Motivation
Indian Numerals and Place Value System
Arithmetic
Ancient Indian Mathematics
Saurabh Panjwani
April 7, 2007
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Why learn about Ancient Indian Math today?
I
Many classical mathematical ideas were invented in India in
the Chriastian era.
I
I
Concept of zero, decimal system, place value, fundamental
theorems in geometry, arithmetic, algebra born in India.
Points of pride for us Indians :)
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Why learn about Ancient Indian Math today?
I
Many classical mathematical ideas were invented in India in
the Chriastian era.
I
I
I
Concept of zero, decimal system, place value, fundamental
theorems in geometry, arithmetic, algebra born in India.
Points of pride for us Indians :)
A lot of this is talked about, little is known
I
Even what we assume to know is often wrong.
I
I
I
e.g., Aryabhatta was not the one to invent the zero!
Did we ever learn this in school? Shouldn’t we teach it today?
Some new math for you (e.g., how quickly can you multiply?)
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Why learn about Ancient Indian Math today?
I
Many classical mathematical ideas were invented in India in
the Chriastian era.
I
I
I
Concept of zero, decimal system, place value, fundamental
theorems in geometry, arithmetic, algebra born in India.
Points of pride for us Indians :)
A lot of this is talked about, little is known
I
Even what we assume to know is often wrong.
I
I
I
I
e.g., Aryabhatta was not the one to invent the zero!
Did we ever learn this in school? Shouldn’t we teach it today?
Some new math for you (e.g., how quickly can you multiply?)
Learn about fascinating aspects of Indian culture
I
I
Mathematics was also rooted in religion
Inclination towards rote learning is in our blood..
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Numbers in Ancient India
Zero and Place Value
Numbers in Ancient India
I
10 used as basis of enumeration since Vedic times.
I
Hindus dealt comfortably with upto 18 denominations.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Numbers in Ancient India
Zero and Place Value
Numbers in Ancient India
I
10 used as basis of enumeration since Vedic times.
I
I
Hindus dealt comfortably with upto 18 denominations.
Could conceive extremely large numbers..
I
I
I
I
Yajurveda Samhita: paraardha = 1012 .
Lalitavistaara (5th Cent. BC): tallakshana = 1053 .
Pali grammar: asaankheya = 10140 .
Anuyogadvaara Sutra (A Jain text, 100BC): Seershaprahelikaa
= 10194 .
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Numbers in Ancient India
Zero and Place Value
Numbers in Ancient India
I
10 used as basis of enumeration since Vedic times.
I
I
Could conceive extremely large numbers..
I
I
I
I
I
Hindus dealt comfortably with upto 18 denominations.
Yajurveda Samhita: paraardha = 1012 .
Lalitavistaara (5th Cent. BC): tallakshana = 1053 .
Pali grammar: asaankheya = 10140 .
Anuyogadvaara Sutra (A Jain text, 100BC): Seershaprahelikaa
= 10194 .
.. and pretty small ones, too!
I
I
Artha Shaastra (4th Cent. BC): pramaanu ≈ 1.3 × 7−10 in.
1
Satapatha Braahmana (2000 BC): praana ≈ 17
th second.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Numbers in Ancient India
Zero and Place Value
Scripts
I
No clear evidence of when Indians started writing numbers.
I
I
Even mathematical and astronomial texts written in prose!
Historians surmise based on inscriptions and excavated data.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Numbers in Ancient India
Zero and Place Value
Scripts
I
No clear evidence of when Indians started writing numbers.
I
I
I
Even mathematical and astronomial texts written in prose!
Historians surmise based on inscriptions and excavated data.
Kharosthi script
I
right to left, no place value, brough in from Afghan.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Numbers in Ancient India
Zero and Place Value
Scripts
I
No clear evidence of when Indians started writing numbers.
I
I
I
Kharosthi script
I
I
Even mathematical and astronomial texts written in prose!
Historians surmise based on inscriptions and excavated data.
right to left, no place value, brough in from Afghan.
Brahmi script
National script of ancient Hindus (1000 BC). Some claim it to be
basis of Nagri script .
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Numbers in Ancient India
Zero and Place Value
Scripts
I
No clear evidence of when Indians started writing numbers.
I
I
I
Kharosthi script
I
I
Even mathematical and astronomial texts written in prose!
Historians surmise based on inscriptions and excavated data.
right to left, no place value, brough in from Afghan.
Brahmi script
National script of ancient Hindus (1000 BC). Some claim it to be
basis of Nagri script .
I
Nagri script
I
I
The mother of the modern decimal system; first one to
incorporate place value
Historians guess it was born in the 1st century BC.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Numbers in Ancient India
Zero and Place Value
Invention of Zero
I
Well accepted that both the concept and the symbol of zero
were invented in India.
I
“The importance of the invention of zero can never be
exaggerated. No single mathematical creation has been more
potent for the general on-go of intelligence and power.”
- Prof. Halsted
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Numbers in Ancient India
Zero and Place Value
Invention of Zero
I
Well accepted that both the concept and the symbol of zero
were invented in India.
I
The zero symbol
I
First occurrence in Chandaahsutra due to Pingala (< 200 BC).
I
Sometimes put on top of numbers to denote place value e.g. 3
◦
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Numbers in Ancient India
Zero and Place Value
Invention of Zero
I
Well accepted that both the concept and the symbol of zero
were invented in India.
I
The zero symbol
I
First occurrence in Chandaahsutra due to Pingala (< 200 BC).
I
Sometimes put on top of numbers to denote place value e.g. 3
◦
I
The zero concept
I
I
I
Bakhshaali manuscript (200 BC): first calculations involving 0.
First treatment as a digit: Pancha Siddhaantika (505 AD).
Sometimes used to depict an infinitesimal value e.g., when
computing limits
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Numbers in Ancient India
Zero and Place Value
Place Value
I
Earliest mention in Anuyogadvaara Sutra (100 BC):
I
“The total number of human beings in the world, when
expressed in terms of denominations, occupies 29 places ”
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Numbers in Ancient India
Zero and Place Value
Place Value
I
Earliest mention in Anuyogadvaara Sutra (100 BC):
I
I
“The total number of human beings in the world, when
expressed in terms of denominations, occupies 29 places ”
Also mentioned in Puranas
I
“In case of multiples from the units place, the value of each
place is ten times the value of the preceding place. ”
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Numbers in Ancient India
Zero and Place Value
Place Value
I
Earliest mention in Anuyogadvaara Sutra (100 BC):
I
I
Also mentioned in Puranas
I
I
“The total number of human beings in the world, when
expressed in terms of denominations, occupies 29 places ”
“In case of multiples from the units place, the value of each
place is ten times the value of the preceding place. ”
Historians: place value invented around 200 BC.
I
I
At this time, Romans and Greeks couldn’t write beyond 104 !
Babylonians did have place value, but base was 60. (:-?)
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Exposition and Teaching
I
Conciseness of exposition, especially in scientific matters, was
highly prized.
I
I
General mentality: compact is cool!
Hard to decipher their mathematical texts, even today.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Exposition and Teaching
I
Conciseness of exposition, especially in scientific matters, was
highly prized.
I
I
I
General mentality: compact is cool!
Hard to decipher their mathematical texts, even today.
Arithmetic done on dust, spread on ground or on a board.
(Use your fingers; get your hands dirty!)
I
I
dhooli-karma ∼ “dust” work; or
patti-ganita ∼ math done on a “board” (patti = board)
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Exposition and Teaching
I
Conciseness of exposition, especially in scientific matters, was
highly prized.
I
I
I
Arithmetic done on dust, spread on ground or on a board.
(Use your fingers; get your hands dirty!)
I
I
I
General mentality: compact is cool!
Hard to decipher their mathematical texts, even today.
dhooli-karma ∼ “dust” work; or
patti-ganita ∼ math done on a “board” (patti = board)
Teaching method emphasized rote learning
I
I
Three-step process of teaching math rules: memorize, practice,
and then, rationalize. (Many students didn’t reach last stage.)
Stress on oral education.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Birth and Death of mathematics
I
Religion was of supreme importance in Vedic times.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Birth and Death of mathematics
I
Religion was of supreme importance in Vedic times.
I
.. and religion necessitated study of astronomy.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Birth and Death of mathematics
I
Religion was of supreme importance in Vedic times.
I
.. and religion necessitated study of astronomy.
I
.. and astronomy required doing hard math. So there!
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Birth and Death of mathematics
I
Religion was of supreme importance in Vedic times.
I
.. and religion necessitated study of astronomy.
I
I
.. and astronomy required doing hard math. So there!
Scientific approach was not deemed a hinderance to spiritual
knowledge. Eventually, math fluorished for its own sake.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Birth and Death of mathematics
I
Religion was of supreme importance in Vedic times.
I
.. and religion necessitated study of astronomy.
I
I
I
.. and astronomy required doing hard math. So there!
Scientific approach was not deemed a hinderance to spiritual
knowledge. Eventually, math fluorished for its own sake.
Most mathematical research occurred upto 12th Cent. AD.
I
Post 12th Cent. AD, we’ve been busy grappling invaders.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
The Basic Operations
I
Addition and Subtraction
I
Two methods: direct (what we use today) and inverse (go
from left to right). Some regarded inverse as easier!
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
The Basic Operations
I
Addition and Subtraction
I
I
Two methods: direct (what we use today) and inverse (go
from left to right). Some regarded inverse as easier!
Multiplication: mentioned in Sulba Sutras (800 BC)
I
I
I
Six methods known: door-junction (kapaat sandhi),
comparment method, cross multiplication, zig-zag, by places
and by parts. (Various short-cuts practised.)
Dhuli-karma necessitated erasing and re-writing in calculations.
Shorter methods for squaring known.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
The Basic Operations
I
Addition and Subtraction
I
I
Multiplication: mentioned in Sulba Sutras (800 BC)
I
I
I
I
Two methods: direct (what we use today) and inverse (go
from left to right). Some regarded inverse as easier!
Six methods known: door-junction (kapaat sandhi),
comparment method, cross multiplication, zig-zag, by places
and by parts. (Various short-cuts practised.)
Dhuli-karma necessitated erasing and re-writing in calculations.
Shorter methods for squaring known.
Division: known to Indians before 4th Cent. AD
I
I
Removing common factors known even before that.
In Europe, division was believed to be hard till 15th Cent. AD
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
The Basic Operations
I
Square Root and Cube Root (yes! these were called basic
operations)
I
Aryabhatta explains method for finding both types of roots in
just 2 shlokas (Sanskrit verses) .
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
The Basic Operations
I
Square Root and Cube Root (yes! these were called basic
operations)
I
I
Aryabhatta explains method for finding both types of roots in
just 2 shlokas (Sanskrit verses) .
Checking by Nines (10th Cent. AD): A method to verify
correctness of all operations.
I
I
Sum the operands to single digit; sum the result
(product/quotient/..) to single digit; the operation performed
on these summed-up values should still be correct.
Necessary, not sufficient, condition!
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Fractions and Proportions
I
Mention of the fraction 3/8 in the Rig Veda (< 1000 BC)
I
I
Artha Shaastra mentions various fractional measures.
cf. Babylonians used fractions but only with unit numerators
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Fractions and Proportions
I
Mention of the fraction 3/8 in the Rig Veda (< 1000 BC)
I
I
I
Artha Shaastra mentions various fractional measures.
cf. Babylonians used fractions but only with unit numerators
Fractions were divided into classes.
I
There were no notations for basic operations! Instead,
notations existed for fraction classes like ba ± dc , ba of dc , etc.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Fractions and Proportions
I
Mention of the fraction 3/8 in the Rig Veda (< 1000 BC)
I
I
I
Fractions were divided into classes.
I
I
Artha Shaastra mentions various fractional measures.
cf. Babylonians used fractions but only with unit numerators
There were no notations for basic operations! Instead,
notations existed for fraction classes like ba ± dc , ba of dc , etc.
Basic operations easily extended to fractions. Division by 0
was not recognized.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Fractions and Proportions
I
Mention of the fraction 3/8 in the Rig Veda (< 1000 BC)
I
I
I
Fractions were divided into classes.
I
I
I
Artha Shaastra mentions various fractional measures.
cf. Babylonians used fractions but only with unit numerators
There were no notations for basic operations! Instead,
notations existed for fraction classes like ba ± dc , ba of dc , etc.
Basic operations easily extended to fractions. Division by 0
was not recognized.
The Rule of Three: The seed of today’s Unitary Method
I
“In the rule of three, multiply the phala (fruit) by the ichchha
(desire), and divide by the pramaana (argument). The result is
- Aryabhtta (499 AD)
the phala of the ichchha.”
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Fractions and Proportions
I
Mention of the fraction 3/8 in the Rig Veda (< 1000 BC)
I
I
I
Fractions were divided into classes.
I
I
I
Artha Shaastra mentions various fractional measures.
cf. Babylonians used fractions but only with unit numerators
There were no notations for basic operations! Instead,
notations existed for fraction classes like ba ± dc , ba of dc , etc.
Basic operations easily extended to fractions. Division by 0
was not recognized.
The Rule of Three: The seed of today’s Unitary Method
I
I
Referred to as the Golden Rule in Europe, later on.
Compound proportions (rule of 5, rule of 7, . . . ) known, too.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Other Notable Contributions
I
Besides arithmetic, much pioneering work was done in algebra
and geometry
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Other Notable Contributions
I
Besides arithmetic, much pioneering work was done in algebra
and geometry
I
Solving bivariate quadratic equations was known during
Aryabhatta’s time.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Other Notable Contributions
I
Besides arithmetic, much pioneering work was done in algebra
and geometry
I
I
Solving bivariate quadratic equations was known during
Aryabhatta’s time.
Aryabhatta (499 AD) made many fundamental contributions:
I
I
I
Computing the area of triangles, pyramids, trapeziums
Area of a circle! He first noted that π is transcendental.
His kuttaka method for solving diophantine equations
(ax ± by = c) used today in grad school math classes.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Other Notable Contributions
I
Besides arithmetic, much pioneering work was done in algebra
and geometry
I
I
Solving bivariate quadratic equations was known during
Aryabhatta’s time.
Aryabhatta (499 AD) made many fundamental contributions:
I
I
I
I
Computing the area of triangles, pyramids, trapeziums
Area of a circle! He first noted that π is transcendental.
His kuttaka method for solving diophantine equations
(ax ± by = c) used today in grad school math classes.
More trivia . . .
I
I
The first proof of Pythogoras’ Theorem appears in the Sulba
Sutras (800 BC).
Aryabhatta gave the first astronomical constant, computed the
exact length of the earth’s day (upto 0.1 secs) and its
circumference (with 0.2% error).
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Think about it. . .
I
The basic tenets of mathematics were, largely, invented in
India.
I
The modern number system was born in India. (Can you
imagine a world without it?)
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Think about it. . .
I
The basic tenets of mathematics were, largely, invented in
India.
I
I
The modern number system was born in India. (Can you
imagine a world without it?)
I ask: “Why aren’t we teaching this stuff in schools today?
I
I
Ignorance, inaccessibility of old texts.. or lack of initiative.
Can Udai help? Two ideas I propose:
I
I
I
(Short-term) Use a medium like DSH (Digital StudyHall) to
convey these ideas quickly to a large network of schools.
(Long-term) Write a book on this topic which is accessible to
school-children. (Books in the market currently not
interesting/understandable for school-goers). Eventually,
affect a change in school syllabi.
Contact me ([email protected]), if interested in
willing to help.
Saurabh Panjwani
Ancient Indian Mathematics
Motivation
Indian Numerals and Place Value System
Arithmetic
Teaching Principles and Philosophies
The Fundamental Operations
Fractions and Proportions
Sources
I
History of Hindu Mathematics. Bibhutibhusan Datta and
Avadesh Narayan Singh. Asia publishing House (1962).
I
The Aryabhatiya of Aryabhatta. Walter E. Clark. University
of Chicago Press (1930).
I
Wikipedia
Saurabh Panjwani
Ancient Indian Mathematics