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Exploring Pascal || Student Center || Teachers' Place
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Pascal's triangle is an arithmetical triangle made up of staggered rows of numbers:
 

 

 

 

 

 

 

 

 1

 

 

 

 

 

 

 

 

    Row 0
 

 

 

 

 

 

 

 1

 

 1

 

 

 

 

 

 

 

    Row 1
 

 

 

 

 

 

 1

 

 2

 

 1

 

 

 

 

 

 

    Row 2
 

 

 

 

 

  1

 

 3

 

 3

 

 1

 

 

 

 

 

    Row 3
 

 

 

 

 1

 

 4

 

 6

 

 4

 

 1

 

 

 

 

    Row 4
 

 

 

 1

 

 5

 

10

 

10

 

 5

 

 1

 

 

 

    Row 5
 

 

 1

 

 6

 

15

 

20

 

15

 

 6

 

 1

 

 

    Row 6
 

 1

 

 7

 

21

 

35

 

35

 

21

 

 7

 

 1

 

    Row 7
 1

 

 8

 

28

 

56

 

70

 

56

 

28

 

 8

 

 1

    Row 8
               

.

                   
               

.

                   
               

.

                   
 
This triangle was studied by Blaise Pascal and Omar Khayyam, and was described about 500 years earlier by the Chinese mathematician, Yang Hui. "Chinese references indicate that the triangle appeared in a work of Chia Hsien about 1050. Indeed, a similar arrangement of binomial coefficients was known to the Arabs about the same time that it was being used in China" (D. M. Burton, The History of Mathematics.)
 

Constructing Pascal's Triangle

Start with the two top rows, which are 1 and 1 1. To find any number in the next row, add the two numbers above it, as shown in the following diagram.

At the beginning and end of each row, where there is only one number above, write a 1. You can think of this rule for placing the 1's as included in the first rule: to get the first 1 in any line, you add the number above and to the left - since there is no number there, pretend it's zero - and the number above and to the right (1), to get a sum of 1.

When people talk about an entry in Pascal's Triangle, they usually give a row number and a place in that row, beginning with row 0 and place 0. For instance, the number 20 appears in row 6, place 3.

Pascal first published his ideas on the generation of the triangle in 1665. It was constructed with each horizontal line formed from the one above it by making each number equal to the sum of the numbers above and to the left in the row above. For example, the third number in the fourth line (10) equals 1 + 3 + 6. See W. W. Rouse Ball, A Short Account of the History of Mathematics (4th edition, 1908).

A figurate number triangle consists of Pascal's triangle in a square grid with zeroes, as written by Jakob Bernoulli.


Finding Numbers in Pascal's Triangle

   Explore these number sets and mathematical patterns.


On The Web

   Site Suggestions

Web sites with lesson plans, Java applets, and more information about Pascal's triangle.
   References
  1. Ask Dr. Math FAQ: Pascal's Triangle
  2. Eric's Treasure Trove of Mathematics: Pascal's Triangle


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