283 = 2^5 + 8 + 3^5.
432 = 4 × 3^3 × 2^2.
3340 = 3333 + 3 + 4 + 0.
3341 = 3333 + 3 + 4 + 1.
3342 = 3333 + 3 + 4 + 2.
3343 = 3333 + 3 + 4 + 3.
3344 = 3333 + 3 + 4 + 4.
3345 = 3333 + 3 + 4 + 5.
3346 = 3333 + 3 + 4 + 6.
3347 = 3333 + 3 + 4 + 7.
3348 = 3333 + 3 + 4 + 8.
3349 = 3333 + 3 + 4 + 9.
Sunday, March 11, 2012
Mathematics in Vedas and Ancient Indian Mathematicians - Mathematics
Mathematics in Vedas
VEDAS are the earliest records of human wisdom, which are being handed down to humanity through oral tradition of knowledge transmission. Basing on the astronomical references available in the Vedic literature we can say the age of VEDAS to be 6000BC – 19000 BC.
Digits and the fractions in VEDAS:
The concept of digits, to the several millions can be clearly seen in all VEDAS.
We take an example from YAJUR VEDA: The existence of bigger numbers in series can be seen in Yajur Veda
ek(1) cha dash (10) cha shatama (10^2)
chashatam(10^2) cha sahastrama (10^3)
chasahastram (10^3) cha ayutam(10^4)
chaayutam(10^4) cha niyutam (10^5)
chaniyutam(10^5) cha prayutam (10^6)
chaarbudam (10^8) cha nyarbudam
chasamudrascamadhyam (10^16)
chaantashchparaddashch (10^17)
yetah meh agneishtikahdenavahsantuamrutramushminloke
(yajurveda/ adhyaya 17 / kandika 2 )
This hymn indicates a series of numbers which starts with 1, the next number being 10, 100, 1000 ….…. Up to 10^17, of course in the form of a geometric progression
Fractions can be clearly understood in the following Rig Vedic hymn
jyesthaahchamasadva (1/2)
karotikaniiyantreen (1/3)
kunavametyahkanishtaahchaturaskaroti (1/4) tvashtribhavastatpanayadvachovah
(rigveda/mandal 4/sukta 33/ mantra 5)
The hymn speaks of a glass of drink in to two, three, four parts , however whether they had a system of representing the fractions in the form of a numerator and a denominator is uncertain.
Geometry in VEDAS:
The simplest form of polygon can be taken as triangle since 3 is the minimum number of sides with which a polygon can be drawn. The name of the triangle (Tribhuja) can be seen in the following lines from AtharvanaVeda, in a non mathematical context.
yoakrandayatsalilammahitvayonimkrutvaatribhujamshayanahvatsahkaamdugoviraajahsaguhachakretanvahparachiah
The Almighty made a triangle (Tribhuja) with earth , the intermediate space and the heaven as its sides and the generated multipleforms of beings, in a mystic way.
Sahastra sheersha purushahsahasrakshahsahasrapat
sabhumimavikshvatovrutvaatyatishtddshangulam
Every point on the circumference of a circle can be treated as its vertex. Thus the circumference of a circle is a set of infinite points. This is the sahasrashirsha property of a circle.
The diameter of a circle can be called as its axis. The axis
of symmetry of the circle is any of its diameter is its sahasrakshat.
A circle can be considered as an infinite sided polygon. This is its sahasrapadat
Note: “Mathematics in Vedas” is something different from “Vedic Mathematics”
“Vedic Mathematics” is introduced by HIS HOLYNESS SWAMI BHARTHI KRISHNA THIRTHAJI (whose is named as venkat raman by his parents)(1884-1960)
Swamiji , who was an accomplished vedic scholar wrote 16 volumes on vedic mathematics, comprehensively in all branches of mathematics, But it is learnt that all volumes were lost mysteriously. Despite of weak eyesight Swamiji with his untiring capacity, strong will and determination wrote a comprehensive book on vedic mathematics.
Vedic mathematics is not just some tricks of doing mathematical calculations quickly, it goes far beyond this and becomes science in itself beneficial in solving problems related to higher mathematics i.e. solutions of simultaneous equations, determinants, transcendental equations, solutions of cubic and higher order equations, solutions of differential equations etc.
Ancient Indian Mathematicians - Mathematics
Let us have a quick glance of list of Ancient Indian mathematicians and their contributions.
1. Lagadha (1350 BC): The ancient Indian astronomer wrote the book titled VedangaJyothisyam. So many wonders of mathematical and astronomical ideas are found in that small book
kalaadashacavimshaasyaat
dvimuhuurtastunaadike. …
naadike dvi muhuurtastu
THE TABLE OF CADIVISONS OF TIME PROPOSED IN HIS BOOK
5 guruvaaksharas = 1 kaashthaa
124 kaashtaas = 1kalaa
(10 + 1/20) kalaas = 1 naadikaa
2 naadikaas = 1 muhurta
30 muhurtas = 1 day
61 days = 1 ritu
3 ritus = 1 ayana
2 ayanas = 1 year
5 years = 1 yuga
This book is written in order to help to determine the actual time of performance of vedic rituals. In the he streamline many mathematical operations and their results.
Lagadha is a leader, with many followers, interpreters, commentators admirers of his mathematical work.
2. SULVA SUTRAS BY BAUDHAYANA
The very purpose of sulvasuthras is to provide technical assistance to design construct VEDIS AND CHITIS (fire altars) for successful Yajnas.
They have discussed on basic geometrical concepts, ideas, properties, propositions. Further we can observe the use of fractions and surds, indeterminate equations.
The time period of sulvasuthras is estimated at around 800BC
These sutras’ explained about mensuration details of 3D objects and Pythagoras theorem.
3. Aryabhata:(476BC-550BC): he is great Indian mathematician and astronomer. He accurately measured the period of earth rotation with respect to the fixed stars in the sky, deduced the shape of earth and its rotation, and made many remarkable achievements in astronomy and mathematics. He gave the formula to calculate the value of PI.
4 Varahamira( 6th century AD): His main astronomical work“PANCHSIDDHANTIKA” as observed early the five systems of astronomy are discussed in this. He gave tables using the formulae R sinθ and R cosθ of the angles 0 to 900 with the intervals3045’. His contributions to combinations are also important.
5.Bhramaguptha: He is the great astronomer and mathematician belongs to fifth century AD. He wrote three texts (1) Bhrama sputa Siddhantha (2) KhandKhadyaka (3) Uttar Kanda-khadyaka. Bhramsputhasiddhantha is the standard and basic text in astronomy. In this book he discussed motions of celestial bodies, their speed, solar and lunar eclipses etc. Also discussed the progressions, permutations,algebra-quadraticpolynomials, properties of cyclic quadrilaterals.
6. BhaskaraI(600 to 680AD) : He wrote two treatise “Mahabhaskariyam” and “laghubhaskariyam” dealing with Mathematical Astronomy. He gave the tables of sine function upto two correct decimals. He also discussed the equations just similar to Pell’s equations of modern period. He is also having credit for numbers and symbolism, classifications of equations.
7.Sridharacharya(750AD)There are two books available, which are written by sridharacharya (1) Trisathika (2) Pathiganitha
His contributions in mathematics are as follows
(1) Place names of numbers in decimal system
(2) Definition and properties of zero
(3) Use of symbol “+”
(4) Methods of multiplication of numbers
(5) Square root and cube root of any number
(6) Sum of the cubes and squares of the terms of Athematic progression
(7) Rational solutions of the equations of the form Nx2+1 = y2
8. Lallacharya (720AD-790AD): Lallacharya was one of the leading and eminent mathematicians, astronomer and astrologers of eighth century. His notable works are SHISHYADHIDATANTRA – A work on astronomy in two volumes namely Grahadhya and goladhyaya. He is the first astronomer to consider Celestial Sphere.
9. GANITHA SARA SANGRHA of Mahaveracharya: Mahaveeracharya was a 9th century mathematician, who wrote Ganithsarasangraha. This book deals with Imaginary numbers, LCM, combinations, solutions to algebraic equations, properties of cyclic quadrilaterals etc.
10: Bhaskaracharya II (: The most popular Indian Astronomer of Twelfth century. His celebrated work, SIDDHANTHA SIROMANI consists of four parts namely
Lilavathy- deals with arithmetic, elementary algebra, Geometry and mensuration
Bijaganitham, - deals with advanced algebra
Grahaganitham,
Goladhayam – deals with computations of planetary motions, eclipses, and spherical astronomy.
11.Binary System of PINGALA:
Chandasutram by Maharishi Pingala contains 18 parichedas (sub-chapters) in 8 adhyayas (Main chapters)
The first parichcheda of six slokas are not sutras. The rest of Chandasutram is composed of sutras.
MaaYaRaaSa TaaJaBhaa Na La GaSammitamBhramativangamayamJagatiyasys.
SajayathiPingalaNagah Siva prasadatvisudhamatih
Gana Pingala Boolean Index
Maa GGG 000 1
Ya LGG 100 2
Raa GLG 010 3
Sa LLG 110 4
Taa GGL 001 5
Ja LGL 101 6
Bhaa GLL 011 7
Na LLL 111 8
The ancient Indians are pioneering in mathematical works.
The above list is partial and nowhere nearer to completeness.
A short list of mathematical works that ancient Indians are pioneered.
1. Decimal Number system and Place value system
2. The number zero.
3. Approximation for square roots 2 and 3
4. Pythagoras theorem
5. Diophantine equations like Pell’s Equation
6. Binary Arithmetic
7. Permutations and Combinations
8. Pascal’s Triangle
9. Trigonometric Functions
10. Calendar Making
11. The value of Pi
12. Infinite series.
VEDAS are the earliest records of human wisdom, which are being handed down to humanity through oral tradition of knowledge transmission. Basing on the astronomical references available in the Vedic literature we can say the age of VEDAS to be 6000BC – 19000 BC.
Digits and the fractions in VEDAS:
The concept of digits, to the several millions can be clearly seen in all VEDAS.
We take an example from YAJUR VEDA: The existence of bigger numbers in series can be seen in Yajur Veda
ek(1) cha dash (10) cha shatama (10^2)
chashatam(10^2) cha sahastrama (10^3)
chasahastram (10^3) cha ayutam(10^4)
chaayutam(10^4) cha niyutam (10^5)
chaniyutam(10^5) cha prayutam (10^6)
chaarbudam (10^8) cha nyarbudam
chasamudrascamadhyam (10^16)
chaantashchparaddashch (10^17)
yetah meh agneishtikahdenavahsantuamrutramushminloke
(yajurveda/ adhyaya 17 / kandika 2 )
This hymn indicates a series of numbers which starts with 1, the next number being 10, 100, 1000 ….…. Up to 10^17, of course in the form of a geometric progression
Fractions can be clearly understood in the following Rig Vedic hymn
jyesthaahchamasadva (1/2)
karotikaniiyantreen (1/3)
kunavametyahkanishtaahchaturaskaroti (1/4) tvashtribhavastatpanayadvachovah
(rigveda/mandal 4/sukta 33/ mantra 5)
The hymn speaks of a glass of drink in to two, three, four parts , however whether they had a system of representing the fractions in the form of a numerator and a denominator is uncertain.
Geometry in VEDAS:
The simplest form of polygon can be taken as triangle since 3 is the minimum number of sides with which a polygon can be drawn. The name of the triangle (Tribhuja) can be seen in the following lines from AtharvanaVeda, in a non mathematical context.
yoakrandayatsalilammahitvayonimkrutvaatribhujamshayanahvatsahkaamdugoviraajahsaguhachakretanvahparachiah
The Almighty made a triangle (Tribhuja) with earth , the intermediate space and the heaven as its sides and the generated multipleforms of beings, in a mystic way.
Sahastra sheersha purushahsahasrakshahsahasrapat
sabhumimavikshvatovrutvaatyatishtddshangulam
Every point on the circumference of a circle can be treated as its vertex. Thus the circumference of a circle is a set of infinite points. This is the sahasrashirsha property of a circle.
The diameter of a circle can be called as its axis. The axis
of symmetry of the circle is any of its diameter is its sahasrakshat.
A circle can be considered as an infinite sided polygon. This is its sahasrapadat
Note: “Mathematics in Vedas” is something different from “Vedic Mathematics”
“Vedic Mathematics” is introduced by HIS HOLYNESS SWAMI BHARTHI KRISHNA THIRTHAJI (whose is named as venkat raman by his parents)(1884-1960)
Swamiji , who was an accomplished vedic scholar wrote 16 volumes on vedic mathematics, comprehensively in all branches of mathematics, But it is learnt that all volumes were lost mysteriously. Despite of weak eyesight Swamiji with his untiring capacity, strong will and determination wrote a comprehensive book on vedic mathematics.
Vedic mathematics is not just some tricks of doing mathematical calculations quickly, it goes far beyond this and becomes science in itself beneficial in solving problems related to higher mathematics i.e. solutions of simultaneous equations, determinants, transcendental equations, solutions of cubic and higher order equations, solutions of differential equations etc.
Ancient Indian Mathematicians - Mathematics
Let us have a quick glance of list of Ancient Indian mathematicians and their contributions.
1. Lagadha (1350 BC): The ancient Indian astronomer wrote the book titled VedangaJyothisyam. So many wonders of mathematical and astronomical ideas are found in that small book
kalaadashacavimshaasyaat
dvimuhuurtastunaadike. …
naadike dvi muhuurtastu
THE TABLE OF CADIVISONS OF TIME PROPOSED IN HIS BOOK
5 guruvaaksharas = 1 kaashthaa
124 kaashtaas = 1kalaa
(10 + 1/20) kalaas = 1 naadikaa
2 naadikaas = 1 muhurta
30 muhurtas = 1 day
61 days = 1 ritu
3 ritus = 1 ayana
2 ayanas = 1 year
5 years = 1 yuga
This book is written in order to help to determine the actual time of performance of vedic rituals. In the he streamline many mathematical operations and their results.
Lagadha is a leader, with many followers, interpreters, commentators admirers of his mathematical work.
2. SULVA SUTRAS BY BAUDHAYANA
The very purpose of sulvasuthras is to provide technical assistance to design construct VEDIS AND CHITIS (fire altars) for successful Yajnas.
They have discussed on basic geometrical concepts, ideas, properties, propositions. Further we can observe the use of fractions and surds, indeterminate equations.
The time period of sulvasuthras is estimated at around 800BC
These sutras’ explained about mensuration details of 3D objects and Pythagoras theorem.
3. Aryabhata:(476BC-550BC): he is great Indian mathematician and astronomer. He accurately measured the period of earth rotation with respect to the fixed stars in the sky, deduced the shape of earth and its rotation, and made many remarkable achievements in astronomy and mathematics. He gave the formula to calculate the value of PI.
4 Varahamira( 6th century AD): His main astronomical work“PANCHSIDDHANTIKA” as observed early the five systems of astronomy are discussed in this. He gave tables using the formulae R sinθ and R cosθ of the angles 0 to 900 with the intervals3045’. His contributions to combinations are also important.
5.Bhramaguptha: He is the great astronomer and mathematician belongs to fifth century AD. He wrote three texts (1) Bhrama sputa Siddhantha (2) KhandKhadyaka (3) Uttar Kanda-khadyaka. Bhramsputhasiddhantha is the standard and basic text in astronomy. In this book he discussed motions of celestial bodies, their speed, solar and lunar eclipses etc. Also discussed the progressions, permutations,algebra-quadraticpolynomials, properties of cyclic quadrilaterals.
6. BhaskaraI(600 to 680AD) : He wrote two treatise “Mahabhaskariyam” and “laghubhaskariyam” dealing with Mathematical Astronomy. He gave the tables of sine function upto two correct decimals. He also discussed the equations just similar to Pell’s equations of modern period. He is also having credit for numbers and symbolism, classifications of equations.
7.Sridharacharya(750AD)There are two books available, which are written by sridharacharya (1) Trisathika (2) Pathiganitha
His contributions in mathematics are as follows
(1) Place names of numbers in decimal system
(2) Definition and properties of zero
(3) Use of symbol “+”
(4) Methods of multiplication of numbers
(5) Square root and cube root of any number
(6) Sum of the cubes and squares of the terms of Athematic progression
(7) Rational solutions of the equations of the form Nx2+1 = y2
8. Lallacharya (720AD-790AD): Lallacharya was one of the leading and eminent mathematicians, astronomer and astrologers of eighth century. His notable works are SHISHYADHIDATANTRA – A work on astronomy in two volumes namely Grahadhya and goladhyaya. He is the first astronomer to consider Celestial Sphere.
9. GANITHA SARA SANGRHA of Mahaveracharya: Mahaveeracharya was a 9th century mathematician, who wrote Ganithsarasangraha. This book deals with Imaginary numbers, LCM, combinations, solutions to algebraic equations, properties of cyclic quadrilaterals etc.
10: Bhaskaracharya II (: The most popular Indian Astronomer of Twelfth century. His celebrated work, SIDDHANTHA SIROMANI consists of four parts namely
Lilavathy- deals with arithmetic, elementary algebra, Geometry and mensuration
Bijaganitham, - deals with advanced algebra
Grahaganitham,
Goladhayam – deals with computations of planetary motions, eclipses, and spherical astronomy.
11.Binary System of PINGALA:
Chandasutram by Maharishi Pingala contains 18 parichedas (sub-chapters) in 8 adhyayas (Main chapters)
The first parichcheda of six slokas are not sutras. The rest of Chandasutram is composed of sutras.
MaaYaRaaSa TaaJaBhaa Na La GaSammitamBhramativangamayamJagatiyasys.
SajayathiPingalaNagah Siva prasadatvisudhamatih
Gana Pingala Boolean Index
Maa GGG 000 1
Ya LGG 100 2
Raa GLG 010 3
Sa LLG 110 4
Taa GGL 001 5
Ja LGL 101 6
Bhaa GLL 011 7
Na LLL 111 8
The ancient Indians are pioneering in mathematical works.
The above list is partial and nowhere nearer to completeness.
A short list of mathematical works that ancient Indians are pioneered.
1. Decimal Number system and Place value system
2. The number zero.
3. Approximation for square roots 2 and 3
4. Pythagoras theorem
5. Diophantine equations like Pell’s Equation
6. Binary Arithmetic
7. Permutations and Combinations
8. Pascal’s Triangle
9. Trigonometric Functions
10. Calendar Making
11. The value of Pi
12. Infinite series.
Friday, November 26, 2010
“Target Centum in X class Mathematics”
BEST PRACTICES FOR HIGH IMPACT:
3. During Revision
2. Before the Day of examination
1. During examination
SIX PRINCIPLES OF EFFECTIVE EXAM-PAPER PRESENTATION (During examination)
1. ENRICH YOUR CONTENT
2. SYSTEMATIC ALLIGNMENT
3. CORDIAL COMMUNICATION – i.e. Meaning full connectivity between steps
4. COMPLETENESS OF THE FORMULAE / DEFINITIONS
5. TEXT HIGH LIGHTING – USAGE OF COLOUR- PENS (SKETCHS?!?)/ SPECIAL BOXES
6. SCIENTIFIC RECHECK OF THE ANSWER PAPER
Before THE DAY OF EXAMINATION
1. Practice thoroughly the problems from the CHECK LIST
2. Read the highlighted parts from your text book
3. Answer Three To Four Bit Papers
4. Special attention to the two mark and one mark questions
During Revision
1. START to STAR
2. Objective defined Practice Hour
3. Reading the Theory content from the Text Book/notes underlining
4. Preparing CHECK LIST of PROBLEMS
5. Practice bits minimum of 15 model papers along with textual bits
All the best
3. During Revision
2. Before the Day of examination
1. During examination
SIX PRINCIPLES OF EFFECTIVE EXAM-PAPER PRESENTATION (During examination)
1. ENRICH YOUR CONTENT
2. SYSTEMATIC ALLIGNMENT
3. CORDIAL COMMUNICATION – i.e. Meaning full connectivity between steps
4. COMPLETENESS OF THE FORMULAE / DEFINITIONS
5. TEXT HIGH LIGHTING – USAGE OF COLOUR- PENS (SKETCHS?!?)/ SPECIAL BOXES
6. SCIENTIFIC RECHECK OF THE ANSWER PAPER
Before THE DAY OF EXAMINATION
1. Practice thoroughly the problems from the CHECK LIST
2. Read the highlighted parts from your text book
3. Answer Three To Four Bit Papers
4. Special attention to the two mark and one mark questions
During Revision
1. START to STAR
2. Objective defined Practice Hour
3. Reading the Theory content from the Text Book/notes underlining
4. Preparing CHECK LIST of PROBLEMS
5. Practice bits minimum of 15 model papers along with textual bits
All the best
Tuesday, October 19, 2010
special properities of quadrilaterals
Nature of the diagonals -----------------------------Type of quadrilaterals
Diagonals are equal --------------------------------------Need not be rectangle or square
Diagonals are perpendicular ---------------------------Need not be a rhombus
Diagonals are perpendicular & equal------------------ Need not be a rhombus, rectangle or square
Diagonals bisect each other -----------------------------Parallelogram
Diagonals bisect & perpendicular----------------------- Rhombus
Diagonals bisect & equal ----------------------------------Rectangle
Diagonals bisect, perpendicular & equal ---------------Square
Name of the quadrilateral whose midpoints are joined-------------The Quadrilateral formed
Quadrilateral---------------------------------------------------------------------------- Parallelogram
Parallelogram ---------------------------------------------------------------------------Parallelogram
Trapezium ------------------------------------------------------------------------------Parallelogram
Rectangle ______________________________----------------------------------Rhombus
Rhombus ---------------------------------------------------------------------------------Rectangle
Square---------------------------------------------------------------------------------------- Square
Diagonals are equal --------------------------------------Need not be rectangle or square
Diagonals are perpendicular ---------------------------Need not be a rhombus
Diagonals are perpendicular & equal------------------ Need not be a rhombus, rectangle or square
Diagonals bisect each other -----------------------------Parallelogram
Diagonals bisect & perpendicular----------------------- Rhombus
Diagonals bisect & equal ----------------------------------Rectangle
Diagonals bisect, perpendicular & equal ---------------Square
Name of the quadrilateral whose midpoints are joined-------------The Quadrilateral formed
Quadrilateral---------------------------------------------------------------------------- Parallelogram
Parallelogram ---------------------------------------------------------------------------Parallelogram
Trapezium ------------------------------------------------------------------------------Parallelogram
Rectangle ______________________________----------------------------------Rhombus
Rhombus ---------------------------------------------------------------------------------Rectangle
Square---------------------------------------------------------------------------------------- Square
A NEW CLASSIFICATION OF NUMBERS
By
P. G. RAJESWARA RAO
p_grrao@yahoo.co.nz
We can’t live without numbers. We need them in our daily activities – big and small.
I would like to share with you some special classification that I have observed with natural numbers
Take any natural number and find its digit sum square it and find the digit sum of the square. If we continue the procedure the outcome is as follows:
Take 22
Sum of digits = 4
Square of 4 = 16
Sum of digits = 7
Square of 7 = 49
Sum of digits = 13
Square of 13 = 169
Sum of digits = 16
Square of 16 = 256
Sum of digits = 13
Square of 13 = 169
22 , 4 , 16 , 7 , 49 , 13 , 169 , 16 , 256 , 13 , 169
The series repeats from 13 to 256 continuously
Take 35
35 , 8 , 64 , 10 , 100 , 1 , 1 , 1 , 1 , ……………………………….
It continues with one only
Take 75
75 , 12 , 144 , 9 , 81 , 9 , 81 , 9 , 81……………………………….
It continues with 9 and 81 alternatively
According to my observation all natural numbers came under any of these three types
So the numbers can be classified into three categories based on the special property
1) C TYPE : Cyclic type which will have repeating series of 13 – 256
2) N TYPE : These are the series of numbers which have 9 , 81 as repeating numbers
3) O TYPE : these are the series if numbers which will have only one recurring
From the above examples
22 is C TYPE
35 is N TYPE
75 is O TYPE
SOME SPECIAL OBSERVATIONS
1) Among the single digit numbers O type are 2, N type are 3, C type are 4.
2) Among the double digit numbers O type are 20, N type are 30, C type are 40.
3) N TYPE has no primes.
4) O TYPE has 7 primes among which 2 pairs are twin primes.
5) C TYPE numbers have 14 primes
6) Multiples of 3 , 6 , 9 belong to N TYPE
7) Multiples of 2 , 4 , 5 , 7 , 8 are distributed in the three categories
8) If a number belongs to one category then the sum of that number and nine multiples belongs to same category
9) This type of classification can also be done with cubes and digit sum there also three categories will exist.
P. G. RAJESWARA RAO
p_grrao@yahoo.co.nz
We can’t live without numbers. We need them in our daily activities – big and small.
I would like to share with you some special classification that I have observed with natural numbers
Take any natural number and find its digit sum square it and find the digit sum of the square. If we continue the procedure the outcome is as follows:
Take 22
Sum of digits = 4
Square of 4 = 16
Sum of digits = 7
Square of 7 = 49
Sum of digits = 13
Square of 13 = 169
Sum of digits = 16
Square of 16 = 256
Sum of digits = 13
Square of 13 = 169
22 , 4 , 16 , 7 , 49 , 13 , 169 , 16 , 256 , 13 , 169
The series repeats from 13 to 256 continuously
Take 35
35 , 8 , 64 , 10 , 100 , 1 , 1 , 1 , 1 , ……………………………….
It continues with one only
Take 75
75 , 12 , 144 , 9 , 81 , 9 , 81 , 9 , 81……………………………….
It continues with 9 and 81 alternatively
According to my observation all natural numbers came under any of these three types
So the numbers can be classified into three categories based on the special property
1) C TYPE : Cyclic type which will have repeating series of 13 – 256
2) N TYPE : These are the series of numbers which have 9 , 81 as repeating numbers
3) O TYPE : these are the series if numbers which will have only one recurring
From the above examples
22 is C TYPE
35 is N TYPE
75 is O TYPE
SOME SPECIAL OBSERVATIONS
1) Among the single digit numbers O type are 2, N type are 3, C type are 4.
2) Among the double digit numbers O type are 20, N type are 30, C type are 40.
3) N TYPE has no primes.
4) O TYPE has 7 primes among which 2 pairs are twin primes.
5) C TYPE numbers have 14 primes
6) Multiples of 3 , 6 , 9 belong to N TYPE
7) Multiples of 2 , 4 , 5 , 7 , 8 are distributed in the three categories
8) If a number belongs to one category then the sum of that number and nine multiples belongs to same category
9) This type of classification can also be done with cubes and digit sum there also three categories will exist.
Sunday, October 3, 2010
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