Welcome To

NAMASTE LECTURE SERIES

Welcome To

NAMASTE LECTURE SERIES

2009

Presents

Something Something NonmathematicalNonmathematical

and and Something Something

MathematicalMathematical

Something Something NonmathematicalNonmathematical

and and Something Something

MathematicalMathematical

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 .

COMMITTEE OF NAMASTE COORDINATORS COMMITTEE OF NAMASTE COORDINATORS 1.1. 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 President (Former), Nepal Mathematical Society, Tribhuvan University

8. Prof. Dr. Siddhi Prasad Koirala Chairman , Higher Secondary School Board, Secondary School Board))

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, andTo work for the implementation of such changes.

NAMASTE's NAMASTE's main objectives aremain objectives are

NAMASTE DOCUMENTS* Mathematics Awareness Movement (MAM)

Advocacy Strategy (A Draft for Preliminary Discussion)

* Mathematics Education for Early ChildhoodDevelopment

(A Discussion paper)

* The Lichhavian Numerals and

The Changu Narayan Inscription

W H E R E

D O

W E

C O M E

F R O M ?

NO MANNO MAN

NO COUNTINGNO COUNTING

ANDAND NO MATHEMATICSNO MATHEMATICS

The universe emerged from a tremendously

dense and hot state about 13.7 billion years ago.

The big bang is often explained using the image

of a two dimensional

universe (surface of a balloon) expanding in

three dimensions

SCIENTIFIC NOTATIONSSCIENTIFIC NOTATIONS

Number Number Exponential FormExponential Form Symbol Symbol PrefixPrefix

1,000,000,000,0001,000,000,000,000 10101212 TT teratera

1,000,000,0001,000,000,000 101099 GG gigagiga

1,000,0001,000,000 101066 MM megamega

1,0001,000 101033 kk kilokilo

11 101000

0.010.01 1010-2-2 cc centicenti0.0010.001 1010-3-3 mm millimilli

0.0000010.000001 1010-6-6 Greek mu)Greek mu) micromicro0.0000000010.000000001 1010-9-9 nn nanonano

0.0000000000010.000000000001 1010-12-12 pp picopico

Age of the Universe in Years :Age of the Universe in Years :

SHAPE OF THE UNIVERSE

Angle sum > 180 degree

Angle sum < 180 degree,

Angle sum = 180 degree,

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

ARE WE ALONE IN THE UNIVERSE?ARE WE ALONE IN THE 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.

Finite non-expanding universe

Hundreds of

Thousands of Stars

200000000000

Our Galaxy

STARS

Age13,600 ± 800 million years

B L A C K H O L E

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 (?)

Age4.560 million years

Distance between the Earth and the Sun

149598000 km

Distance between the Earth and the Sun

149598000 km

Born4,560 million years ago

Rotation and Revolution of the Earth

Born 4.5 billion years ago

OR

THE WORLD

ANCIENT CIVILIZATIONS

WORLD CIVILIZATIONS

Nepal

The land where a well developed number system existed as

early as the beginning of the first

millennium CE.

107 AD

What Do We

KnowAboutOur

AncientNumbers ?

BRAMHI SCRIPTBRAMHI SCRIPTININ

ASHOKA STAMBHA INSCRIPTION (249 BCE)ASHOKA STAMBHA INSCRIPTION (249 BCE)LUMBINI, NEPALLUMBINI, NEPAL

Number Words In

The Brahmi Script InscriptionOf

Ashoka Stambha (249 BCE),Lumbini

Brahmi Script Devanagari Script

jL;

c7-efluo]_read as read as

Brahmi Numerals

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

Some Numerals In Some Other Ancient Inscriptions

First Phase : Numerals for 4, 6, 50 and 200

No numeral for 5 but for 50Second 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 MoreAbout

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.

Numerals in

Ancient Nepal

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

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

WHY DO WE FOCUS ON THE NUMBERS

Lichhavian Number 1 to Lichhavian Number 1 to 9999

Lichhavian Numbersand

Major Number Systems

Table I(A)

Numbers

Brahmi Chinese Lichhavian Tocharian

100200300400500

Table I(B)

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 Something About Lichhavian Number Lichhavian Number

SystemSystem

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 Something About Lichhavian Number Lichhavian Number

SystemSystem

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 Something About Lichhavian Number Lichhavian Number

SystemSystem

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 hundrednumeral for 5 attached to symbol for 100 by a ligature stands for 500 to be read as 5 times 1 hundrednumeral 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 Something About Lichhavian Number Lichhavian Number

SystemSystem

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.

Something About Something About Lichhavian Number Lichhavian Number

SystemSystem

COLLECTION CLASSIFICATION

COMPREHENSION

MANIPULATION MANIFESTATION MYSTIFICATION

Saka 107 (185 AD)Saka 107 (185 AD) or or

Saka 207 (285 AD) Saka 207 (285 AD)

Saka 386 Saka 386 Interpreted Interpreted

by asby as

Babu Ram (Nepali)Babu Ram (Nepali) 464 AD464 AD

Bhagwan Lal (Indian)Bhagwan Lal (Indian) 329 AD329 AD

Levi (French)Levi (French) 496 AD496 AD

Flit (British)Flit (British) 705 AD 705 AD

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

Three unit symbols attached to the symbol

One six unit symbol attached to the symbol is being interpreted as 600

NOT REPORTED SO FARNOT REPORTED SO FAR

SEQUENCIAL GAP AND

INCONSISTENCY

Two unit symbols attached to the symbol is being interpreted as 500 also

One four unit symbol attached to the symbol is being interpreted as 400

One five unit symbol attached to the symbol is being interpreted as 500 also

MANIPULATION, MANIFESTATION, MYSTIFICATION

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

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

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 ?CRITICAL ISSUES ?

CRITICAL ISSUES ?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?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 ? ”

WHAT IS TO BE DONE?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 Indrajit464 AD by Babu Ram Acharya496 AD by Levi705 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 ?”

WHAT IS TO BE DONE?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:

Hindu-Hindu-ArabicArabic

BabylonianBabylonian ChineseChinese Egyptian Greek RomanEgyptian Greek Roman Nepali Nepali

100100

200200

300300

400400

500500

600600

A BA B