Pascal Triangle & Its Applications.

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Discussion Overview

The discussion revolves around the origin and historical development of Pascal's Triangle, as well as its applications beyond binomial expansion. Participants express curiosity about the mathematical and historical context of the triangle and its patterns.

Discussion Character

  • Historical
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses a desire to understand the origin of Pascal's Triangle and its emergence as a mathematical concept.
  • Another suggests that there is ample information available online, including a specific book that discusses the history of the triangle.
  • A participant notes that the entries in Pascal's Triangle can be derived from the binomial coefficients and highlights the recursive relationship between the entries.
  • There is a question raised about the motivations behind the creation of such an array by mathematicians like Pascal and Omar Khayyam, emphasizing the interesting patterns it contains.
  • One participant asserts that the triangle's origins are linked to binomial expansion.
  • Another participant mentions that Khayyam had studied the triangle before Pascal, suggesting that it could be referred to as the "Khayyam-Pascal triangle." This statement is met with acknowledgment and a note about historical revisionism by some authors.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the historical origins of Pascal's Triangle, with some attributing its discovery to Khayyam and others to Pascal. The discussion includes multiple competing views regarding the contributions of different mathematicians.

Contextual Notes

There are unresolved questions about the historical timeline and the extent of contributions made by various mathematicians to the development of Pascal's Triangle.

mubashirmansoor
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Hello,

I have searched through out the internet but I couldn't find any information about the origin of Pascal Triangle & how it came into being... Because I am sure that such a precise & accurate numerical array didn't come out of no where !

I will be very thankful for your guidance.

My second question is about the applications of this array, except Binomial expansion.

Many thanks for giving me the time.
 
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I think there is a lot of information about this topic on the internet.
As so often, a useful starting point is Wikipedia (see "Pascal's triangle").
You can also search Amazon.com; for instance, a review on 'Pascal's Arithmetical Triangle: The Story of a Mathematical Idea' (Edwards, A.W.F, 2002) describes the book as "an impressive culmination of meticulous research into original sources, this definitive study constitutes the first full-length history of the Arithmetic Triangle."
 
Many thanks for introducing the book, huba.

I will try to find it from somewhere :)

Any further comments will be highly appreciated.
 
once you know the binomial coefficients satisfy (n,k) = "n choose k"

= (n-1,k) + (n-1,k-1), what else is there to say about pascal's triangle?

i.e. each entry is obtained from adding the two previous ones.
 
What I would like to know is what made Pascal, Omar Khayam or the Chinese mathematicians to build such an array which is obtained from adding the previous ones...

Because it simply contains dozens of very interesting patterns!
 
The obvious thing is that they got it from the binomial expansion.
 
did you know a iranian mathmatican (khayam)had found this before pascal
 
hadi amiri 4 said:
did you know a iranian mathmatican (khayam)had found this before pascal

Yep, I knew that. He didn't find it but he studied it many centuries prior to Pascal. Actually it's supposed to be called "Khayyam-Pascal triangle"!

Mind you, some western authors have tried to revise the history...
 

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