The accuracy of the following binomial expressions can be examined by the direct multiplication: (a+b)0 = 1 (a+b)1 = 1*a + 1*b (a+b)2 = 1*a2 + 2*ab + 1*b2 (a+b)3 = 1*a3 + 3*a2b + 3*ab2 + 1*b3 (a+b)4 = 1*a4 + 4*a3b + 6*a2b2 + 4*ab3 + 1*b4

If the computational board fields are used for the numbers from the above binomial expressions, the following number triangle is formed :

There are some proofs that this number triangle was familiar to the Arab astronomer, poet and mathematician Omar Khayyam as early as the XI century. Most probably the number triangle came to Europe from China through Arabia. The Chinese representation of the binomial coefficients, often equally called Pascal`s Triangle being found in his work published for the first time after his death ( in 1665 ) and dealing with figurate numbers, is found for the first time on the title page of the European Arithmetic written by Appianus, in 1527. Late in the XVII century this arithmetical triangle became the key point of the development of the following three mathematical branches : investigation of the infinite series, computation of the finite differences and the theory of probabilities.

The first column of the arithmetical triangle is the figure " one" column. Figuratively the figure "one" column represents points. Here is the alternative form of the number triangle where all its columns are the number "one" columns:

The addition by rows results in the column#2 i.e. the column of the natural numbers : 1,2,3,4,5,6,... Figuratively the series of natural numbers is a straight line. Thus the next triangle is obtained:

Furher addition by rows results in the column#3, the column of the so - called triangular numbers of : 1,3,6,10,15,21,28 ... The triangular numbers symbolize a triangle and an area. The number triangle, whose columns consist of the triangular numbers is shown as follows:

This time adding by the rows of the upper triangle results in the tetrahedral numbers . These numbers are the column#4 of the Pascal Triangle. The set of tetrahedral numbers symbolize pyramids and dimensionaly the volume : 1,4,10,20,35,56,84 ... The same procedure can be applied to the other columns of the Pascal Triangle, too, whereby a n th dimensional area is formed. The integration is the connection between the neighbouring columns.

In the book, Liber abaci ( meaning Book of the Abacus or Book of Calculating ), the practical arithmetic problem is presented : A pair of rabbits is put in a limited area. This pair of rabbits produces another pair each month. If the rabbits do not die, the question is : How many pairs of rabbits there would be ? The answer from the book is this sequence of numbers : 1 1 2 3 5 8 13 21 34 55 ..... The series of numbers was named " Fibonacci numbers" by Edouard Lucas ( 1842-1899 ).

Lucas invented numerous significant applications of these.The Fibonacci sequence is a recursive sequence where the first two values are 1 and each successive term is obtained by adding together the two previous terms. The definition of the Fibonacci series is :

By adding diagonal numbers of the Pascal Triangle Fibonacci sequence can be obtained :

It can be pressumed where on his journeys Fibonacci met this sequence if "rows" in double Pascal`s Triangle are summed :

For the sake of playfulness among numbers and areas let us point out the connection between the arithmetical triangle and Fibonacci numbers. The Pascal Triangle is shown as follows and the numbers in rows are summed:

R. Knott found the Fibonacci numbers appearing as sums of " rows" in Pascal`s Triangle. By drawing Pascal`s Triangle with all the rows moved over by 1 place, we have a clearer arrangement which shows the Fibonacci numbers as sums of columns :

Here is the alternative form of Pascal`s Triangle, with the double rows re-aligned as columns and the sums of the new columns are the Fibonacci numbers :

There is a reciprocal connection between Fibonacci numbers and arithmetical triangle. There are also numerous recursive relations for the Fibonacci numbers :

It is obvious that the structure of Pascal`s Triangle is built in these recursive relations, which certainly indicates the existing connection between the numbers of Pascal`s Triangle and Fibonacci numbers .

The French mathematician Edouard Lucas found a similar series :

The Fibonacci rule of adding the latest two to get the next is kept, but here we start from 2 and 1 (in this order) instead of 0 and 1 for the (ordinary) Fibonacci numbers :

Here is one combination of Fibonacci numbers the sum of which gives the Lucas number :

Let us form the series of four consequent Fibonacci numbers :

i.e.

The series of Lucas numbers, L(n+3) , is obtained :

The connection between Fibonacci and Lucas numbers will be plastically shown and thus the series of four consequent Lucas numbers is formed :

i.e.

The series of Fibonacci numbers, 5 * F(n+3) ,is obtained :

The Fibonacci numbers play a significant role in Nature, in art and architecture. Many plants show Fibonacci numbers in the arrangements of the leaves around their stems. The leaves are often arranged so that leaves above do not hide leaves below. This means that each one gets a good share of the sunlight and catches the most rain to channel down to the roots as it runs down the leaf to the stem.

A Fibonacci number is on many plants : buttercups have 13 petals; some delphinum have 8; corn marigolds have 13 petals; some asters have 21 ... Fibonacci numbers can also be seen in the arrangement of seeds on flowerheads.

Back to contents  Back to main page