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 Academician (Nepal Academy of Science and Technology) Member Secretary (Namaste)
7. Prof. Dr. Shankar Raj Pant 8. Prof. Dr. Siddhi Prasad Koirala Chairman , Higher Secondary School Board, Secondary School Board)
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.
NAMASTE DOCUMENTS * Mathematics Awareness Movement (MAM) Advocacy Strategy (A Draft for Preliminary Discussion) Development (A Discussion paper) * The Lichhavian Numerals and The Changu Narayan Inscription
NO MAN
SCIENTIFIC NOTATIONS
SHAPE OF THE UNIVERSE
ARE WE ALONE IN THE UNIVERSE?
WORLD CIVILIZATIONS
BRAMHI SCRIPT IN ASHOKA STAMBHA INSCRIPTION (249 BCE) LUMBINI, NEPAL
Number Words In The Brahmi Script Inscription Of Ashoka Stambha (249 BCE),Lumbini Brahmi Script Devanagari Script jL; c7-efluo]_
Brahmi Numerals The best known Brahmi numerals used around 1st Century CE.
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.
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.
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.
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 : 800 - 200 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.
WHY DO WE FOCUS ON THE NUMBERS
Lichhavian Number 1 to 99
Table I(A)
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, … .
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.
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
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 1 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.
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, 4(100) 1(60) 2(1) = 4100 + 1 60 + 2 1 and 469 is to be read as 4 times hundred or 4 hundreds and 1 sixty and 1 nine or, 4(100) 1(60) 1(9) = 4100 + 1 60 + 1 9.
COLLECTION COLLECTION CLASSIFICATION COMPREHENSION MANIPULATION MANIFESTATION MYSTIFICATION
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
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
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.
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.
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
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 229 AD, 364 AD, 396 AD and 605 AD ?”
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:
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