Binomial Theorem and the Pascal 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 (as in diagram 6) in 1303. Yang Hui attributes the triangle to Jia Xian, who lived in the eleventh century' (Stillwell, 1989, p136). 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' (Katz, 1993, p191).

Consider the example (x cubed) = 12,812,904. A reasonable guess would be a three digit number starting with a 2. Why? We know (200 cubed) is equal to 8,000,000 and (300 cubed) is equal to 27,000,000, therefore (200 cubed) < (x cubed) < (300 cubed). In other words, the closest integer solution can be written as x = 200 + 10b + c. Ignoring c temporarily we need to find largest b such that

(200 + 10b cubed) = 1x(200 cubed) + 3x (200 sq)x10b + 3x200x(10b sq) + 1x(10b cubed) <= 12,812,904.

Taking away 1x(200 cubed) from each side we end up with :

3x(200 sq)x10b + 3x200x (10b sq) + 1x(10b cubed) <= 4,812,904

By trying in turn b = 1,2,3... one discovers that b = 3 is in fact the highest possible number to satisfy this inequality. Actually for b = 3 you obtain 4,167,000.

Next subtract this value from 4,812,904 to derive an inequality for c :

3x(230 sq)xc + 3x230x(c sq) + 1x(c cubed) <= 645,904.

In this case c = 4 satisfies the equality.

Therefore the result is x = 234.

Jia Xian realised that this solution could be generalised to nth order roots for n > 3 by using the binomial expansion. Katz reports (1993, p191) 'Yang Hui stated that he [Jia Xina] not only wrote out the Pascal triangle of binomial coefficients through to the sixth row but also developed the usual method of generating the triangle - add the numbers in the two places above in order to find the number in the place below.'

If Pascal was not around until 1623 (Struik, 1967, p192 ) why was he credited it? 'It is of course not the only instance of mathematical concept being named after a rediscoverer rather than a discoverer, but in any case Pascal deserves credit for more than just rediscovery' (Stillwell, 1989, p136). He showed that it could be used in 2 ways;

  1. as the binomial coefficients
  2. to divide the stakes of a game of chance between two players

It was therefore credited to Pascal due to the ingenious use he made of it in probabilities.

Clearly recognition should not be dismissed or ignored to the Chinese as they did discover it some 300 years before Pascal.


Diagram 6. Khu Shijiei triangle, depth 8, 1303.

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