Pascal Triangle & Its Applications.

In summary, Pascal's triangle is a numerical array that is obtained from adding the two previous ones.
  • #1
mubashirmansoor
260
0
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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  • #2
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."
 
  • #3
Many thanks for introducing the book, huba.

I will try to find it from somewhere :)

Any further comments will be highly appreciated.
 
  • #4
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.
 
  • #5
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!
 
  • #6
The obvious thing is that they got it from the binomial expansion.
 
  • #7
did you know a iranian mathmatican (khayam)had found this before pascal
 
  • #8
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...
 

1. What is the Pascal Triangle?

The Pascal Triangle is a triangular array of numbers named after the French mathematician Blaise Pascal. It starts with a 1 at the top, and each subsequent row is formed by adding the two numbers above it and putting the sum in between. It is a visual representation of the binomial coefficients, which are used in combinatorics and probability.

2. How is the Pascal Triangle used in mathematics?

The Pascal Triangle has many applications in mathematics. One of the most well-known is its use in calculating binomial expansions, which are used in the study of algebra and calculus. It is also used in probability theory to calculate the chances of an event occurring based on different outcomes. Additionally, the triangle can be used to find patterns in numbers and to solve problems in combinatorics.

3. Can the Pascal Triangle be extended beyond its traditional 7 rows?

Yes, the Pascal Triangle can be extended beyond its traditional 7 rows. In fact, the triangle can be infinitely expanded, with each row representing the coefficients of the (n+1)th power of a binomial. However, as the number of rows increases, the size of the triangle becomes increasingly unwieldy and difficult to work with.

4. What are some real-life applications of the Pascal Triangle?

The Pascal Triangle has various real-life applications, including in computer programming, where it is used to generate fractals and to optimize algorithms. It is also used in genetics to understand the genetic inheritance of traits. In addition, the triangle has been used in music composition to create musical patterns and in art to create symmetrical designs.

5. Are there any other similar triangular arrays in mathematics?

Yes, there are other similar triangular arrays in mathematics, such as the Tartaglia Triangle and the Yanghui Triangle. These triangles share similar properties with the Pascal Triangle and are used in different areas of mathematics. However, the Pascal Triangle remains the most well-known and widely used triangular array in mathematics.

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