Figurative Numbers
Pythagoreans discovered the figurative
numbers. The Greeks were deeply interested in numbers especially to those
connected with the geometric shapes, and given the name therefore figurative
numbers. Since Pythagoreans as the early custom of Greeks used to play with
the pebbles to form the different shapes, so they were more fascinated with the
relationship that emerged with the different shapes of pebble like Triangular,
Square, Cubic, Pyramid, Hexagonal ...etc
The
Greek word for pebble was pséphoi,
meant to calculate. The pebbles made it possible for Pythagoreans to identify
different shapes, the simplest being the two dimensional figure the triangle
and simplest three dimensional figures was the tetrahedron.
Aristotle in his Metaphysics writes “They
(the Pythagoreans) supposed the elements of numbers to be the elements of all
things and the whole heaven to be a musical scale and a number ...Evidently
then these thinkers also consider that number as the principal both as matter
for things and as forming both their modification end their permanent states.”
This part of the chapter deals with only the
figurative numbers and its different properties.
Triangular
Numbers:- This is a kind of Polygonal number. It is the
number of dots required to draw a triangle. The triangular numbers are formed
by the partial sum of the series 1+ 2 + 3+ … + n.
The Greeks also noted that these triangular
numbers are the sum of consecutive natural numbers, as they appear in the
number sequence. If the process continues till n th array then numbers of
pebbles in the nth array is 1+2+3+...+n=n* (n+1)/2
1 first
triangular number
1+2=3 second triangular
number
1+2+3=6 third triangular number
1+2+3+4=10 fourth triangular number
1+2+3+4+5=15 fifth triangular number
And so on...
Here
is a picture of first few triangular numbers.
Properties
of Triangular Numbers:-
v A triangular number can never end with 2, 4,
7, or 9.
v The sum of the two consecutive triangular
numbers is always a square number.
T1 +T2 = 1 + 3 =
4 = 22 T2 + T3 = 3 + 6 =
9 = 32 T3 + T4 = 6 + 10 = 16 =
42
v All perfect numbers are triangular numbers.
v A triangular number greater than 1 can not
be a cube, a fourth Power or a fifth Power.
v The only triangular number which is also a
prime is 3.
v The only triangular number which is also a
Fermat number is 3
v The only Fibonacci numbers which are also
triangular are 1, 3, 21, and 55.
v Some triangular numbers are the product of
three consecutive numbers. T3 = 6 = 1* 2
* 3 T15 = 120 = 4* 5* 6 T20 = 210 = 5 * 6 * 7 T44
= 990 =9 * 10 * 11 T608 = 185136 = 56 * 57 *
58 ----------------------
---------------------
v
|
Triangular number can
be seen in Pascal’s triangle. Look at the Pascal’s Triangle and you will find
that the third diagonal is all triangular numbers.
Square
Numbers:- The number 1, 4,
9,16,25,36... are called the square
numbers. It is the numbers of dots arranged in such a way that it represent a
square shape. These are the square of the natural numbers 1, 2, 3, 4, 5,
6….. respectively.
The
Greeks also have discovered that if consecutive odd numbers are added they
become square numbers. 1=1*1
1+3=4=22
1+3+5=9=32
1+3+5+7=16=42
1+3+5+7+9=25=52
Note that the square number can be found by
addition of all triangular number in the following manner—
1 3 6 10 15 21 28 36...
1 3
6 10 15 21 28 36
...
1 4
9 16 25 36 49 64....
Properties of Square Numbers:-
o Every square number can end with 00, 1, 4,
5, 6, or9.
o No square number ends in 2, 3, 7, or 8.
o
Look at
the following pattern 12 = 1 112 = 121 and 1 + 2 + 1 = 4 = 22 1112 = 12321 and 1 + 2 + 3 + 2 + 1 =9 = 32
11112 = 1234321 and 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 = 42
111112 = 123454321 and 1 + 2 + 3 + 4 + 5 + 4 + 3 +2 +1 = 25 = 52 --------------------------------------------------------------------------------------------------------------------------------------------------------------------
Cube Numbers:- The
numbers which can be represented by three dimensional cubes are called cubic
number. 1,8,27,64,125...are cubic numbers which are obviously the cubes of
1,2,3,4,5,....
Properties
of Cubic Numbers:-
- 13=1
first odd number
23=8=3+5
sum of next two odd numbers
33=27=7+9+11
sum of next three odd numbers
43=64=13+15+17+19 sum of next four odd numbers
53=125=21+23+25+27+29 sum of next five odd numbers
- Between 1
and 100 there are only two numbers 1 and64 that are also square numbers.
- If C1, C2, C3 ….are the first, second, third… cubic number then they exhibit a unique property:- C1 = ( T1)2 C1 + C2 = 1 + 8 =( T2) 2 C1 + C2 +C3 = 1 + 8 + 27 = 36 = ( T3) 2 C1 + C2 +C3+C4 = 1 + 8 + 27 + 64 = 100 = (T4)2
- Tetrahedral Numbers:- The numbers that can be represented by the layers of triangles forming a tetrahedron shape are called tetrahedral numbers. It is a figurative numbers of the form Tn = nC3 where n = 3, 4, 5,….4, 10, 20...are the example of tetrahedral numbers.
Properties of Tetrahedral Numbers:-
- The tetrahedral numbers are the sums of the consecutive triangular numbers beginning from 1. T1= 1 T2 = 1 + 3 = 4 T3 = 1 + 3 + 6 = 10 T4 = 1 + 3 + 6 + 10 = 20 T5 =1 + 3 + 6+ 10 + 15 = 35 T6 = 1 + 3 + 6 + 10 + 15 + 21 = 56 -------------------------------------------------
- The sum of two consecutive numbers is a Pyramidal number. T1 + T2 = 1 + 4 = 5 T2 + T3 = 4 + 10 = 14 T3 + T4 = 10 + 20 = 35 T4 + T5 = 20 + 35 = 55 T5 + T6 = 35 + 56 = 91
-
The tetrahedral numbers can be seen in the fourth diagonal of a Pascal’s triangle
11 11 2 11 3 3 11 4 6 4 11 5 10 10 5 11 6 15 20 15 6 1
Pentagonal
Numbers:- Those numbers which
represent the shape of pentagon are called pentagonal number. In the pentagonal
numbers the lower base is a square with a triangle on the top. 1,
5,12,22,35...are its example. The nth pentagonal number Pn is given by the
formula:-
Pn = n ( 3n – 1 )
If we represent the pentagonal numbers by
P1,P2 ,.... then
the n th number Pn =n(n-1)/2+n2
Properties:-
- Every nth
pentagonal number is one third of the 3n – 1 th triangular number.
Hexagonal
Number:- Those numbers which form
a shape of hexagon are called hexagonal numbers. 1, 6, 15, 28, 45…. are the few examples of
hexagonal numbers.
Hexagonal numbers are of the form n (2n-1).
Properties:-
·
Every
hexagonal numbers is a triangular number.
· - 1,7,19,37,61,91... are the centered hexagonal numbers.
·
11 and 26 are the only numbers that can be
represented by the sum using the maximum possible of six hexagonal numbers.
Pyramidal
Number:- Those
numbers which can be represented as layers of squares forming a pyramid are
called pyramidal numbers. The pyramid
class can be formed by adding successive layers of which the next above the nth
is the (n-1)th member of the same figurative number series.
35
55
There are many more figurative numbers which
are not discussed here but one thing is clear that they are really very- very
interesting. Though in the initial phase; the study of such numbers produced no
immediate results but certainly they are important as it led to the study of
series, which provided the clue to an understanding of numbers which are not
full grown. The credit certainly goes to the Pythagoreans who dealt with such
numbers. Even in the history triangular numbers played an important role in
suggesting rules for forming and adding the terms of series. A relic of such
numbers is seen in the problems relating to the pilling of round shot, still to
be found in algebras. Ovid in his poem De Nuce talks about pyramidal
number. So the journey which
Pythagoreans began with pebbles has now reached many mile stone in the
mathematics and mathematicians are also looking for other figurative numbers
making their journey endless.
Drop your comments here
Rajesh Kumar Thakur
rkthakur1974@gmail.com
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