Importance of Number 1

                    Important Characteristics of 1

The Greek did not consider 1 as a number. German Kobel in 1537 wrote in his book on computation –wherefrom thou understandest that 1 is no number, but it is a generatrix, beginning, and foundation for all other numbers. The Greek considered that 1 was both even and odd because when 1 is added to even number it produces odd and vice versa. 2 is an even number but 2 + 1 = 3 is an odd. On the other hand 3 is odd and 3+ 1 = 4 is even. Let us see some features of one.

1.      In the real number system, 1 can be represented in two ways as a recurring decimal.          

   0.9999… = 1            and 1.000 =1

2.      If a ZERO means nothing, number ONE is the opposite. Everybody wants to grab number position in the world in their respective fields.           

3.      It is one of the first non- zero Natural number. A natural number is a counting number that begins with 1 and goes to infinity. It is denoted by N. Hence, N = {1 , 2, 3, 4, ….) is a set of natural number.

4.      It is the first odd number. A number which is written in the form of 2n + 1 is called an odd number. In simple word, a number which is not divisible by 2 is called an odd number. For example – 1, 3, 5, 7, 9, …

5.      It is the only number other than 0(ZERO) whose square is the same as itself.                 

        0² = 0 and 1² = 1

6.      The root of modern 1 traces back to Indian Devnagri numeral.

             

7.      A , B, C, D, E, M, T, U, V, W,Y have one line of symmetry. The line of symmetry is a line which divides the Pictures into two parts and they form a mirror image of the other, that is the images in the both the side of the line are reflection of other, when it folded one side over the another along the line of symmetry are equal.

8.      A Mobius strip has one (1) edge and one (1) surface. The Mobius strip has the mathematical property of being non –orientable. It can be realized as a ruled surface. It was discovered by German mathematicians August Ferdinand Mobius and Johann Benedict Listing in 1858.

  

9.      One can’t be used as the base of a positional numeral system.

10.  The logarithm base 1 is undefined.

11.   One (1) is neither prime nor composite number. A prime number has two factors 1 and the number itself whereas a composite number has more than 2 factors.

12.  Factorial of zero (0) is one, i.e. 0! = 1.

13.  One (1) is called the multiplicative or reflexive identity of multiplication. If you multiply any number say a by 1 you will get no change in the result obtained. In simple word, a x 1 = a.

14.  One can be partitioned in one way.

15.   In the Peano axiom, 1 is the successor of 0.

16.  One is the first figurative number of every kind, such as Triangular number, Pentagonal number and Centered Hexagonal number.

17.  One is the magnitude of a unit vector. ǀ ˆi ǀ = ǀ jˆ ǀ = ǀ kˆ ǀ = 1

18.   Probability of an event that is almost certain is 1 (one) i.e. P (A) = 1. Such event is called Sure event.

19.  Number 1 , 11 , 111 , 1111 are all triangular numbers in the base – 9.

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Importance of ZERO

                                       Important characteristics of Number Zero

1.      Roman Numbers have no symbol for zero.

2.      The ancient Greek didn’t recognize 0 as a number.

3.      The word ZERO was originated in Italy. Leonardo Fibonacci who made the use of zero popular in European countries through his book Liber Abaci (Books on Number) used the word Zefiro meaning empty which was later abbreviated to zero.

                     

4.      There is no year zero in our Gregorian calendar.

5.       In Mayan calendar, time began at day 0, a day that has been calculated to correspond with August 11, 3114 BC.

6.      It is the only Complex Number ( a number written in the form of z = a + ib , where a and b are real numbers) which is both real and imaginary. 0 = 0 + i.0

7.      0 is the additive identity. Additive identity is a number which when added to a number gives the same number as a result.                                                                                                                              A + 0 = A

8.       −273⁰ C is called the ABSOLUTE ZERO. This is the lowest possible temperature at which molecules have zero heat energy and all molecules motion stops.

9.      Any number multiplies with zero gives the result 0, however big the number is. For example – 987654321 x 0 = 0.

10.  If you subtract zero from any number the result remain unchanged. For example :- 987 – 0 = 987

11.  You can’t divide any number by zero. Any number divided by zero is undefined. Now the question is why can’t we divide any number by zero? The simple reason is that if you divide 1 by 1/10 you get the result 10. If you divide 1 by 1/1000, you will get 1000 and if you divide 1 by 1/10000000000, you get 10000000000. The fact of the matter is when the denominator decreases and comes closer to zero we get the quotient larger and larger. In mathematical term it tends towards infinity. But infinity is not a number and so the rule of arithmetic simply says that you are not allowed to divide any number by zero.

12.  In telephone, there are no letters assigned to 0 and 1. These numbers remain unassigned because they are so called FLAG NUMBERS kept for the special purposes such as emergency or operator services.

13.  It is the only integer that is neither positive nor negative.

14.  It is the first member of Whole number.

15.  It can be portioned in 1 way.

16.  The word ZERO is the only number name in English that can be traced back to Arabic.

17. Power zero to any number is one: a⁰ = 1 provided a ≠ 0.

18.  Factorial of zero is 1.i.e. 0! = 1.

19.  Nikhilism is the belief that nothing has any value, purpose or meaning.

20.  Log0 is not defined.

21.  The importance of zero can’t be ruled out in sports. In Tennis LOVE means zero; in Cricket DUCK means zero and in Golf SCRATCH means zero.

22.   The scalene triangle, Parallelogram and Trapezium have 0 line of symmetry.

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Journey of Figurative Number

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 = 5^{2                       --------------------------------------------------------------------------------------------------------------------------------------------------------------------}                                                                                                                                                                        

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 C_1, C_2, 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:-

1. 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                                                                            -------------------------------------------------
2. 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           

4. 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:-

              P_n = n ( 3n – 1 )

               If we represent the pentagonal numbers by P₁,P₂ ,.... then the n th number P_n =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.  

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Rajesh Kumar Thakur

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