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(From Georges Ifrah's The Universal History of Numbers)
FAQ in Ancient
Indian Mathematics
If you have
questions on Vedic Mathematics or related topics, don't be
bashful, send them to
Aryabhatta@indicethos.org.
No
question will be deemed too simple or obvious.
Contents
Answers

What were the
strengths of the ancient Indian Mathematicians
The Indian
Mathematicians of the ancient era were primarily number
theorists. Their interest in this field arose from a need to do
astronomical calculations, which in turn were needed to advise
the farmers on the proper timing to plant crops and for
performing rituals . In particular they excelled in Diophantine
Equations (probably erroneously credited to Diophantus , because
the Vedics were adapt at it long before Diophantus), algebraic equations in which only solutions in
integers are permitted. Examples of Diophantine equations
that the ancients dealt with are

ax
+ by = 1: this
is a linear Diophantine.

x^{n}
+ y^{n} = z^{n}:
For n = 2 there
are infinitely many solutions (x,y,z),
the
Pythagorean triples. For larger values
of n,
Fermat's last theorem states that no
positive integer solutions x,
y, z
satisfying the
above equation exist.

x^{2}
 n y^{2} = 1:
(Pell's
equation) which is named, mistakenly,
after the English mathematician
John Pell. It was studied by
Brahmagupta
and much later by
Fermat.
Another field or
approach which the ancients favored was algebraic geometry,
where geometry was studied predominantly using algebraic
equations. Again the need was for designing ritual altars and
their proper orientation with respect to the heavenly bodies and
for following the rules of Vastu sastra (architecture) .
The use of concise mathematical symbolism
(<,>,=,*,(), inf., sup., the integral sign, the derivative sign,
the sigma sign etc is a relatively recent revolution in
mathematics, When calculus was invented by Newton and Leibniz
independently, they used different notations. Newton used dots
on top of the quantity and called them fluxions, while Leibniz
used the 'd' notation. Unfortunately when all the major
advancements in science and mathematics were happening in Europe
in the 17th and 18th century, India was stagnant, caught in
interminable wars and conquests.
Ironically it was the Indian place value notation that triggered
the advance in Europe. Prior to this it was common to express
all problems in wordy sentences and verse.
As to the value of PI (cannot be expressed as a fraction) it is
impossible to calculate it without the use of some variant of
the 'method of exhaustion'. In this particular instance it was a
matter of using increasingly larger number of isosceles
triangles( forming an n sided polygon) within the circle. This
is the germ of the method of series expansions. The ancient
Indians were well aware of this technique and used it for
several trigonometric calculations. The felicity with which
Indians did series expansions extended to Srinivasa Ramanujam in
recent times. He was the
incomparable master and there may be none like him on the face
of this earth again.
Again there is no claim that the Indians did everything. For
example there is no evidence that the Indians were familiar with
the representation of complex numbers and complex variables or
advanced topics such as the calculus of variations.
The ancient vedic Indics were interested in very practical
aspects of mathematics, namely the positions of the stars,
developing a Panchanga (calendar), ordinary mathematics for
everyday use, measurements, such as that of land and weights
etc.. There is no evidence that they were familiar with the
science of mechanics for instance, which developed in Europe in
the 18th century after Newton.
Of course, the most
important single concept that they developed was the decimal
place value system. While other civilizations like the Maya and
the Oaxaca of ancient Mexico developed place value systems
independently, there is no evidence that they mastered the
technique of using numbers with such facility as the
Vedics did. The Vedics also developed facility with areas such
as Trigonometry and there is mention that they manipulated
arrays of numbers in the same manner as we use matrices today.
Even with all these
skills, it must be recognized that this forms only a small part
of what constitutes the corpus of modern mathematics today. It
must also be recognized that the Greeks and the Babylonians also
had developed mathematical prowess, if one accepts that the
axiomatic approach to geometry as primarily a Greek development.
Despite all these
caveats, it must be admitted that for the era in which
they lived the Vedic contributions to the sum of human knowledge
especially in mathematics was considerable and should therefore
be a matter of great inspiration for those of us who consider
themselves part of the Indic civilization. Further there is much
study yet to be done and many more manuscripts which remain
uninvestigated so the quest for what was the state of knowledge
of the ancients in fields such as mathematics has just begun

Why do you use the term Vedic
mathematics
Strictly speaking the later contributors
like
Varahamihira did not live in the Vedic era, but the methods
used show a continuity in development till the Modern age and
hence the use of the term Vedic , signifying techniques which
came into use in the distant past seems justified

Why even bother with Vedic mathematics
when we have progressed much farther in the intervening
millennia ?
I will quote the answer I gave in a
discussion forum (BharatRakshak)a few years ago. "
First, do we understand the corpus
of VM and Vedic Science (i certainly
don't know enough about VM to make an
authoritative statement). What part
of what we know to be contributions,
were purely Vedic and what part were
contributions by mathematicians like
Aryabhatta and Bhaskaracharya who
came much later ?
Second what part of it is already
accepted in modern mathematics ( and
I am not talking about arithmetical
tricks to do multiplication and
division). I remember in my Hall and
Stevens text on geometry which i
used in India, the proof of
Pythagoras theorem took up a whole
page. I would have loved to have the
4 alternate proofs offered by the
VMB. I am interested for example in
the development of Astronomy.
Ptolemy used a concept called
ecliptics (if i recall the use of
epicycles  pl. don't harangue me on
this, as this is of the top of my
head) to get around the fact that the
earth went around the sun and not
the other way around. Did he borrow
that from Vedic or was that a
different stream of thought (Ptolemy
of course predates alKhwarismi and
his 'Sindhind zij'). Given that
Ptolemy post dates Vedic why did alKhwarismi
choose to rely more on Vedic ( a
more ancient technology) when he
presumably had access to both ?
Third the point made by James is
valid. The point is not to judge
Vedic Science by the standards of
21st century and then trash it
saying it is outmoded. Of course it
is outmoded, in certain respects,
you should expect that after 6
millennia. If it was not, that is
tantamount to admitting that we have
made no progress in the intervening
millennia. The point is, does it give
an alternate model at looking at
nature (e.g. AyurVeda) that is
equally valid, useful and perhaps
more elegant.
Fourth, these efforts at
understanding our past, should not
be restricted to Vedic systems but
to other systems in India's past
such as Yunani (etymology Ionian =
Greek) which is attributed to
Islamic savants. I am interested for
example in tracing through the
development of algebra (coined by
al Khwarismi as algibr wal
Maquaballah) from its ancient Vedic
origins because of the efforts of
Islamic savants in the middle ages,
at a time when Europe was in the
dark ages and was struggling with
Roman numerals.
My own view is that Vedic science
and math is a forgotten science and
for the most part does not
contradict what has been discovered
subsequently. First let us
understand what it says before
getting an anxiety neurosis that it
is going to replace Western science
in Indian schools.




Where are these results to be found in
the ancient texts
The earliest works on Mathematics by the
Vedic savants are recorded in the Sulvasutras, the sacred books
of altar construction in the Vedas, in particular the Apastambha
Sulvasutra, the Baudhayana Sulvasutra and the Katyayana
Sulvasutra (named after the mathematicians who developed the
mathematics and wrote the sutras. The Sulvasutras are appended to a particular Veda
(see FAQ on Hinduism
for the typical contents of a Veda) can be translated as rope
rules or "manuals of measurement", the modern term for which
would be metrology. But in reality the scope of the
investigations in the Sulvasutras is far broader and comprises
among other fields, number theory, trigonometric, algebra,
algebraic geometry, series expansions, the concept of a
rational number, etc. etc.. The dating of these Sulvasutras, while
occurring after the main corpus of the Veda was compiled,
is of great antiquity, greater than that of Babylonian
mathematics. It is only now that we are beginning to understand
the extent of the antiquity of these ancient mathematicians. See
for instance The origin of Mathematics by Lakshmikantham and
Leela. in the list of references later in this page.
A word needs to be said about the use of
Sutras (Aphorisms) as a means of communication and recording of
results. The dictionary says that a sutra is Any of various
aphoristic doctrinal summaries produced for memorization
generally during the millennia before the common era and
later incorporated into Hindu literature.
We
must recall that writing materials during that period of
history, were not plentiful and had to be laboriously produced
probably by the author of the sutras himself . Whatever needed
to be communicated had to be in as brief a form as possible.
Hence the need for economy in language and the use of sutras. In
fact there is evidence from which one can infer that the Vedics
were the first to use symbols and mathematical equations and
hence their prowess with Algebra .
Brevity has its downside
however and the charge has been made by Europeans that the
Vedics rarely provided proof. The reason was that the proof was
generally terse and incomprehensible to the majority of the
people. The notion that those who mastered these topics used
brevity as a means to restricting this knowledge, overlooks the
fact few in the populace would have the capability to pursue
this rigorously without the intellectual discipline that
comes from years of study. Then as now Mathematics had the
reputation of being a difficult subject to master.
Later texts were known as Siddhantas

Who were the main contributors to
Vedic Mathematics
The main contributors during the Vedic
period were, as we mentioned earlier Baudhayana, Apastambha, and
Katyayana. The following list of Mathematicians who were born in
the geographical region corresponding to present day India was
compiled by researchers at the University of St. Andrews,
Scotland, who have done an outstanding job in compiling a lot of
information (this list includes some modern day
mathematicians born in the Indian sub continent also).
We will discuss the
contributions of some of the main mathematicians of the ancient
era.

Why are the contributions of the Vedic
savants not as widely known as those of Greek, Arab
mathematicians
In the early years of colonial rule by the
British (the attitude persists among Western Indologists even
today, although less so among mathematicians in the West)
there was great reluctance to believe the sacred texts, after
they were first recognized by Sir William Jones in the
1770's. A typical reaction was that of W. W Rouse Ball in
History of Mathematics . I posted the following in the Bharat
Rakshak forum in2000
"Typical
of the racism exhibited by the Brits and other Europeans is W.W.
Rouse Ball in 'A short account of the History of mathematics'
Dover Publications,1960, (originally appeared in 1908), page
146 'The Arabs had considerable commerce with India, and a
knowledge of one or both of the two great Hindoo works on
algebra had been obtained in the Caliphate of AlMansur (754775 AD)though it was not until fifty or seventy years later that
they attracted much attention. The algebra and arithmetic of the
Arabs were largely founded on these treatises, and I therefore
devote this section to the consideration of Hindoo
mathematics. The Hindoos like the Chinese have pretended that
they are the most ancient people on the face of the earth, and
that to them all sciences owe their creation. But it is probable
that these pretensions have no foundation; and in fact no
science or useful art (except a rather fantastic architecture
and sculpture) can be definitely traced back to the inhabitants
of the Indian peninsula
prior to the Aryan invasion. This seems
to have taken place at some time in the fifth century or in the
sixth century when a tribe of Aryans entered India by the north
west part of their country. Their descendants, wherever they
have kept their blood pure, may still be recognized by their
superiority over the races they originally conquered; but as is
the case with the modern Europeans, they found the climate
trying and gradually degenerated. Note the blatant racism in the
second paragraph and the venom that this author exhibits. [This
message has been edited by Kaushal (edited 15062000).]
"
There are
a lot of facile and unsubstantiated assumptions that are made
here, later to be proved false, but we will deal
with them on the chapter where we discuss
AIT. Thus
there was great reluctance to admit that the dark skinned
natives of the Indian subcontinent could be capable of
intellectual effort. Even after the advent of the legendary
Srinivasa Ramanujam, the great number theorist in the
early years of the 20th century, from what is now Chennai,
Tamil Nadu , such attitudes among British and European scholars
were hard to dispel. We will have a lot more to say about
Srinivasa Ramanujam later in these pages.
With
the coming of the internet, and the great proficiency of the
Indics in matters related to Information Technology, this state
of affairs has begun to change. Both the Indics and Western
savants have begun to realize the profound importance of these
early developments in mathematics to the advancement of human
civilization. See for instance a recent column on
Place Value systems.

Did
the Vedics use symbols when they expressed their algebraic
equations ?
The use of
concise mathematical symbolism (<,>,=,*,(), inf., sup., the
integral sign, the derivative sign, the sigma sign etc is a
relatively recent revolution in mathematics, When calculus was
invented by Newton and Leibniz independently, they used
different notations. Newton used dots on top of the quantity and
called them fluxions, while Leibniz used the 'd' notation.
Unfortunately when all the major advancements in science and
mathematics were happening in Europe in the 17th and 18th
century, India was stagnant, caught in interminable wars and
conquests and was finally subjugated by the British.
Ironically it was the Indian place value notation that triggered
the advance in Europe. Prior to this it was common to express
all problems in wordy sentences and verse.
As to the value of PI (cannot be expressed as a fraction) it is
impossible to calculate it without the use of some variant of
the 'method of exhaustion'. In this particular instance it was a
matter of using increasingly larger number of isosceles
triangles( forming an n sided polygon) within the circle. This
is the germ of the method of series expansions. The ancient
Indians were well aware of this technique and used it for
several trigonometric calculations. The felicity with which
Indians did series expansions extended to Ramanujam. He was the
incomparable master and there may be none like him on the face
of this earth again.
Again there is no claim that the Indians did everything. For
example there is no evidence that the Indians were familiar with
the representation of complex numbers and complex variables or
advanced topics such as the calculus of variations (mathematics
enthusiasts will recall that
Queen Dido
of Carthage, who had the correct intuition for the solution
of the Isoperimetric problem, was also the first recorded
instance of a woman mathematician).
The ancient vedic Indics were interested in very practical
aspects of mathematics, namely the positions of the stars,
developing a Panchanga (calendar), ordinary mathematics for
everyday use, measurements, such as that of land and weights
etc.. There is no evidence that they were familiar with the
science of mechanics for instance, which developed in Europe in
the 18th century after Newton. However, their use of distant
stars as invariants and their familiarity with the precession of
the equinoxes , hints of an understanding if not the inklings of
an inertial frame of reference and this is an area in which I
need to educate myself further

Can
we see examples of what you mean by Vedic Mathematics
First we will
make a distinction between the corpus of Vedic Mathematics (VM)
from the 16 extant sutras that are still available to us
courtesy of the book on Vedic mathematics (VMB) by Jagadguru
BhaarathiKrishnatirtha, the late Sankaracharaya of Puri. here we will
restrict ourselves to VMB, which is largely a text at High
school level, although i will wager very few school children
even in India are familiar with these techniques. So what does
VMB comprise of ?
Here is
a good description
if this site does not work try this
An example of the proof of
Pythagoras theorem (originally given by Baudhayana)
(from Pacific Institute of Mathematical
Sciences, by
Jeganathan
Sriskandarajah

It is now commonly acknowledged that the
Vedics invented the place value system of numerals we use
today including the symbol for zero. Do we know the name of
the person who invented the zero
Civilization centers where the Zero was
used in the ancient world
The place value system has been used by
people in the ancient world in disparate locations such as
Babylonia , India, China and by the Maya in what is now
Central America. The popular conventional view in the west goes
thus
The
final independent invention of the
zero was in India. However, the time
and the independence of this
invention has been debated. Some say
that Babylonian astronomy, with its
zero, was passed on to Hindu
astronomers but there is no absolute
proof of this, so most scholars give
the Hindus credit for coming up with
zero on their own.
The reason
the date of the Hindu zero is in
question is because of how it came
to be.
Most
existing ancient Indian mathematical
texts are really copies that are at
most a few hundred years old. And
these copies are copies of copies of
copies passed through the ages. But
the transcriptions are error
free…can you imagine copying a math
book without making any errors? Were
the Hindus very good proofreaders?
They had a trick.
Math
problems were written in verse and
could be easily memorized, chanted,
or sung. Each word in the verse
corresponded to a number. For
example,
viya
dambar akasasa sunya yama rama veda
sky (0) atmosphere (0) space (0)
void (0) primordial couple (2) Rama
(3) Veda (4)
0 0 0 0 2 3 4
Indian
place notation moved from left to
right with ones place coming first.
So the phrase above translates to
4,230,000.
Using a
vocabulary of symbolic words to note
zero is known from the 458 AD
cosmology text Lokavibhaga.
But as a more traditional numeral—a
dot or an open circle—there is no
record until 628, though it is
recorded as if wellunderstood at
that time so it’s likely zero as a
symbol was used before 628.
Which it
probably was, considering that 30
years previously, an inscription of
a date using a zero symbol in the
Hindu manner was made in Cambodia.
A striking note about the Hindu
zero is that, unlike the Babylonian
and Mayan zero, the Hindu zero
symbol came to be understood as
meaning “nothing.” This is probably
because of the use of number words
that preceded the symbolic zero.

Part of the reason
that unquestioned precedence is not accorded
the Vedics is the confusion with the dating
of Baudhayana ,Aryabhatta and Panini, who in
my opinion are the leading contenders for
the privilege of having invented the
zero. It is only recently that it has been
universally accepted that the Vedics were
the first to use the place value system
extensively.
"There is
wide ranging debate as to when
the decimal place value system
was developed, but there is
significant evidence that an
early system was in use by the
inhabitants of the Indus valley
by 3000 BC. Excavations at both
Harappa and Mohenjo Daro have
supported this theory. At this
time however a 'complete' place
value system had not yet been
developed and along with symbols
for the numbers one through
nine, there were also symbols
for 10, 20, 100 and so on.
The formation of the numeral
forms as we know them now has
taken several thousand years,
and for quite some time in India
there were several different
forms. These included Kharosthi
and Brahmi numerals, the latter
were refined into the Gwalior
numerals, which are notably
similar to those in use today
(see Figure 7.1). Study of the
Brahmi numerals has also lent
weight to claims that decimal
numeration was in use by the
Indus civilisation as
correlations have been noted
between the Indus and Brahmi
scripts.
It is uncertain how much longer
it took for zero to be invented
but there is little doubt that
such a symbol was in existence
by 500 BC, if not in widespread
use. Evidence can be found in
the work of the famous Indian
grammarian Panini (5th or 6th
century BCE ;
ed. note
 we believe Panini must predate
any text written in Classical
Sanskrit and at a minimum must
have lived prior to 1700 BCE) and later the work
of Pingala a scholar who wrote a
work, ChhandasSutra (c. 200
BCE). The first documented
evidence of the use of zero for
mathematical purposes is not
until around 2nd century AD (in
the Bakhshali manuscript). The
first recorded
'nonmathematical' use of zero
dates even later, around 680 AD,
the number 605 was found on a
Khmer inscription in Cambodia.
Despite this it seems certain
that a symbol was in use prior
to that time. B Datta and A
Singh discuss the likelihood
that the decimal place value
system, including zero had been
'perfected' by 100 BC or
earlier. Although there is no
concrete evidence to support
their claims, they are
established on the very solid
basis that new number systems
take 800 to 1000 years to become
'commonly' used, which the
Indian system had done by the
9th century AD. 
My investigations to date lead me to
conclude that the honor of inventing
the zero and the place value system belongs
to one of three individuals  Apastambha.
Baudhayana, and/or Panini. Of course
Aryabhatta was and is a perennial favorite
for being the main suspect However, we are
closing in on the target and should have the
answer soon in short order. We will examine
the evidence related to each and reach a
conclusion. See also the reasons adduced by
Ifrah cited by me in the section on
Linguistics
In the
Proposed
skeleton of Indian Chronology Baudhayana is dated at
3200BCE, which would make him
contemporaneous or earlier than the
Babylonians.

What do we know about
Panini
References
Books:
 G Cardona, Panini : a survey
of research (Paris, 1976).
 G G Joseph, The crest of the
peacock (London, 1991).
Articles:
 P Z Ingerman, 'PaniniBackus
form' suggested, Communications of the ACM 10
(3)(1967), 137.

What do we know about the life and
mathematical contributions of Bhaskaracharya (or Bhaskara I)

Life and contributions of
Aryabhatta
the Elder (under construction)

What did the Brahmi script look like
An
example of Brahmi script 
Ashoka's first rock
inscription at
Girnar.

Indianstandard silver
drachm of the
GrecoBactrian king
Agathocles (190
BC180
BC), the obverse showing
the Greek legend BASILEOS
AGATOKLEOUS "King Agathocles",
the reverse RAJANE
AGATHUKLAYASA "King
Agathocles" in Brahmi.


To compare this with
Babylonian see for instance the following
Babylonian tablets

If the current chronology is wrong ,
what do you suggest that the
correct chronology be
Proposed
Hypothesis for a consistent
chronology
Event 
Date 
Birth of Dhirgatmas
Birth of Yajnavalkya
Shatapatha Brahmana
Brihadaranyaka Upanishad
Birth of
Panini 
4000 BCE
?
?
3500 BCE ? 
Birth of Pingala
Birth of Veda
Vyasa
Birth of Baudhayana
Birth
of Apastambha
Beginning of
Kali Yuga
Birth of
Aryabhatta
Aryabhatteeyum written by
Aryabhatta
Gautama
Buddha
Coronation of
Chandragupta Maurya
Coronation of
Asoka Maurya (may not
be the same
Asoka who built
the edicts)
Kanishka’s
time
Panchatantra
composed by Vishnusharman
Birth of Adi
Sankara
Coronation of
Chandragupta of Imperial
Gupta Dynasty
Varahamihira
wrote Pancha Siddhanta
Vikramaditya
Bhartrihari
Vikram era calendar
Birth of
Brahmagupta.
Salivahana
Era Calendar Saka
Bhaskara
II wrote Siddhanta
Siromani

?
3374 BCE
3200 BCE
3102 BCE
2765 BCE
2742BCE
18881807 BCE
1554 BCE
1472 BCE
1294 – 1234
BCE
? 1470 510
BCE
509 BCE
327 BCE
123 BCE
102 BCE to 78
CE
100 BCE
30 BCE
78 CE
486 CE 


Partial Source : The
Origin of Mathematics,
Lakshmikantham and Leela, Page30,
University Press of America
Inc.,2000 



References on Vedic mathematics

Links on Vedic mathematics
Excellent Link on Vedic Mathematics
ŠKosla Vepa,2006 
