Number 11

Eleven (11)

1.      Its etymology originates from a German compound ainlif meaning one left.

2.      Eleven is the only palindromic prime with an even number of digits.

3.      It can be portioned in 56 ways

4.       It is a Lucas Number. The numbers in Lucas series are 1, 3, 4, 7, 11…

5.      It is the fourth number that stays the same when written upside down. The first three are 0, 1 and 8. The others such numbers less than 100 are 69, 88, 96.

6.      The sun spot cycle repeats every 11 years. Sun spots are dark areas on the surface of the sun which have strong magnetic field.

7.Any power of 11 ends with 1.                                                                                                                       11² =121                                  11³ =1331  

     

8.The magical sum of this magic triangle is 11.                    

                                                                 

9.      It is the smallest three digits prime in ternary system and four digit prime in binary system. 11 in ternary system is written as (102)₃ and in binary as (1011)₂.

10.  It is the fourth Sophie Germain Prime. A prime number p is a Sophie Germain prime if 2p+ 1 is also a prime. The first ten Sophie Germain primes are 2, 3, 5, 11, 23, 29, 41, 53, 83 and 89.

11.   It is a Stormer number. A Stormer number is a positive integer n for which the greatest prime factor of n² + 1 meets or exceeds 2n. It is named after Carl Stormer. The first eight Stormer numbers are 1, 2, 4, 5, 6, 9, 10, and 11.

12.  When you divide a number with 11 in denominator it will have two digit repeating sequences in their decimal expansion.                                                                                            

    1/11 = 0.090909..                                                    2/11 = 0.181818..                                

     3/11 = 0.272727...                                                 4/11 = 0.363636..

13.  Divisibility by 11:- There are several methods to check the divisibility of a number by 11 but the most common amongst them is to alternately add and subtract digits starting from the right. If the result thus obtained is the multiple of 11 then the original number will be divisible by 11.                                                                                                             Example:-Is 1331 divisible by 11?                                                                                           Here, 1 – 3 + 3 – 1 = 0, hence it is divisible by 11.

Rajesh Kumar Thakur

rkthakur1974@gmail.com

Number 10

                                            Number 10

       1.      It is the base of our numerical system.

       2.      It is the fifth even number. The other four preceding 10 are 2, 4, 6 and 8.

       3.      It is the fourth triangular number. The first three are 1, 3 and 6.

      4.      It is the sum of square of two consecutive odd numbers.                

                                                  10 = 1² + 3²

      5.      October 10, 2010 is written as 10/10/2010. The occurrence of such number is special because the first tow number added together equals the first two digits of the year. Here 10 + 10 = 20 is the first two digit of the year 2010. Such date occurs in every 201 years. The next such combination will occur on 11/11/2211.

        6.      It is the fifth composite number. The first four are 4, 6, 8 and 9.

       7.      Divisibility by 10:- If a number ends with zero it is exactly divisible by 10.         

                          Example:- 100, 560540

       8.      Roman numerals for ten is X. It represents two crossed hands.

     9.      Take any triangle and divide each side into three equal parts. Draw lines from these to the opposite corners. A new shape will appear in the middle of the triangle whose area is 1/10^th of the area of whole triangle. This is known as Marion Walter Theorem. This theorem state - the area of the central hexagonal region determined by trisection of each side of a triangle and connecting the corresponding points with the opposite vertex is given by 1/10 the area of the original triangle.

10.  The strength of earth quake is measured on the Richter scale. Each point on the Richter scale represents a tenfold increase in energy. The Richter scale is a logarithmic scale used to express the total amount of energy released by an earthquake. Its values typically fall between 0 and 9, with each increase of 1 representing a 10 fold increase in energy.      

Value	1	2	3	4	5	6	7	8	9	10
Potential Hazard	Extra-ordinary	Outstanding	Far reaching	High	Note worthy	Intermediate	Moderate	Minim	Low	insignificant

     11.  It is the sum of first three primes.                                                                                                

       10 = 2 + 3 + 5

    12.  It is the sum of three consecutive (3^rd, 4^th and 5^th) Fibonacci numbers.

   13.  The only known solution to                                                                                                            

             n! = a! b!                                                                 is  10! = 6! X 7!

There are only 11 powers of 10 that are products of two integers without any zero. These are--                                                                                                                                           

           10⁰ = 1                         10¹ = 2 x 5                            10² = 4 x 25                             

            10³ = 8 x 125               10⁴ = 16 x 625                         10⁵ = 32 x 3125                                                 10⁶ = 64 x 15625         10⁷ = 128 x 78125                   10⁹ = 512 x 1953125             

14. This beautiful formula according to Ramanujan came into his dream

15. 10! seconds = 6 weeks

16. Ten is the smallest semi-prime that is the sum of all the distinct prime numbers from its lower factor through its higher factor. 
10 = 2 x 5 
10 = 2 + 3 + 5

17. It is the sum of first four factorials
0! + 1 ! + 2 ! + 3 ! = 10 

18. It is a tetrahedral number or triangular pyramidal  number.
1 4,  10, 20 

      Read more on Mathematical Number in my book THE POWER OF MATHEMATICAL NUMBER available on website 

Rajesh Thakur

rkthakur1974@gmail.com

       

Importance of Number 9

                              Mathematical  Characteristics of Nine

      It is the least number that can be expressed as the sum of cubes of first two numbers.              

                                  1³ + 2³ = 9

2.      It is the largest one digit number in Hindu- Arabic Number system.

3.      Casting Out Nines is the popular method to check the four fundamental operations ( + , _ , x and ÷)

4.      If the sum of the digits of a number is the multiple of nine it is divisible by nine.              

                    58527 is divisible by 9 as 5 + 8 + 5 + 2 + 7 = 27 is divisible by 9.

5.      If a number from 1 to 8 is divided by 9 the number which is being divided will be repeated infinitely after decimal.                                                                                  

                1/9 =0.1111….                    2/9 =0.2222….                     3/9 =0.3333….                                                    4/9 =0.4444….                    5/9 =0.5555….                    6/9 =0.6666….                                                                7/9 =0.7777….                    8/9 =0.8888….              

6.      It is a composite number with its proper divisor being 1 and 3.

7.      A polygon with nine sides is called a nonagon.

8.      If you multiply any number by 9 the sum of all digits of the product adds to 9.                                                   578329 × 9 = 5204961 and 5 + 2 + 0 + 4 + 9 + 6 + 1 = 27; 2 + 7 = 9   

               482729235601 × 9 = 4344563120409 and  

                                            4 + 3 + 4 + 4 + 5 + 6 + 3 + 1 + 2 + 0 + 4 + 0 + 9 = 45; 4 + 5 = 9

9.      The difference between a base 10 positive integer and the sum of its digits is a multiple of nine.                                                                                                                                               

     Example:- Let us take the number 768. The sum of its digit is 7 + 6 + 8 = 21. The difference between 768 and 21 is 768 – 21 = 747 which is divisible by 9.

10.  Nine is the atomic number of Fluorine.

11.  In Geometry, a circle that passes through the nine points of a triangle is known as Nine point circle. These nine points are- the midpoint of each side of triangle, the foot of each altitude and the midpoint of the line segment from each vertex of the triangle to the ortho center. The nine point circle is also known as Euler’s circle or Feuerbach’s circle.                        

12.  The first 9 numbers can be arranged in a magic square so that all rows, columns and both diagonals have the same sum 15.

4	9	2
3	5	7
8	1	6

13.  Nine is the maximum number of cubes that one needed to sum to any positive integer.

14.  Take any 3 digit number, invert it and subtract the smallest from the larger. The middle digit will always be 9.                                                                                                               

                        542 – 245  =  297                                                                                                                       

                       861 – 168 = 693

15.  It is the sum of factorials of 3 consecutive numbers.                                                       

                  1! + 2! + 3! = 1 + 2 + 6 = 9

16.  It is the fourth composite number. A composite number has more than 2 factors. 

                    9 has the factors 1, 3 and 9.

17.  It is the sum of first 3 odd numbers.                                                                                                  

                    1 + 3 + 5 = 9

18.  The nine cubes on the size dish stamp make an impossible figure. This arrangement can’t exist in real world.  This is a Swedish postal stamp.

19.  The following pattern shows the use of 0 to 9 used once.                                                

             



       9          =  



97524 / 10836
                                                                                                                                          
= 95823 / 10647
                                                                                                                                           
= 95742 / 10638
                                                                                                                                       
= 75249 /08361
                                                                                                                                         
= 58239 / 06471
                                                                                                                                      
= 57429 / 06381

20.  Let us play an interesting puzzle on multiplying of a number by 9.                                                   Multiply -- 10,112,359,550,561,797,752,808,988,764,044,943,820,224,719 by 9 and you will get the result by simply moving the last nine up to the front. Hence the answer is 910,112,359,550,561,797,752,808,988,764,044,943,820,224,71

Properties of Number 8

        Importance of  Eight (8)

1.      It is a composite number with its proper divisor being 1, 2 and 4.

2.      It is the base of the octal number system. In octal number system one digit represents 3 bits.

3.      The atomic number of oxygen is eight.

4.      Scorpio is the 8^th astrological sign of the Zodiac.

5.      A map consists of 8 directions:- North, Northeast, Northwest, East, Southeast, Southwest, West, South.

6.      In three dimensional spaces there are 8 octants and the eight octants have the different co-ordinates shown here.

I	II	III	IV	V	VI	VII	VIII
+	_	_	+	+	_	_	+
+	+	_	_	+	+	_	_
+	+	+	+	_	_	_	_

7.      Apart from the trivial cases of 0 and 1, 8 is the smallest number for which the sum of the digits of its cube is equal to the number.                                                                                    

                    8³         = 512  and 5 + 1 + 2 = 8

8.      It is the largest cube in the Fibonacci sequence. A Fibonacci number is a numbers in which the next number is the sum of previous two. 1, 1, 2, 3, 5, 8… are the numbers of the Fibonacci sequence.

9.      It is the second cube number. 2³ = 8

10.  One can split a pie into 8 pieces with three straight cuts.

11.  A cube or cuboids has 8 vertices.

12.  An umbrella is octagonal in shape.    

13.  The digital sum of a prime pair product is 8.                                                                 

                          5 x 7 = 35 and 3 + 5 = 8                                                              

                          11 x 13 = 143 and 1 + 4 + 3 = 8                                                                               

                          17 x 19 = 323 and 3 + 2 + 3 = 8                                                                                 

                       29 x 31 = 899 and 8 + 9 + 9 = 26 and 2+ 6 = 8

14.  A number is divisible by 8 if it ends with 000 or its last three digit is divisible by 8. 

                          Example – 1000, 456640

15.  Many spiders, bees and flies have eight legs.

16.  Eight is the third number that remains the same written upside down.

17.  The amount of cloud in the sky is measured in oktas on a scale from zero to eight. 8 oktas means the sky is totally clouded.

18.  It is the second honest number.                                                                                                      

                2³ = Two Cubed = 8 has eight letters.

19.  It is the smallest sum of two factorials of distinct primes.                                                              

                       2! + 3! = 2 + 6 = 8

20.  It is the only cube that is 1 less than a square.                                                                                  

                           8 = 3² – 1 

21.  Digit sum of prime pair product comes out 8.                                                                                  

                5 x 7 = 35                           and        3 + 5 = 8                                                                      

              11 x 13 = 143  and                                   1 + 4 + 3 = 8                                                                

              17 x 19 = 323  and      3 + 2 + 3 = 8

22.  Eight can be portioned in 22 ways.

23.  8 times any triangular number is 1 less than a square. 1, 3, 6, 10,… are the examples of triangular number. If we take 3 as a triangular number than 8x 3 = 24 is 1 less than 25 which is a square number.

24.  It is the second octagonal number represented by the formula n(3n – 2 ), where n is any natural number. The sequence is 1, 8, 21, 40….