1. Welcome To NAMASTE LECTURE SERIES
2009

2. GARUD AT CHANGU NARAYAN TEMPLE
gd:]t
NAMASTE ?
GARUD AT CHANGU NARAYAN TEMPLE

3. Something Nonmathematical and Something Mathematical
NAMASTE'S
NEW NEPAL
MATHS CENTRE
Presents
Something Nonmathematical and Something Mathematical

4. SOMETHING NONMATHEMATICAL
AND
SOMETHING MATHEMATICAL

5. AFTERNOON
DAY
GOOD
MORNING

6. PART
ZERO

7. A Non-profit Service Team
NAMASTE
At a Glance
NAMASTE
National Mathematical Sciences Team
A Non-profit Service Team
Dedicated to
MAM
Mathematics Awareness Movement
Established: March 22, 2005, at the premise of NAST .

8. COMMITTEE OF NAMASTE COORDINATORS
1. Prof. Dr. Bhadra Man Tuladhar
Mathematician (Kathmandu University)
2. Prof. Dr. Ganga Shrestha
Academician (Nepal Academy of Science and Technology)
3. Prof. Dr. Hom Nath Bhattarai,
Vice Chancellor (Nepal Academy of Science and Technology)
4. Prof. Dr. Madan Man Shrestha,
President, (Council for Mathematics Education)
5. Prof. Dr. Mrigendra Lal Singh
President, Nepal Statistical Society
6. Prof. Dr. Ram Man Shreshtha
Member Secretary (Namaste)
7. Prof. Dr. Shankar Raj Pant
President (Former), Nepal Mathematical Society, Tribhuvan University
8. Prof. Dr. Siddhi Prasad Koirala
Chairman , Higher Secondary School Board, Secondary School Board)

9. NAMASTE's main objectives are
To launch a nationwide
Mathematics Awareness Movement
in order to convince the public in recognizing the need for better mathematics education for all children,
To initiate a campaign for
the recruitment, preparation, training and retaining teachers with strong background in mathematics,
To help promote
the development of innovative ideas, methods and materials in the teaching, learning and research in mathematics and mathematics education,
To provide a forum for free discussion on all aspects of mathematics education,
To facilitate the development of consensus among diverse groups with respect to possible changes, and
To work for the implementation of such changes.

10. NAMASTE DOCUMENTS * Mathematics Awareness Movement (MAM)
Advocacy Strategy
(A Draft for Preliminary Discussion)
* Mathematics Education for Early Childhood
Development
(A Discussion paper)
* The Lichhavian Numerals
and
The Changu Narayan Inscription

11. PART ONE

12. W H E R E D O W E C O M E F R O M ?

13. AGO
LONG
LONG
NO MAN
NO MATHEMATICS
AND
NO COUNTING

14. LONG BEFORE MAN CAME
The big bang is often explained using the image of a two dimensional universe (surface of a balloon) expanding in three dimensions
THEORY OF BIG BANG
Some Mathematical
The universe emerged from a tremendously dense and hot state about 13.7 billion years ago.

15. SCIENTIFIC NOTATIONS Age of the Universe in Years : Number
Exponential Form
Symbol
Prefix
1,000,000,000,000
1012 T
tera
1,000,000,000
109 G
giga
1,000,000
106 M
mega
1,000
103 k
kilo 1
100
0.01
10-2 c
centi
0.001
10-3 m
milli
10-6
m (Greek mu)
micro
10-9 n
nano
10-12 p
pico
Age of the Universe in Years :

16. SHAPE OF THE UNIVERSE Angle sum > 180 degree

17. Closed surface like a sphere, positive curvature, Finite in size but without a boundary, expanding like a balloon, parallel lines eventually convergent
Flat surface, zero curvature, infinite and no boundaries, can expand and contract, parallel lines always parallel
Saddle-shaped surface, negative curvature, infinite and unbounded, can expand forever, parallel lines eventually divergent

18. ARE WE ALONE IN THE UNIVERSE?
GENERAL BELIEF : NO
Finite non-expanding universe ?
With about 200 billion stars in our own Milky Way galaxy and some 50 billion other similar galaxies in the universe, it's hardly likely that our 'Sun' star is the only star that supports an Earth-like planet on which an intelligent life form has evolved.

19. MILKY WAY STARS WE COME FROM Age 13,600 ± 800 million years Hundreds
Our Galaxy
STARS
Age
13,600 ± 800 million years
Hundreds of
Thousands of Stars

20. B L A C K H O L E

21. Sun Mercury Venus Ea rth Mars Jupiter Saturn Uranus Neptune Pluto (?)
T H E S O L A R S Y S T E M
Sun Mercury Venus Ea rth Mars Jupiter Saturn Uranus Neptune Pluto (?)
Distance between
the Earth and the Sun km
Age
4.560 million years

22. SOLAR SYSTEM Distance between the Earth and the Sun
Born
4,560 million years ago
Distance between
the Earth and the Sun km

23. Rotation and Revolution of the Earth
SOLAR SYSTEM
Rotation and Revolution of the Earth

24. Born
4.5 billion years ago
Rotating Earth
EARTH

25. TECHTONIC MOVEMENT
LAURASIA
GONDAWANALAND

26. TECHTONIC MOVEMENT OR
CONTINENTS FORMATION

27. THE WORLD

28. ANCIENT CIVILIZATIONS

29. WORLD CIVILIZATIONS

30. INDUS CIVILIZATION
NEPAL

31. WE COME FROM NEPAL Nepal
The land where a well developed number system existed as early as the beginning of the first millennium CE.
WE COME FROM
107 AD
NEPAL
Maligaon Inscription

32. PART TWO

33. What Do We Know About Our Ancient Numbers ?

34. BRAMHI SCRIPT IN ASHOKA STAMBHA INSCRIPTION (249 BCE) LUMBINI, NEPAL
THE BEST KNOWN
AND
THE EARLIEST OF THE KIND

35. Brahmi Script Devanagari Script
Number Words In The Brahmi Script Inscription Of Ashoka Stambha (249 BCE),Lumbini
No Numerals
Brahmi Script Devanagari Script
jL;
c7-efluo]_
read as
read as

36. Brahmi Numerals
The best known Brahmi numerals used around 1st Century CE.

37. Some Numerals In Some Other Ancient Inscriptions
First Phase :
Numerals for 4, 6, 50 and 200
No numeral for 5 but for 50
Second Phase :
Numerals for 1, 2, 4, 6, 7, 9, 10, 20, 80, 100, 200, 300, 400, 700; 1,000; 4,000; 6,000; 10,000; 20,000.
No Numeral for 3 but for 300
Third Phase:
Numerals for 3, 5, 8, 40, 70, …, 70,000.

38. Hypotheses About The Origin of Brahmi Numerals
The Brahmi numerals came from the Indus valley culture of around 2000 BC.
The Brahmi numerals came from Aramaean numerals.
The Brahmi numerals came from the Karoshthi alphabet.
The Brahmi numerals came from the Brahmi alphabet.
The Brahmi numerals came from an earlier alphabetic numeral system, possibly due to Panini.
The Brahmi numerals came from Egypt.

39. Something More About Brahmi Numerals
The symbols for numerals from the Central Asia region of the Arabian Empire are virtually identical to those in Brahmi.
Brahmi is also known as Asoka, the script in which the famous Asokan edicts were incised in the second century BC.
The Brahmi script is the progenitor of all or most of the scripts of India, as well as most scripts of Southeast Asia.
The Brahmi numeral system is the ancestor of the Hindu-Arabic numerals, which are now used world-wide.

40. EPIGRAPHY VERSUS VEDIC MATHEMATICS
Total lack of Brahmi and Kharoshthi inscriptions of the time before 500 BCE
Much of the mathematics contained within the Vedas is said to be contained in works called Vedangas.
Vedic Period : Time before 8000 /1900/ BCE etc.
Vedangas period: 1900 – 1000 BCE.
Sulvasutras Period : BCE.
Origin of Brahmi script : Around 3rd century BCE
No knowledge of existence of any written script during the Ved- Vedangas period.
Numerical calculation based on numerals(?) during the so-called early Vedic period highly unlikely.

41. translated by Kashinath Tamot and Ian Alsop:
Numerals in
Ancient Nepal
MALIGAON INSCRIPTION
Read as
“sam*vat a7 gri- pa 7 d(i)va pka maha-ra-jasya jaya varm(m)a(n*ah*)”
and
translated by Kashinath Tamot and Ian Alsop:
“(In) the (Shaka) year 107 (AD 185), (on) the 4th (lunar) day of the 7th fortnight of the summer (season), of the great King Jaya Varman
Read as Saka Samvat 107(185 AD by Tamot, Alsop, Rajbanshi; and as 207(285 AD) by Garbini

42. WHY DO WE FOCUS ON THE NUMBERS?
in the Maligaon inscription
in the Changu Narayan inscription
Earliest of the available number-symbols.
Concrete evidences of the knowledge of the concept of number and the existence of numerals and a well-developed number system in Nepal at a time (around 2nd century CE) when a civilization like Greek civilization worked with very primitive or alphabetic numerals
Beginning of the recorded history of ancient Nepal

43. Lichhavian Number 1 to 99

44. Lichhavian Numbers and Major Number Systems

45. Table I(A) Table I(B) Numbers Brahmi Chinese Lichhavian Tocharian 100
200
300
400
500

46. Something About Lichhavian Number System
Lichhavian numerals for 1, 2 and 3 consist of vertically placed 1, 2 and 3 horizontal strokes like the Chinese 14th century BCE numerals, Brahmi numerals of the 1st century CE and Tocharian numerals of the 5th century CE.
The Lichhavian numerals for 1, 2, 3, 40, 80 and 90 look somewhat similar to the corresponding Brahmi numerals.
There is a striking resemblance between the Lichhavian and Tocharian numerals for, 1, 2, 3, 20, 30, 80 and 90; just like many Tocharian albhabet.
Several other Tocharian numbers appear to be some kind of variants of the Lichhavian numbers.
Each of the three systems uses separate symbols for the numbers, 1, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 1000, … .

47. Something About Lichhavian Number System
Compound numbers like 11, 12, …, 21, 22, …, 91, 92, … are represented by juxtaposing unit symbols without ligature.
Hundred symbol is represented by different symbols and is often used with and without ligature .
Non-uniformity in the process of forming hundreds using hundred symbol and other unit symbols.
Several variants of numerals are found during a period of several centuries.

48. Something About Lichhavian Number System
Each of the three systems uses separate symbols for the numbers, 1, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 1000, …
Compound numbers like 11, 12, …, 21, 22, …, 91, 92, … are represented by juxtaposing unit symbols without ligature.
Hundred symbol is represented by different symbols and is often used with and without ligature .
Non-uniformity in the process of forming hundreds using hundred symbol and other unit symbols.
Several variants of numerals are found during a period of several centuries.
Available Lichhavian numbers are lesser than 1000.
No reported number lies between 100 and 109, 201 and 209, …, 900 and 909. Numbers for 101, 102, …, 109, 201, 202, …, 209, …, 901, 902, …, 909 are missing .

49. Something About Lichhavian Number System
Arithmetic of Lichhavian is not known
Formation of two and three indicate vertical addition, while formation of 11, 12, …, indicate horizontal addition in the expanded form – a kind of horizontal addition.
Lichhavian system is additive
Lichhavian system is a decimal system.
Liichhavian system is multiplicative:
numeral for 4 attached to symbol for 100 by a ligature stands for 400 to be read as 4 times hundred
numeral for 5 attached to symbol for 100 by a ligature stands for 500 to be read as 5 times 1 hundred
numeral for 6 attached to symbol for 100 by a ligature stands for 600 to be read as 6 times 1 hundred,
Existence of some kind of arithmetic in Tocharian number system may provide some clue in this direction.

50. Something About Lichhavian Number System
Lichhavian numbers like
462
is to be read as
4 times hundred or 4 hundreds and 1 sixty and 2 ones
or, (100) 1(60) 2(1) = 4   1
and
4 times hundred or 4 hundreds and 1 sixty and 1 nine
or, (100) 1(60) 1(9) = 4   9.

51. COLLECTION
CLASSIFICATION
COMPREHENSION
MANIPULATION
MANIFESTATION
MYSTIFICATION

52. CHANGU NARAYAN INSCRIPTION
MALIGAON INSCRIPTION
Saka 107 (185 AD) or
Saka 207 (285 AD)
CHANGU NARAYAN INSCRIPTION
Interpreted
by as
Read as Saka Samvat 107(185 AD by Tamot, Alsop, Rajbanshi; and as 207(285 AD) by Garbini
Saka 386
Babu Ram (Nepali)
464 AD
Bhagwan Lal (Indian)
329 AD
Levi (French)
496 AD
Flit (British)
705 AD

53. SEQUENCIAL GAP AND INCONSISTENCY LIGATURE NOT REPORTED SO FAR
One unit symbol attached to the symbol
is being interpreted as 200
Two unit symbols attached to the symbol
is being interpreted as 300
One five unit symbol attached to the symbol
is being interpreted as 500
LIGATURE
One six unit symbol attached to the symbol
is being interpreted as 600
NOT REPORTED SO FAR
Three unit symbols attached to the symbol

54. How to justify such ambiguous interpretations?
MANIPULATION, MANIFESTATION, MYSTIFICATION
One four unit symbol attached to the symbol
is being interpreted as 400
Two unit symbols attached to the symbol
is being interpreted as 500 also
One five unit symbol attached to the symbol
is being interpreted as 500 also
How to justify such ambiguous interpretations?

55. One Possible Solution
Adopt a uniform system in which the hundred symbol attached to one of the first nine numbers is considered as the next hundred: e.g.,
as 200
as 300
as 500
as 600
as 700
1000 would look like
What is the symbol for 400 ? Naturally, it must look something like

56. Best Solution
Adopt an internationally accepted uniform system in which the hundred symbol attached to one of the first nine numbers is interpreted as the same hundred as the attached unit number : e.g.,
as 100
as 200
as 400
as 500
as 600
1000 would have a new symbol
What is the symbol for 300 ? Naturally, it must look something like

57. CRITICAL ISSUES ? in the number as 100 and as 200 by the epigraphers.
The interpretation of the number-symbol
in the number
as 100 and as 200 by the epigraphers.
The interpretation of the number
in the Changu Narayan inscription as
the number 386.

58. CRITICAL ISSUES ? The unfortunate interpretation of the same symbol
both as the number 300 as well as the number 500 by the same experts in a large number of inscription (as can be seen from the earlier slides).
The hesitation of a great section of epigraphers and ancient history of Nepal in rectifying their old interpretation of the number on the basis of a logical reason and the procedure followed by many ancient civilizations in forming such numbers.

59. WHAT IS TO BE DONE?
Since Changu Narayan Inscription is considered as the starting point for interpolating and extrapolating the ancient history; and hence that of the whole history of Nepal, the date
inscribed in the inscription and read even today as the number 386 needs a careful reexamination on the basis of various facts pointed so far.
We must first of all decide
“ Whether the Lichhavian number stands for
a) both 386 and 586
or, b) 386 only but not for 586
or, c) 586 only but not for 386
or, d) 286 ? ”

60. WHAT IS TO BE DONE?
Since the number of kings and the average period of the rule of known and unknown kings vary from expert to expert, the same process of interpolation and extrapolation of available information yield totally unacceptable imaginary inferences. This is further aggravated by interpretations of the Samvat 386 such as
329 AD by Bhagwan Indrajit
464 AD by Babu Ram Acharya
496 AD by Levi
705 AD by Flit.
In such a situation, we have to decide
“ Whether we have to change these dates, at least, to
229 AD, 364 AD, 396 AD and 605 AD ?”

61. WHAT IS TO BE DONE?
Collection of information, classification and comprehension become meaningless at a time when manifestation of unreasonable manipulation takes place in the form of obvious mystification as can be seen from the following table:

62. ANCIENT CIVILIZATIONS
HUNDREDS IN
ANCIENT CIVILIZATIONS
Hindu-Arabic
Babylonian
Chinese
Egyptian Greek Roman
Nepali
100
200
300
400
500
600
A B
A B

63. THANKS