Exploring Pascal TriangleMade By :-

SANKALP SINGH

In Mathematics Pascal's Triangle is a Triangular Array of Binomial Coefficient.

It is named after the French Mathematician Blaise Pascal.

He was not the first mathematician to study about this type of triangle it was already studied by other Mathematician of countries :- India, Iran, China, Germany and Italy.

What is Pascal’s Triangle ?

Images of Pascal Triangle

Blaise Pascal

Blaise Pascal ( 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, writer and Christian philosopher. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pascal's earliest work was in the natural and applied sciences where he made important contributions to the study of fluids, and clarified the concepts of pressure and vacuum by generalizing the work of Evangelista Torricelli. Pascal also wrote in defence of the scientific method.

About Blaise Pascal

The so called 'Pascal' triangle was known in China as early as 1261. In '1261 the triangle appears to a depth of six in Yang Hui and to a depth of eight in Zhu Shijiei in 1303. Yang Hui attributes the triangle to Jia Xian, who lived in the eleventh century'. They used it as we do, as a means of generating the binomial coefficients. It wasn't until the eleventh century that a method for solving quadratic and cubic equations was recorded, although they seemed to have existed since the first millennium. At this time Jia Xian 'generalised the square and cube root procedures to higher roots by using the array of numbers known today as the Pascal triangle and also extended and improved the method into one useable for solving polynomial equations of any degree'.

History Of Pascal’s Triangle

There are some proofs that this number triangle was familiar to the Arab astronomer, poet and mathematician Omar Khayyam as early as the XI century. Most probably the number triangle came to Europe from China through Arabia. The Chinese representation of the binomial coefficients, often equally called Pascal`s Triangle being found in his work published for the first time after his death ( in 1665 ) and dealing with figurate numbers, is found for the first time on the title page of the European Arithmetic written by Appianus, in 1527.

History Of Pascal’s Triangle

Blaise Pascal was not the first man in Europe to study the binomial coefficients, and never claimed to be such; indeed, both Blaise Pascal and his father Etienne had been in correspondence with Father Marin Mersenne, who published a book with a table of binomial coefficients in 1636. Many authors discussed the ideas with respect to expansions of binomials, answers to combinatorial problems and figurate numbers, numbers relating to figures such as triangles, squares, tetrahedra and pyramids.

History Of Pascal’s Triangle

History Of Pascal’s Triangle

Edition of Jourdanus ’de Arithmetica Triangle

Pascal’s Version Of Triangle

If all the Even numbers are coloured white and all odd numbers are coloured black, a pattern similar to the sierpinski gasket would appear.

In 6 Rows of Pascal's Triangle the second row from the left forms a sequence of TRIAMGULAR NUMBERS.

In 6 Rows of Pascal's Triangle the third row from the left forms a Tetrahedral sequence.

Some Properties of Pascal’s Triangle

If a line is drawn vertically through the middle of the Pascal's Triangle it is a mirror image excluding the centre line.

When diagonals across the Triangle are drawn out the following sums are obtained. They follow the formula of X=(3n-1) with n being the number before X

When the Triangle is left justified the sum of the same coloured diagonals lined out forms the Fibonacci numbers.

Some Properties of Pascal’s Triangle

Step to Construct Pascal’s Triangle

At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1) contains two 1's, both formed by adding the two numbers above them to the left and the right, in this case 1 and 0 (all numbers outside the Triangle are 0's). Do the same to create the 2nd row: 0+1=1; 1+1=2; 1+0=1. And the third: 0+1=1; 1+2=3; 2+1=3; 1+0=1. In this way, the rows of the triangle go on infinitely. A number in the triangle can also be found by nCr (n Choose r) where n is the number of the row and r is the element in that row. For example, in row 3, 1 is the zeroth element, 3 is element number 1, the next three is the 2nd element, and the last 1 is the 3rd element. The formula for nCr is:

n!--------r!(n-r)!

The sum of the numbers in any row is equal to 2 to the nth power or 2n, when n is the number of the row. For example:

20 = 1 21 = 1+1 = 2 22 = 1+2+1 = 4 23 = 1+3+3+1 = 8 24 = 1+4+6+4+1 = 16

Uses of Pascal’s Triangle

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