Number 6

                           Important Feature  of 6

1.      It is the second smallest composite number its proper divisor are 1, 2 and 3. A number is composite if it has more than two factors.

2.      There are six (6) basic trigonometric function- Sine, Cos (ine), Tan(gent), Cot (angent), Sec (ant) , Cosec (ant). These ratios define the ratio of two sides of a right angled triangle.

3.       Six (6) is the only even prime number that is not the sum of successive odd cubes because six is the sum of first three natural numbers.    
                                                                     6 = 1 + 2 + 3

4.      Bee hive is hexagonal in shape.

                                                                                           

5.      A benzene molecule has a ring of 6 carbon atom.                 

            

6.      Atomic number of Carbon is six.

7.      It is the first number which is divisible by  1, 2, 3                                                                              6 = 1 + 2 + 3

8.      Six is the only number whose sum of factor and product comes out to 6.                                         6 = 1 + 2 + 3             and 6 = 1 x 2 x 3

9.      It is the first Perfect Number. Perfect number is a number whose sum of factors excluding itself is equal to the number itself.                                                                                                                    6 = 1 + 2 + 3                                                                                                      
 Saint Augustin writes that although God could have created the world all at once, he preferred to take six days as 6 is symbolized by the Perfect Number.  

10.  It can be portioned in 11 ways.

11.  Six also has a special geometrical relationship with circles. If you take six identical circular coins and place them around other coins of the same size so that they touch it, they will all touch each other too whatever the size of the coin you take.

12.   It is a third triangular number.

13.  A cube has 6 faces it is also known as hexahedron.

14.  It is the highest number on a normal die. A die has number 1 to 6 written on it.

               

15.   Six (6) is a congruent number because it is the area of a right angle triangle having side 3, 4 and 5. A congruent number is an integer that is the area of a right triangle with three rational number side.

16.  The Pythagoreans associated 6 with marriage and health, because it is the product of their first even and first odd numbers, which were female and male respectively. 

17.  It is the only number which is the sum and product of same 3 numbers.                                           6 = 1 + 2 + 3  and 6 = 1 x 2 x 3

18.  Any number of the form of 6m – 1 has two factors whose sum is divisible by 6, where m is any integer.

19. Every prime greater than three is either one more, or one less than a multiple of six.

20. There are five equable (area and perimeter "equal" and integer side lengths) triangles . All of them have area/perimeter divisible by six.

Example, a right angled triangle with sides 5, 12 and 13 has area 30 square unit and perimeter 30 units.

There are infinitely many Pythagorean triples describing integer-sided right triangles, and there are infinitely many equable right triangles with non-integer sides; however, there are only two equable integer right triangles, with side lengths (5,12,13) and (6,8,10). There are exactly three solutions, beyond the right triangles already listed, with sides (6,25,29), (7,15,20), and (9,10,17).
[It was proved in 1904 by W. A. Whitworth and D. Biddle]





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Number 5

                                        Importance of 5

1.      It is the fifth number of Fibonacci sequence. In Fibonacci sequence the next number is the sum of the previous two. Example: - 1, 1, 2, 3, 5, …

2.      With naked eye we can see 5 planets – Mercury, Venus, Mars, Saturn and Jupiter.

3.      Plato the Greek Philosopher reckoned the classic elements (Earth, Air, Fire and Water) were composed of regular shaped solids of which there are five. They are called Platonic solids. The five Platonic solids are – Tetrahedron (4), Cube (6), Octahedron (8),

Dodecahedron (12), and Icosahedrons (20). These are the only convex regular solids possible to construct.

                                          

4.      Any power of 5 ends with 5 except the power 0.                                                                            

                      5² = 25                                                   5³ = 125                                                                                     5⁴ = 625                                                5⁵ = 3125

5.      A five sided polygon is known as Pentagon. Pentagon is the only shape in which the number of sides and number of diagonals are same. 

6.      A number that ends with 0 or 5 is divisible by 5. Example- 1000, 645

7.      Pentagram, Pentangle, and Pentacle are all names for a five pointed star. The Pentagram or five folded star was endorsed by Pythagoreans because in its regular form it contains the Golden ratio (1.618)

                                

8.      Five pointed symmetry is found in an apple.

9.      Five can be partitioned in 7 ways.

      10.  It is a Fermat Prime.

11.  It is the first Wilson Prime. A Wilson Prime is a prime number p such that p² divides     (p – 1 )! + 1, where! denotes the factorial of a number. This is named after the English mathematician John Wilson.

12.  Vulgar fractions with 5 or 2 in the denominator don’t yield infinite decimal expansion. A real number having the denominator in the form of 2ⁿ x 5^m will have a terminating decimal expansion.

13.  Atomic number of Boron is 5

14.  5 is the hypotenuse of the smallest Pythagorean Triangle with its other two sides 3 and 4.                                                                         

          5² = 3² + 4².                                                                                                         

   This is the only Pythagorean Triangles with its sides arranged in Arithmetic Progression.

15.   It is the first prime which is in the form of 4n + 1 from which it follows that it is the sum of 2 squares expressed in one way only.                                                                                        

16.  It is the first prime of the form 6n – 1. All other primes are one more or one less than a multiple of 6 except 2 and 3.

17.  Every number can be expressed as the sum of 5 positive or negative cubes in an infinite numbers of ways.

18.  The general algebraic equation of the 5^th degree cannot be solved in radicals. This was proved by Abel in 1824.

19. One of math's perplexing mysteries. A sphere in five dimensional space has a larger volume (8π^2/15) than in any other dimension for a unit radius. From two dimensions up to five the volume increases, then decreases forever after.

20. 1084 is the smallest integer whose spelling, one thousand eighty-four, contains the 5 vowels (a, e, i, o, u) in order (Jim Wilder)

21. The sum of the first five integers raised to their own power, is prime, 

1^1+2^2+3^3+4^4+5^5=3413

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Number 3

                      Important characteristics of 3

1.      Three (3) is the smallest odd prime number.

2.      It is the first odd prime number.

3.      It is the second ODD number. A number which is written in the form of 2n + 1 is called odd number. Such numbers are not divisible by 2. e.g. 1, 3, 5, 7, 9, 11, 13, 15, 17 and 19 are the first ten odd numbers.

4.      Greeks did not considered 1 as an Odd so according to them 3 is the first odd number. Pythagoreans considered 3 and other odd numbers as male and all even numbers as female.

5.      According to Proclus, it is the first number which increased more by multiplication than by addition. Example:- 3 + 3 = 6 and 3 x 3 = 9

6.      It is the only number that is both the first Fermat Number and first Mersenne number.      Pierre de Fermat was a lawyer by profession and an amateur French mathematician. He is considered the founder of Probability theory along Pascal. He was also considered to be the co- inventor of Analytic Geometry along with Rene Descartes.                                           He is better remembered for his Last Theorem.                                                       Mersenne Numbers are integers of the form M_n = 2ⁿ – 1. Marin Mersenne was a monk in a church in France who worked on prime numbers. Mersenne number M_n is valid for n = 2, 3, 5, 7, 13, 17, 19, 31, 67 and 257.

7.     
It is the second triangular number and only prime triangular number.                            

                                                  

8.      It is the only prime number which is one less than the perfect square. 3 = 2² – 1.

9.      The numbers of Primary colors are three. They are Red, Yellow and Blue.

10.  Divisibility Rule of 3:- A number is divisible by 3 if the sum of digits can be divided by 3.  Example: 345 is divisible by 3 because the sum of digits 3 + 4 + 5 = 12 is divisible by 3.

11.  If the denominator of a rational number is not divisible by 3 then the repeating part of its decimal expansion is divisible by 9.                                                                                              1/7 = 0.142857… 

Here the denominator 7 is not divisible by 3 but the repeating decimal 142857 is divisible by 9.

12.  Any number expressed on the base 3 is called Ternary number. Ternary number has three digits 0, 1 and 2.

13.   Three (3) is the smallest number of side required to make a polygon. Triangle is the smallest polygon with three sides. There are 3 types of triangle – Scalene, Equilateral and Isosceles.

14.  With just ruler and compass, it is impossible to trisect an angle. Trisecting an angle was one of the three famous problems of antiquity.

15.  Three (3) can be partitioned in three ways.

16.  A, F, H, K, N, Y, Z are all made up of three lines.

17.  Three (3) is the number of spatial dimension we live in.

18.  According to Pythagoras, 3 is the first male number.

19.  Three dimensional figures have length, breadth and height. Example: - Cube, Cuboid, Prism, Sphere, Cylinder, Cone etc.

20.  Three non- collinear points determines a plane or a circle.

21. It is the only prime number followed by a square number.

22. The solution of Ramanujan Nested radical that you see in the image is 3.

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Important of Number 4

1                                                   Important Characteristics of Number 4    

It is a square number, i.e 2² = 4                      

2.      The word FOUR has four letters. In the English language there is no other number whose number of letters is equal to its value.                                                                            

        FOUR = 4 (Number of Letters)

3.      A tetrahedron has four triangular faces.         

                                  

4.      In DNA molecules Thymine (T), Adenosine (A), Guanine (G) and Cytosine (C) are the four bases

5.      Four dimensional has four dimensions namely- Length, Breadth, Height and Time. It is denoted by (x, y, z, t).

6.      There are four fundamental operations in math. They are – Addition (+), subtraction (− ) , Multiplication (x ) and Division (÷).

7.      It is the smallest number of colours sufficient to colour all planar maps with no adjoining countries showing the same colour.

8.      4 is the smallest composite that is equal to the sum of its prime, i.e. 4 = 2 + 2

9.      4 is the smallest squared prime ( p² ) and the only even in the form 4 = 2².

10.  Lagrange’s four square theorem states that every composite integer can be written as the sum of at most four square numbers.   This theorem was proved in 1770.                                   

                   Example :         3 = 1² + 1² + 1² + 0²

31 = 5² + 2² + 1² + 1²

310 = 17² + 4² + 2² + 1².

11.  In human there are four blood groups namely A, B, AB and O.

12.  Humans have four canines, four incisors and four wisdom teeth.

13.  Atomic number of Beryllium is 4 and valency of Carbon is also four.

14.  A 4 x 4 square has an area equal to its perimeter.

            Perimeter = 4 + 4 + 4 + 4 = 16 unit                                                                                                           Area = 4 x 4 = 16 square unit

15.  x² – y² is divisible by 4 only when x – y is even.

16.  Any prime of the form 4k + 1 is the sum of two square numbers.          

                               13 = 4 x 3 + 1 = 2² + 3²                                                 

                               73 = 4 x 18 + 1 = 3² + 8²

17.  It can be portioned in 5 ways.

18.  4 is the smallest honest number. Numbers n that can be described using exactly n letters in standard mathematical English. Four has 4 letters. It is the smallest honest number.

19.  4 is the smallest composite number. A number is said to be composite if it has more than 2 factors. 4 has the factors 1, 2 and 4.

20.  A number is divisible by 4 if it last two digits is either 00 or divisible by 4. Example: - 100, 1465784 is divisible by 4.

21.  There are four suits in playing cards. They are – Clubs, Diamonds, Spade and Heart.

22. 4² = 2⁴ is the solution of the equation    a^b =b^a

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Importance of Number 2

                                        Important Characteristics of Number 2

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1.      Two is the only number x such that the sum of the reciprocals of the powers of x equal to itself.              1 + ½ + ¼ + ¹/₈ + --- = 2

2.      Square root of 2, i.e √2 is the first known irrational number discovered by Pythagoras.

3.      Two (2) is the only even prime number.

4.      There are no integers x, y, z for which xⁿ + yⁿ = zⁿ  is valid if n > 2. This is known as Fermat’s last Theorem. This theorem was proposed by Piere de Fermat in the year 1670 and it remained a mystery for the mathematicians for over 350 years until it was proved in year 1995 by Andrew Wiles. In case n = 2, the equation turns out to be a Pythagoras Theorem.

5.      N² ± N is divisible by 2 where N is a natural number.

6.      H, I, O and X have two lines of symmetry.                                                                                                                                                                                                                                                          7.      The common symmetry found in nature is bilateral. A single axis has two sides one goes in positive direction and another goes in negative direction. Our bodies are bilaterally symmetrical and we naturally distinguish the thing in two ways , e.g Right and Left, Up and Down etc.

8.      A number is divisible by 2 if it ends with 0, 2, 4, 6 and 8.                                 

         Example: 248, 98634, 666, 900 are divisible by 2 but 41, 333, 79 are not divisible by 2.

9.      2 can be partitioned in two ways.

10.  A binary code is written to base 2 and has just 2 digits 0 and 1. In binary system the odd number ends with 1 and even number ends with 0. This system was first used by Leibniz in 1679 though it is referred to in a Chinese book which supposedly dates from about 3000BC. Leibniz associated 0 with nothingness and 1 with odd.  In binary number system 14 is written as 1110 and is represented as (14)_{10 =} (1110)_2.

11.  For any polyhedron, the Euler’s formula is V – E + F = 2, where V , E and F represent Vertices, Edges and Faces respectively.

12.  Atomic number of Helium is 2.

13.  Two has a unique property such that 2 + 2 = 2 x 2.

14.  Chess, Squash and Sumo wrestling are all games played by two competitors.

15.  A figure having length and breadth but no depth is called a 2- dimensional object.

16. There are 20 digits in the expansion of 2⁶⁴ but surprisingly the base 2 is disappeared.                2⁶⁴ = 1,84,46,74,40,73,70,95,51,616
17. All known Perfect Numbers are Even.
18. Goldbach conjecture says that every even number greater than 2 is the sum of two prime numbers. Example : 8 = 3 + 5