Chaos and Fractals, Part 1
The Pascal Triangle
Blaise Pascal (1623-1662) was a great French mathemetician and scientist. When he was twenty years old, he built some 10 mathematical machines for the addition of integers, a precursor to modern computers. What is known as the arithmetic triangle (or Pascal's Triangle) was not originally his discovery. The first printed form of the arithmetic triange in Europe dates back to 1527. A Chinese version had already been published in 1303 (see image.)
But Pascal used the arithmetic triangle to solve some problems related to chances in gambling, which he had discussed with Pierre de Fermat in 1654. This research later became the foundations of probability theory.
The arithmetic triangle is a triangular array of numbers composed of
the coefficients of the expansion of the polynomial (1 + x)n. Here n denotes the row starting from n = 0. Row n has n + 1 entries. For example, for n = 3 the polynomial is (1 + x)3 = 1 + 3x + 3x2 + x3. Thus, row number 3 reads: 1,3,3,1.
There are two ways to compute the coefficients. The first one inductively computes one row based on the entries of the previous row. Assume that the coefficients a0,...,an in row n are given:(1 + x)n = a0 + a1x + ...+ anxn, and the coefficients b0,...,bn + 1 of the following row are required: (1 + x)n + 1 = b0 + b1x + ...+n + 1xn + 1
Comparing coefficients we obtain the result b0 = a0, bk = ak - 1ak for k = 1,...,n, bn + 1 = an.
Wikipedia Binomial coefficient In Mathematics, the binomial coefficient 
is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n. is often called the choose function of n. k is the number of k-element subsets (the k-combinations) of an n-element set; that is, the number of ways that k things can be 'chosen' from a set of n things.
and 
where n denotes the factorial (factorial: the product of all positive integers less than or equal to n of n.
Blaise Pascal (1623-1662) was a great French mathemetician and scientist. When he was twenty years old, he built some 10 mathematical machines for the addition of integers, a precursor to modern computers. What is known as the arithmetic triangle (or Pascal's Triangle) was not originally his discovery. The first printed form of the arithmetic triange in Europe dates back to 1527. A Chinese version had already been published in 1303 (see image.)
But Pascal used the arithmetic triangle to solve some problems related to chances in gambling, which he had discussed with Pierre de Fermat in 1654. This research later became the foundations of probability theory.
The arithmetic triangle is a triangular array of numbers composed of
the coefficients of the expansion of the polynomial (1 + x)n. Here n denotes the row starting from n = 0. Row n has n + 1 entries. For example, for n = 3 the polynomial is (1 + x)3 = 1 + 3x + 3x2 + x3. Thus, row number 3 reads: 1,3,3,1.
There are two ways to compute the coefficients. The first one inductively computes one row based on the entries of the previous row. Assume that the coefficients a0,...,an in row n are given:(1 + x)n = a0 + a1x + ...+ anxn, and the coefficients b0,...,bn + 1 of the following row are required: (1 + x)n + 1 = b0 + b1x + ...+n + 1xn + 1
Comparing coefficients we obtain the result b0 = a0, bk = ak - 1ak for k = 1,...,n, bn + 1 = an.
Wikipedia Binomial coefficient In Mathematics, the binomial coefficient is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n. is often called the choose function of n. k is the number of k-element subsets (the k-combinations) of an n-element set; that is, the number of ways that k things can be 'chosen' from a set of n things.
Given a non-negative integer n and an integer k, the binomial coefficient is defined to be the natural number
and where n denotes the factorial (factorial: the product of all positive integers less than or equal to n of n.
Alternatively, a recursive defition can be written as
where 
.
(TO BE CONTINUED)
posted by Mission Heavens at
6:53 AM
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