The document explores Pascal's triangle, named after French mathematician Blaise Pascal. While Pascal did not discover it, he popularized it in Europe. The triangle arranges the binomial coefficients and was known in other ancient cultures as well. It exhibits several mathematical properties like generating triangular and tetrahedral numbers.

1. Exploring Pascal Triangle
Made By :-
SANKALP SINGH

2. What is Pascal’s Triangle ?
 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.

3. Images of Pascal Triangle

4. Blaise Pascal

5. About 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.

6. History Of Pascal’s Triangle
 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'.

7. 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.

8. 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.

9. History Of Pascal’s Triangle
Edition of Jourdanus ’de Arithmetica Triangle

10. Pascal’s Version Of Triangle

11. Some Properties of Pascal’s 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.

12. 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.

13. 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)!

14.  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

15. Uses of Pascal’s Triangle