Probably the most well-known contribution to the field of mathematics by Pascal deals with the Arithmetic Triangle, or, as it is now known, Pascal's Triangle.

Pascal was not the discoverer of the Arithmetic Triangle or of the numbers that comprise its rows. He was, however, the first to formally study and explain, with proof, certain "consequences" or "relationships" that are inherent to its structure.

The modern representation of Pascal's Triangle is constructed a row at a time. The first row consists of a single "1" At the right and left end of each row, there is also a "1". Each other entry in the triangle is found by adding together the two entries directly above it. Like so:

1
1     1
1     2     1
1     3     3     1
1     4     6     4     1
1     5     10     10     5     1 ... and so on, without end.

These numbers comprise what are known as the binomial coefficients. These numbers can be expressed as _nC_k, where n is the row of Pascal's Triangle, and k is the number of items from the left you are (where the top most row is the 0th row, and the left most item is the 0th item.)

For example, ₄C₂, the 2nd entry on the 4th row, is 6. These numbers also represent the number of ways to choose k items from a set of n items.

Pascal discovered and proved 19 separate "consequences" of the construction of the triangle.

A few of his "consequences" are as follows:

• _nC_r = _n-1C_r-1 + _n-2C_r-1 + _n-3C_r-1 + ... + _r-1C_r-1  (Consequence II)
  In layman's terms, the sum of the numbers running northeast from the entry above and to the left of any entry a is equal to entry a.
  
  
• _nC_r = _nC_r-n  (Consequence V)
  In layman's terms, this states the the triangle is symmetric from left to right.
  
  
• 2ⁿ = _nC₀ + _nC₁ + _nC₂ + ... + _nC_n  (Consequence VIII)
  In layman's terms, the sum of any row is equal to 2 raised to that row's number.
  
  
• nCr+1/nCr  =   (n-r)/(r+1)  (Consequence XII)

This last consequence is of particular import, because it was the first explicit formulaic mathematical inducation, a particularly powerful method of proof.

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Relations of Pascal's Triangle to other Famous Constructions

Hidden in Pascal's Triangle are a couple of other famous constructions.

First, if we change all of the numbers to Modulo 2 (in layman's terms, look at the odds vs. the evens), The pattern formed by the numbers is that of the fractal known as Sierpinski's Triangle or Sierpinski's Sieve. Every line that consists of all odd entries is the bottom of one further iteration of the fractal.

Second, seen here, the sum of the shallow diagonals of Pascal's triangles are the Fibonacci numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...).

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