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The Journey of Figurative Number

Posted on 22 June, 2015 No comments
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 

                                    1
  1   1
         1   2   1
  1   3   3   1     
1   4   6   4   1
 1  5  10   10   5   1
1  6    15    20  15 6  1        
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

 More interestingly each higher square number is formed by adding L shaped set of pebbles to the previous number. The L-shape was called gnomon by the Greeks which referred to an instrument imported to Greece from Babylon for measuring time.
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:-                                                                                                                                                                            C= ( T1)2                                                                                                                                C1 + C2 = 1 + 8 =( T22                                                                                                       C1 + C2 +C3 = 1 + 8 + 27 = 36 = ( T32                                                                               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:-
    1. The tetrahedral numbers are the sums of the consecutive triangular numbers beginning from 1.                                                                                                                                                                  T1= 1                                                                                                           T= 1 + 3 = 4                                                                                                                    T= 1 + 3 + 6 = 10                                                                                                               T4 = 1 + 3 + 6 + 10 = 20                                                                                                        T5 =1 + 3 + 6+ 10 + 15 = 35                                                                                       T= 1 + 3 + 6 + 10 + 15 + 21 = 56                                                                            -------------------------------------------------
  1. The sum of two consecutive numbers is a Pyramidal number.                                                                                                T1 + T2 = 1 + 4 = 5                                                                                                       T2 + T = 4 + 10 = 14                                                                                                           T+ T = 10 + 20 = 35                                                                                                        T+ T5 = 20 + 35 = 55                                                                                                                 T+ T= 35 + 56 = 91           

  2.                                     1
      1   1
             1   2   1
      1   3   3   1     
    1   4   6    4   1  
       1   5   10    10   5   1
     1   6   15    20   15   6   1 
            
    The tetrahedral numbers can be seen in the fourth diagonal of a Pascal’s triangle               

                                   





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:-
  1. 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.                                                                                   
   
 11 = 1 + 1 + 1 + 1 + 1 + 6                                                                                                                     26 = 1 + 1 + 6 + 6 + 6 + 6

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




Fascinating Number Pattern

Posted on 17 June, 2015 No comments
Hello Readers,

Please read mathematical blogs of mine at -- www.mathspearl.blogspot.in


Fascinating Number Pattern

Number pattern is my weakness. The moment I see the pyramid of number pattern I start jumping and these pattern which I have collected from different sources is presented here for your enjoyment. I do hope you will love it and enjoy.

Amazing Number 142857
While solving the problem of mensuration we generally take the value of pi to be equal to 22/7 which in mixed fraction gives 3 1/7. This fractional part 1/7 when changed it will be a non- terminating decimal which is equal to 0.142857 142857…
The number 142857 shows an explicit feature; when multiplied by 1, 2, 3, 4, 5 and 6.
142857 × 1 = 142857
142857 × 2 = 285714
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285
142857 × 6 = 857142
This is not the end of the story. The sum of every digit of the number (if written in a table) column wise or row wise comes to be equal to 27.
×
1
4
2
8
5
7
Row wise sum
1
1
4
2
8
5
7
27
2
2
8
5
7
1
4
27
3
4
2
8
5
7
1
27
4
5
7
1
4
2
8
27
5
7
1
4
2
8
5
27
6
8
5
7
1
4
2
27
Column wise sum
27
27
27
27
27
27


You will be astonished to see more feature of this number. Let’s do the division process of number 1, 2, 3, 4, 5 and 6 by 7 and see the beauty of the number obtained.
1/ 7 = 0.142857 …                  and                   142 + 857 = 999
2/7 = 0.428571 …                  and                   428 + 571 = 999
     3/7 = 0.285714                      and                    285 + 714 = 999
    4/ 7 = 0. 857142                    and                     857 + 142 = 999
    5 / 7 = 0.571428                    and                    571 + 428 = 999
    6/ 7 = 0.714285                   and                     714 + 285 = 999
So far, we did multiplication of number 142857 by number from 1 to 6, let’s do the multiplication of this number by 7.
142857 × 7 = 999, 999
Moreover, 142 + 857 = 999 and 14 + 28 + 57 = 99
(142857)2 = 20408122449 and 20408 + 122449 = 142857.

Pattern 2:- Multiplication of Number in recurrence of 9 with 2
Numbers like 9, 99, 999, 9999 … etc. are amazing numbers and it shows a beautiful relation when multiplied by 2. Enjoy the beauty here:-
9 × 2 = 18
99 × 2 = 198
999 × 2 = 1998
9999 × 2 = 19998
99999 × 2 = 199998
999999 × 2 = 1999998
9999999 × 2 = 19999998
99999999 × 2 = 199999998
999999999 × 2 = 1999999998


Pattern 3:- Multiplication of 987654321 by multiples of 9

987654321 is a number written in reverse order from the highest one digit number 9 to the least one digit number 1. This number when multiplied by the multiples of 9 it shows an exceptional beautiful pattern of number. Are you ready to enjoy the beauty?


Did you notice the beauty of pattern?
The sum of digit at both the extreme is 9 and the multiple is placed at both the extreme. As far as the middle part of the product is concerned it is the recurring of number that is 1 less than the number at the unit digit of the multiplier.

Pattern 4:  Here is a pyramid of multiplication of two numbers with equal number of 1’s and 8’s and the result obtained is amazing.
1 × 8 = 8
11 × 88 = 968
111 × 888 = 98568
1111 × 8888 = 9874568
11111 × 88888 = 987634568
111111 × 888888 = 98765234568
1111111 × 8888888 = 9876541234568
11111111 × 88888888 = 987654301234568

Pattern 5:- The below pattern is a very beautiful pattern of numbers that involves the multiplication of number 1, 12, 123 …. 123456789 by 8 and adding number 1, 2, 9 to it making a beautiful decorative pattern.
1 × 8 + 1 = 9
12 × 8 + 2 = 98
123 × 8 + 3 = 987
1234 × 8 + 4 = 9876
12345 × 8 + 5 = 98765
123456 × 8 + 6 = 987654
1234567 × 8 + 7 = 9876543
12345678 × 8 + 8 = 98765432
123456789 × 8 + 9 = 987654321

Pattern 6:- Pyramid of 8
This beautiful pyramid of 8’s is formed with the number 9, 98, 987 … multiplied by 9 and adding 7, 6, 5 ….
9 × 9 + 7 = 88
98 × 9 + 6 = 888
987 × 9 + 5 = 8888
9876 × 9 + 4 = 88888
98765 × 9 + 3 = 888888
987654 × 9 + 2 = 8888888
9876543 x 9 + 1 = 88888888
98765432 × 9 + 0 = 888888888

Pattern 7:- Multiplication of number 12345679 with the multiples of 9’s:- If you multiply 12345679 with the multiples of 9’s the result will be the recurring of the as many times the multiples of 9’s.
12345679 × 9 = 111 111 111
12345679 × 18 = 222 222 222
12345679 × 27 = 333 333 333
12345679 × 36 = 444 444 444
12345679 × 45 = 555 555 555
12345679 × 54 = 666 666 666
12345679 × 63 = 777 777 777
12345679 × 72 = 888 888 888
12345679 × 81 = 999 999 999

Pattern 8:- Multiplication of number 65359477124183 with multiples of 17:- Number 65359477124183 makes astonishing pattern when multiplied with the multiples of 17.
65359477124183 × 17 = 1111 1111 1111 1111
 65359477124183 × 34 = 2222 2222 2222 2222
65359477124183 × 51 = 3333 3333 3333 3333
65359477124183 × 68 = 4444 4444 4444 4444
65359477124183 × 85 = 5555 5555 5555 5555
65359477124183 × 102 = 6666 6666 6666 6666
65359477124183 × 119 = 7777 7777 7777 7777
65359477124183 × 136 = 8888 8888 8888 8888
65359477124183 × 153 = 9999 9999 9999 9999

Pattern 9:- Palindrome Number Pattern with the multiples of 11, 111 ….:- Here is a beautiful pattern formed with the multiples of 11, 111, 1111 …. with itself.
11 × 11 = 121
111 × 111 = 12321
1111 × 1111 = 1234321
11111 × 11111 = 123454321
111111 × 111111 = 12345654321
1111111 × 1111111 = 1234567654321
11111111 × 11111111 = 123456787654321
111111111 × 111111111 = 12345678987654321

Pattern 10:- Beautiful pattern of multiplication of 9, 99 …:- Multiplication of number 9 shows a beautiful pattern. Enjoy the beauty of pattern.


Pattern 11:- Pattern of square making a beautiful pyramid: - There are certain numbers which presents a beautiful pyramid pattern when its square is formed. Let’s enjoy the beauty.
42 = 16                                     72 = 49                                     92 = 81
342 = 1156                               672 = 4489                               992 = 9801
334= 111556                         667= 444889                         9992 = 998001
33342 = 1115556                     66672 = 4444889                     9999= 99980001
            333342 = 1111155556             666672 = 4444488889             999992 = 9999800001

Pattern 12:- Multiplication pattern with number 76923:- If you multiply 76923 with 1, 10, 9, 12, 3 and 4 you will see a beautiful pattern of result obtained. The number obtained in each case contains the same number of digit.
76923 × 1 = 076923
76923 × 10 = 769230
76923 × 9 = 692307
76923 × 12 = 923076
76923 × 3 = 230769
76923 × 4 = 307692

Pattern 13:- Pyramid of square of number in the recurring of 9:- Square of number 9, 99, 999, 9999, 99999, 999999 … etc. will form a pattern that looks like a pyramid. See the beauty.
92 = 81
99 = 9801
999= 998001
9999= 99980001
999992 = 9999800001
9999992 = 999998000001
99999992 = 99999980000001
Pattern 13: - Circular Prime Pattern: - This is a circular Prime pattern :-A circular prime is a prime number that remains prime as each leftmost digit in turn is moved to the right hand side. For more information you can visit www.
19937
99371
93719
37199
71993
19937
Pattern 14:- Multiplication pattern with the primes makes a beautiful pattern.


                                                    
 Pattern 15:- Division of number 1 to 9 by 11:- Number 1 to 9 divided by 11 gives non ending decimal expansion.
1/11 =0.09090909….                                                  2/ 11 = 0.18181818…
3/11 = 0.27272727…                                                  4/11 = 0.36363636—
5/11 = 0.45454545…                                                  6/11 = 0.54545454….
7/11 = 0.63636363…                                                  8/11 = 0.72727272…

Pattern 16:- Division of number 1 to 9 by 9:- Number 1 to 9 divided by 9 gives non ending decimal expansion.
1 /9 = 0.11111…                                                         2 /9 = 0.22222…
3/9 = 0.33333…                                                          4/9 = 0.44444….
5/9 = 0.55555….                                                         6/9 = 0.66666…..
7/ 9 = 0.77777….                                                        8/9 = 0.88888…

Pattern 17:- Palindrome Pattern:-


Pattern 18:- Square of Palindrome number giving the palindrome result
112 =121
1012 = 10201
10012 = 1002001
100012 = 100020001
1000012 = 10000200001
10000012 = 1000002000001
100000012 = 100000020000001
1000000012 = 10000000200000001
Pattern 19: Product of Palindrome number making a beautiful pyramid: Here palindrome number made with recurring of 1 and 8 are shown making a beautiful pyramid of number.


Pattern 20:- Pattern with number 987654321 :- Number 987654321 when multiplied by the multiple of 9 gives a beautiful pattern of number where the number placed at the  extreme left and extreme right are the digit by which the number is multiplied whereas the middle recurring digits are one less than the number placed at the extreme right.


Pattern 21:- Pascal Triangle: - In Algebra, you must have used the Pascal triangle to obtain the coefficient of binomial expansion. But this is not the end of the game, Pascal triangle when expended for 9 -10 rows shows different number pattern itself. If I say that Pascal triangle has natural number, triangular number, square number, Pentagonal number, Hexagonal number, Fibonacci number etc. then you will simply not believe my words but it is true.

In the above diagram, you can see the second diagonal showing you the Natural numbers like 1, 2, 3, 4, 5 …. Whereas the third diagonal from either side shows you the triangular number 1, 3, 6, 10 … The Square number (1, 4, 9, 16, 25 …), Pentagonal number (1, 5, 12, 22, 35,…)  Hexagonal number (1, 6, 15. 28, 45,…) and Fibonacci sequence (1, 1, 2, 3, 5, 8, 11, …) can be witnessed in this triangle.

Pattern 22:- Pyramid of 1:- See the beauty of number in increasing order making the pyramid of 1’s.
1 × 9 + 2 = 11
12 × 9 + 3 = 111
123 × 9 + 4 = 1111
1234 × 9 + 5 = 11111
12345 × 9 + 6 = 111111
123456 × 9 + 7 = 1111111
1234567 × 9 + 8 = 11111111
12345678 × 9 + 9 = 111111111
123456789 × 9 + 10 = 1111111111

 So Explore the beauty and never forget to inform me about your sweet reaction.

Rajesh Kumar Thakur
rkthakur1974@gmail.com
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