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Posted on October 11th, 2012, by

I have a choice among different themes to choose one and I think that the best I can tell you about Pascal’s triangle. After my previous words it becomes understandable that this work will be denoted to Pascal’s Triangle and several main questions connected with it  will be revealed. I want to explain in this work when it was appeared, why it is so popular and how it can be used

First of all we should stop on the notion Pascal’s triangle and understand what it means. In mathematics, Pascal’s triangle is a geometric arrangement of the binomial coefficients in a triangle. If to be short then Pascal’s triangle include a system of numbers arranged in rows resembling a triangle with each row consisting of the coefficients in the expansion of (a + b)n for n = 0, 1, 2, 3, But such explanation is not show us the Pascal’s triangle characteristics.

It is named after Blaise Pascal in much of the western world, although other mathematicians studied it centuries before him in India, Persia, China, and Italy. The rows of Pascal’s triangle are conventionally enumerated starting with row zero, and the numbers in odd rows are usually staggered relative to the numbers in even rows. A simple construction of the triangle proceeds in the following manner. On the zeroth row, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left with the number directly above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place. For example, the first number in the first row is 0 + 1 = 1, whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row. (www.dic.academic)

I can say that it is magic triangle. Frankly speaking, if numbers of Pascal’s triangle possess the special characteristics, that it possible consider magic.

Looking for the words of other scientist on this topic I found such words: Pascal’s triangle is so simple, that even 10-years child can draw it. At the same time it includes unknown treasures and connected together various mathematics aspects, which have no similar characteristics at first look. So, the unusual characteristic allow to consider the triangle Pascal one of the the most graceful schemes in the whole mathematician. Martin Gardner Math novels

Pascal’s triangle is an endless numeric table triangular form, in which on lateral sides stand the units and any number, except these lateral units.

1

1     1

1     2      1

1    3      3    1

1     4    6     4   1

1   5    10    10    5   1

.  .  .  .  .  .  .  .  .  .  .  .  .  .  .

Reveal most magic numbers characteristics triangle realize, what else characteristic possesses the Pascal’s triangle. Each number is an amount two located on it numbers. Triangle possible to continue unlimited

 

 

 

 

 

Triangle possesses the symmetry for vertical axis, getting through its top.

Along direct, the parallel sides of the triangle (on drawing noted green line) situated triangular numbers and their generalizations on event space all dimensionality.

 

 

 


How can we use this triangle? Of course there are a lot of variant and we will take one of them for example.  Let, for instance, we want to calculate the amount of natural numbers row from 1 before 9.

Having Come down diagonally before number 9, we see on the left from below from it number 45. It that and gives the sought amount.

 

 

 

One more example: binominal factors there is a factors decompositions of the polynomial on degree x and y

 

 

 

Summarizing the work I want to say that in it was revealed numbers characteristic, being included in Pascal’s triangle and define using a number characteristics of Pascal’s triangle. When we learn more information my presentations on this topic changed and I begin to think about world around me using mathematics theories. It help me to knew many new aspects and it is strange that when you begin something learn you are sometimes not  very much interested in it, but then you understand that topic so interesting that you want to learn it more.

There are a lot of math riddles what can be solved with Pascal’s triangle help.

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