problem of Points was also a game about probabilities. The question was determining how a game ’ s winnings should be divided if the game was ended prematurely. Questions about games like these stirred the development of probability theory, and the need to understand binomial numbers completely.
What is Pascal's Triangle?
What is Pascal's triangle? Pascal's triangle is an array of the binomial coefficients, which are the numbers that appear as coefficients of terms when expanding a power of a binomial. The first few rows of Pascal's triangle are shown below.
Pascal's Triangle Patterns
The numbers in Pascal's triangle form many different patterns. The simplest of the Pascal's triangle patterns is a pattern that can be used to construct Pascal's triangle row by row. Firstly, the outermost numbers of every row are always equal to 1. Numbers within the row can be calculated by adding the two adjacent numbers from the row above.
Pascal's Triangle Example
Pascal's triangle can be used to more easily simplify binomial powers. As a Pascal's triangle example, consider:
Pascal's Triangle Fibonacci Sequence
Another well-known numerical pattern that is hidden within Pascal's triangle, is the Pascal's triangle Fibonacci sequence:
What is Pascal's triangle?
Pascal's triangle helps to determine the coefficients of binomial expansion of ( x + y) n , where n is the number of row in the triangle above starting from 0.
What is the name of Pascal's treatise on the triangle?
Pascal's Traité du triangle arithmétique ( Treatise on Arithmetical Triangle) was published posthumously in 1665. In this, Pascal collected several results then known about the triangle, and employed them to solve problems in probability theory. The triangle was later named after Pascal by Pierre Raymond de Montmort (1708) who called it "Table de M. Pascal pour les combinaisons" (French: Table of Mr. Pascal for combinations) and Abraham de Moivre (1730) who called it "Triangulum Arithmeticum PASCALIANUM" (Latin: Pascal's Arithmetic Triangle), which became the modern Western name.
How long did Nepal rule Kumaon?
So nepal ruled kumaon for about 23 years and garhwal for 11 years.now taking only garhwali example those 11 years of Gurkha rule were the most horrendous years of garhwal history.villages were looted and burned.women raped.children killed.a large no of garhwali people were taken to Nepal as slaves.they were auctioned in haridwar.there are tales of how these people were sold cheaper than the cattle.
How were people punished in the Gurkha rule?
People were punished in most brutal ways.chopping of hands was common. If gurkhas thought someone was lying they would put a person hands in boiling oil.they said if a person is telling the truth his hands won't burn.whenever the soldiers needed food or any essential items they were free to loot any village .many villages were abandoned by the people due to this.this space is not enough to write all the stuff that happened.but Gurkha rule is remembered with so much anger by garhwali people that this question of Uttarakhand people joining Nepal is insulting.
When did Pingala learn binomial coefficients?
From later commentary, it appears that the binomial coefficients and the additive formula for generating them, ( n r) = ( n − 1 r) + ( n − 1 r − 1), were known to Pingala in or before the 2nd century.
How to find the coefficient of the first variable in the expansion of a Pascal triangle?
So we get, ( x + y) 2 = 1 ⋅ x 2 + 2 x y + 1 ⋅ y 2
How to do triangular arrangement?
To get this following triangular arrangement, start with 1 in the first row, and imagine two 0 s at its side. Now add one 0 (that you imagined) and 1 and write 1 below and again add another 0 and 1 and write 1 below. So now in the next row, below 1 in the first row, you get 1 1. Now again imagine two 0 s at the side of two 1s. So add the first 0 and 1 and write 1 below. Then add 1 and 1, write 2 below and finally add the 0 on the right and 1 to write 1 below. Keep doing this process for each row.
