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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Quotation on Numbers

Quotation on Numbers

1.       There is safety in numbers. --- Anonymous

2.       Numbers are intellectual witness that belongs only to mankind. ---Aristotle

3.       The numbers are a catalyst that can help turn raving mad men into polite humans.—Philip. J .Davis.

4.       It is not once nor twice but times without number that the same ideas make their appearance in the world. --- Aristotle

5.       Perfect numbers like perfect men are very rare. – Rene Descartes

6.       God created the natural numbers and all the rest is the work of man.--- Leopold Kronecker

7.       All the effects of nature are only mathematical results of a small numbers of immutable laws. – Pierre Simon Laplace.

8.       The imaginary number is a fine and wonderful resource of the human spirit, almost an amphibian between being and not being.--- Gottfried Wilhelm Leibniz.

9.       God created everything by  number, weight and measure.--- Isaac Newton

10.   Zero is the number of object that satisfy a condition that is never satisfied. But as never means- in no case. I don’t see that any progress has been made. --- Henri Poincare

11.   Wherever there is a number, there is a beauty.—Diadochus Proclus

12.   All was numbers.—Pythagoras

13.   Numbers was the substance of all things.--- Pythagoras

14.   Number rules the universe. – Pythagoras

15.   Number if the ruler of forms and ideas, and the cause of gods and demon. --- Pythagoras

16.   Women have a passion for mathematics. They divide their age in half, double the price of their clothes, and always add at least five years to the age of their best friend. – Marcel Achard

17.   I am x years old in x² years.--  Augustus De Morgan

18.   A good decision is based on knowledge and not on numbers. --- Plato

19.   The creator of the universe works in mysterious ways. But he uses a base counting system and likes round numbers. --- Scott Adams

20.   I would play with numbers in a way that other kids would play with their friends. --- Danniel Tammet.

21.   Squaring numbers is a symmetrical process that I like very much. And when I divide one number by another, say 13 divided by 97, I see a spiral rotating downwards in larger and larger loops that seems to wasp and curve. The shapes coalesce into the right number. I never write anything down. --- Daniel Tammet

22.   I think my numbers speak for themselves. --- Jack Youngblood

23.   Torture numbers, and they will confess to anything. --- Gregg Easter Brook

24.   Life is not measured by the numbers of breaths we take, but by the moments that take our breath away. --- Anonymus

25.   Anyone can count the seed in an apple, but only God can count the number of apples in a seed—Robert H Schuller

26.   The intelligence of the creature known as a crowd; is the square root of the people in it. --- Terry Pratchetl

27.    The primary source of all mathematics is the integers. --- Herman Minkowski

28.   The trouble with integers is that we have examined only the small ones. Maybe all the exciting stuff happens at really big numbers, ones we can’t get our hands on or even begin to think about in any very definite way. So maybe all the action is really inaccessible and we are just fiddling around. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions.--- Ronald Graham.

29.   The study of the infinite is much more than a dry, academic game. The intellectual pursuit of the Absolute Infinite is a form of the soul’s quest for God. Whether or not the goal is ever reached, an awareness of the process brings enlightenment. – Rudy Rucker

30.   We think of the number Five as applying to appropriate groups of any entities whatsoever – to five fishes, five children, five apples, five days--- We are merely thinking of those relationships between those two groups which are entirely independent of the individual essences of any of the members of either group. This is a very remarkable fear of abstraction; and it must have taken ages for the human race to rise of it. --- Alfred Norsh Whitehead.

31.   Never dismiss the intuition of the ancients, who believed that number is the essence of all things. Number is the secret source of entire cultures, and men have been killed for their heresies and seductive credos. The whole history of mathematics is subterranean, taking place beneath history itself, a shadow- world scarcely perceived even by the learned. --- Don De Lillo

32.   There can be no dull numbers, because if there were, the first of them would be interesting on account of its dullness. – Martin Gardner

33.   A person who can within a year solve x² – 92y² = 1 is a mathematician. --- Brahmagupta

34.   I love to count. Counting has given me special pleasure down through the years. I can think of innumerable occasions when I stopped what I was doing and did a little counting for the sheer intellectual pleasure of it. --- Don De Lillo

35.   Number is the bond of the eternal continuance of things. – Plato

36.   I had been to school most of the time, and could spell and read, and write just a little, and could say the multiplication table up to six times seven is thirty five, and I don’t reckon I could ever get any further than that if I was to live forever. I don’t take no stock in mathematics, anyway.---  Mark Twain

37.   While the abacus extended the usefulness of the number system and made possible elaborate calculations, it had the distinct advantages of being a thing quite apart from the mind that used it. The training of mind which uses an abacus is not complete, because the process of combinations which such a mind uses are mechanical and external. The abacus served useful purpose in that stage of civilization when the mind of a man has not attained a number system which is detached from all mechanical devices and yet possesses all the virtues that mechanical devices contributes. --- C H Judd

38.   The theory of numbers is the last great uncivilized continent of mathematics. Out of this theory as out of Africa, there is always something new. For the last 2500 years amateurs as well as professionals have explored it yet there is every reason to expect that future discoveries with or without the help of machines will far surpass those of past. ---E T Bell

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