Pascal
Triangle

The accuracy of the
following binomial expressions can be examined by the direct
multiplication:
(a+b)^{0} 
= 
1 








(a+b)^{1} 
= 
1*a 
+ 
1*b 






(a+b)^{2} 
= 
1*a^{2} 
+ 
2*ab 
+ 
1*b^{2} 




(a+b)^{3} 
= 
1*a^{3} 
+ 
3*a^{2}b 
+ 
3*ab^{2} 
+ 
1*b^{3} 


(a+b)^{4} 
= 
1*a^{4} 
+ 
4*a^{3}b 
+ 
6*a^{2}b^{2} 
+ 
4*ab^{3} 
+ 
1*b^{4} 

If the computational board
fields are used for the numbers from the above binomial expressions,
the following number triangle is formed :
1 











1 
1 
1 










2 
1 
2 
1 









4 
1 
3 
3 
1 








8 
1 
4 
6 
4 
1 







16 
1 
5 
10 
10 
5 
1 






32 
1 
6 
15 
20 
15 
6 
1 





64 
1 
7 
21 
35 
35 
21 
7 
1 




128 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
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. Late in the XVII
century this arithmetical triangle became the key point of the
development of the following three mathematical branches :
investigation of the infinite series, computation of the
finite differences and the theory of
probabilities.
Pascal employed his
arithmetical triangle in 1653, but no account of his method
was printed till 1665. The triangle is constructed as in the figure
below, each horizontal line being formed form the one above it by
making every number in it equal to the sum of those above and to the
left of it in the row immediately above it; ex. gr. the
fourth number in the fourth line, namely, 20, is equal to 1 + 3 + 6
+ 10.
The numbers in each line are what are now
called figurate numbers. Those in the first line are called
numbers of the first order; those in the second line, natural
numbers or numbers of the second order; those in the third line,
numbers of the third order, and so on. It is easily shewn that the
mth number in the nth row is (m+n2)! / (m1)!(n1)!

Pascal's arithmetical
triangle, to any required order, is got by drawing a diagonal
downwards from right to left as in the figure. The numbers in
any diagonal give the coefficients of the expansion of a
binomial; for example, the figures in the fifth diagonal,
namely 1, 4, 6, 4, 1, are the coefficients of the expansion
Pascal used the triangle partly for this
purpose, and partly to find the numbers of combinations of
m things taken n at a time, which he stated,
correctly, to be (n+1)(n+2)(n+3) ... m / (mn)!

The first column of the arithmetical triangle is the figure "
one" column. Figuratively the figure "one" column
represents points. Here is the alternative form of the number
triangle where all its columns are the number "one" columns:
1 









1 
1 
1 








2 
1 
1 
1 







3 
1 
1 
1 
1 






4 
1 
1 
1 
1 
1 





5 
1 
1 
1 
1 
1 
1 




6 
1 
1 
1 
1 
1 
1 
1 



7 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
The addition by rows results in the column#2 i.e. the column of
the natural numbers : 1,2,3,4,5,6,... Figuratively the series
of natural numbers is a straight line. Thus the next triangle is
obtained:
1 









1 
2 
1 








3 
3 
2 
1 







6 
4 
3 
2 
1 






10 
5 
4 
3 
2 
1 





15 
6 
5 
4 
3 
2 
1 




21 
7 
6 
5 
4 
3 
2 
1 



28 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
Furher addition by rows results in the column#3, the column of
the so  called triangular numbers of : 1,3,6,10,15,21,28 ...
The triangular numbers symbolize a triangle and an area. The number
triangle, whose columns consist of the triangular numbers is shown
as follows:
1 









1 
3 
1 








4 
6 
3 
1 







10 
10 
6 
3 
1 






20 
15 
10 
6 
3 
1 





35 
21 
15 
10 
6 
3 
1 




56 
28 
21 
15 
10 
6 
3 
1 



84 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
.... 
This time adding by the rows of the upper triangle results in
the tetrahedral numbers . These numbers are the column#4 of
the Pascal Triangle. The set of tetrahedral numbers symbolize
pyramids and dimensionaly the volume : 1,4,10,20,35,56,84 ... The
same procedure can be applied to the other columns of the Pascal
Triangle, too, whereby a n th dimensional area is formed. The
integration is the connection between the neighbouring columns.
An interesting number`s triangle can be generated by adding up
natural numbers :






1 
1 
....... 
2* 01 





2 
2 
2 
....... 
2* 03 




3 
3 
3 
3 
....... 
2* 06 



4 
4 
4 
4 
4 
....... 
2* 10 


5 
5 
5 
5 
5 
5 
....... 
2* 15 

6 
6 
6 
6 
6 
6 
6 
....... 
2* 21 
7 
7 
7 
7 
7 
7 
7 
7 
....... 
2* 28 
Also an interesting number`s triangle can
be generated by adding up triangular numbers :






01 
01 
01 
....... 
3* 01 





03 
03 
03 
03 
....... 
3* 04 




06 
06 
06 
06 
06 
....... 
3* 10 



10 
10 
10 
10 
10 
10 
....... 
3* 20 


15 
15 
15 
15 
15 
15 
15 
....... 
3* 35 

21 
21 
21 
21 
21 
21 
21 
21 
....... 
3* 56 
28 
28 
28 
28 
28 
28 
28 
28 
28 
....... 
3* 84 
The number triangle, whose columns consist of the tetrahedral
numbers is shown as follows. We get pyramidal numbers:





01 
01 
01 
01 
....... 
4* 01 




04 
04 
04 
04 
04 
....... 
4* 05 



10 
10 
10 
10 
10 
10 
....... 
4* 15 


20 
20 
20 
20 
20 
20 
20 
....... 
4* 35 

35 
35 
35 
35 
35 
35 
35 
35 
....... 
4* 70 
56 
56 
56 
56 
56 
56 
56 
56 
56 
....... 
4*126 
