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so far we looked at the mathematical concepts
that were available in some of the most ancient
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text so vedas and sulvasutras the next 3-4
talks we will be focusing our attention on
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a single text called aryabhatiya of aryabhatta
aryabhatta who was in the later part of the
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5th century and his text aryabhatiya is one
the most seminal text on indian astronomy
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and mathematics so we will start with the
period
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of aryabhatta and then we will have a brief
description on the work aryabhatiya aryabhatiya
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as i was mentioning so while discussing this
decimal place value system as we compose in
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a very first time and therefore without it
will be extremely difficult for us to understand
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the contents so the earliest commentary that
is available for this is one by bhaskara so
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bhaskara lived in the early part of 7 century
and it is a very profound commentary on aryabhatiya
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we
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will see a brief description of where bhaskara
lived and what was his period and then we
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will move on to this text percy so aryabhatta
first introduces the notational places then
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he start discover describing about the fundamental
operations square squaring finding out the
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square root cube root and so on then he moves
on to various other topics the text aryabhattiya
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as such is composed in four parts we will
discuss all that in great detail but
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before i move on to that i just wanted to
say about the very name aryabhatta so aryabhatta
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so this is how one can derive the word aryabhatta
so bhata normally means a soldier or guardian
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and so on so the term arya refers to a noble
person in fact very interesting definition
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has been presented in one of the text called
shabdakalpadruma so that have noted down this
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is a very beautiful definition so the one
who performs what is to be performed by
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thing so this is not sufficient and you also
process in the negative form so this what
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is called have to be done but i will also
engage myself in a serious activities that
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is not acceptable and therefore that is acceptable
in the society at that particular point of
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time so the one who so the aryabhatta cab
be derived in 2 ways so the one who protects
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that and one who save guardian and so on and
so forth any way so term can be derived and
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the very name
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of the text is based on the name of the author
so sometimes it just happens the text could
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be completely different so here it is called
so that which belongs to aryabhatta or that
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which has been composed by aryabhatta so that
is why it is called the name aryabhata appears
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twice in the text so which is not quite common
in many of the text in the indian literature
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but here write in the beginning of deepika
pada which is the first section so he
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mention his name and again once more in the
next section called ganitha pada the time
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of aryabhatta is also pretty clear to us that
is no ambiguity because there is a verse in
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iii section goes like
this
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i was 23-years-old at a certain point of time
so that is mention in the first part of
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the solga so so when 3600 years of the i was
23-year-old so when we go back and then find
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out so what was the beginning of so it just
happens that these 499 ad
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so people are opinion that this verse has
been composed in and therefore aryabhatta
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was 23-year-old and he compose this work so
this actually clearly tells that aryabhatta
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was born in 4676 ad coming to aryabhatiya
as i was mentioning so this is the text which
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is very clearly datable and it was composed
in 499 and as far as we know this is the earliest
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datable text that is available in full form
for us so they have been in earlier words
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in fact aryabhatta himself
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towards the end the work he says he says i
entered into the ocean of knowledge with the
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intellect as board and i lifted up gym so
which means i mean he is trying to point out
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that there has been earlier literature so
from which he has carried out some important
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things and presented in this work so this
aryabhatiya is available for us in complete
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form and the earlier works which were available
for aryabhatta have been shot of compiled
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by varahamihira in his
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text called panchasiddhantika will come little
later more or less contemporary so the systematically
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discuss the procedures for planetary computation
and it has been composed in a very cryptic
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style and therefore some people prefer though
it is in the form of verses it is in the form
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of aryametre so the text is many times refer
to as sutra itself and there are 4 padas as
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i said 4 sections deepikapada 13 verses ganitapada
33 verses kalakriyapada 5 verses
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golapada 50 verses in all we find 121 versus
composed in arya meter which consists of the
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text aryabhatiya adn in fact yesterday you
might have that srinivas was mentioning that
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so is also called as 108 so this 108 says
it just drop out the 13 verses so it just
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happens to be 108 so we will see it little
more in greater detail later why this 108
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so to give you an idea of what are the contents
of this 4 chapter so gotikapada so basically
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he llays
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out all the parameters that are essential
for doing astronomical computations in the
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to be described in the later part of the work
then ganitapada as a name itself implies discuss
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all the necessary mathematics that required
for doing this planetary competition starts
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with square square root cube cube root and
then it moves on to discuss the solution for
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indeterminate equations of first order and
which is called which will be discuss the
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great length ahh by
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professor sriram and also myself with you
later in kalakriyapada we primary find the
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geometrical picture and implies by the computational
procedure it has been discussed in great detail
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and in golapada the variety of topics which
are discuss so it is in fact almost occupy
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50% of 50 verses in golapada so it discuss
various details regarding the shape of the
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earth so the source of light and planets calculation
of eclipse visibility of planets and so forth
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so these are the various topics which are
discussed in golapada before i proceed into
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the text
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aryabhatiya i will say a couple of words on
bhaskara commentary because i will be more
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or less dealing both of them together so i
am not going to take a bhaskara separately
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so we will see a brief note on bhaskar bhaskar
as actually return three major works one is
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aryabhatta bhartiya as i was mentioning the
other 2 are sort of independent works but
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bhaskara describes them as aryabhatta means
so it a primarily exposition on what has been
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described in
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aryabhatta that is how we describe those who
are called that these 3 words put together
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so a lot of light on the kind of mathematical
knowledge as well as the astronomical theories
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which were present around that period so bhaskara's
time as we estimated to the 629 ad and his
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work actually displays a great amount of scholarship
and it is really in intellectual peace to
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be his bhasya so i had an occasion to go through
in great detail in connection with this
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course in fact in one place he says so he
refers to a certain number and you can take
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this as an exercise now yesterday we discussed
0 and then agni rama so 3 rama and therefore
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3 and and so on you can see this so based
on the last four digits ahh 1000 times itself
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is 4320000 years so based on some ahh analysis
for it has been shown by that this corresponds
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to 629 ad bhaskara was just about 140 years
after aryabhatta very close
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to the period of composition of aryabhatiya
then there are references in this work as
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well as the work of brahmagupta that aryabhatta
had many so one of the most famous one was
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refers to him in various places and also brahmaputra
refers to them ok this was an introduction
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now we move on to the text aryabhatiya itself
so the first section gitikapada consists of
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13 versus as i was mentioning so verse 1 is
indicatory work and
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it goes like so we find a very clear statement
where aryabhatta himself says aryabhatta states
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so what does he state so so this gitikapada
so which essentially present numbers certain
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parameters so is considered to be out of the
text in some set so the details of person
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so the numbers because which are required
and the kind of find aa sin table etc are
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all essential for doing computation but they
need not be integrated with the text perse
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and you want to
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understand astronomy so it is in this sense
i mean that it has been segregated out and
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the last verse goes like this
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this is called the so leaving out the first
verse and the last verse and the kind of colours
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with all that she says so once a person knows
this so it is a kind of but what is the point
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that i want to convey here was this so this
very clearly tell that the basic text of aryabhatiya
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is leaving out gitikapada and just consists
of 108 verses and
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there is also a palace with the end in fact
one find another invitation at the beginning
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of ganitapada so ganesha see invocation basically
marks the beginning of the text so we find
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2 invocation so one is completely separated
out and then again in ganitapada i will find
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invocation so we have basically 3 part of
aryabhatiya ganita and gola what we are going
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to discuss here is only ganitapada ok the
ganitapada commences with this verse
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basically offering my veneration my veneration
to what you can easily guess so
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these are the names for the planets so leave
it out who starts from refers to moon then
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group of stars constellations so to all the
celestial bodies and then i start ahh this
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work aryabhatta means states states what so
aryabhatta does not playing that i am going
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to say completely everything out of my head
so it is not that so all that he says is so
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the knowledge which was highly rewired in
place called and going to narrate here
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so commenting up on these word in bihar and
earlier it was called place of great learning
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where even nalanda university was existing
at a point of time aryabhatta says that i
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actually narrate the knowledge which was highly
review in this place and so further bhaskara
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says is being hurt so what is hurt in fact
here we find reference to all the five which
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has been compiled in ah so he says means bramha
so is
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brahma siddhanta so we having other so by
the people and basing on aryabhatta has presented
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his work after the invocatory verse the next
work basically presents the ahh notational
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places so he says so the names of the various
places in fact this is extremely essential
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for even deciphering the number which has
been given by aryabhatta so he basically tells
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the means so 10 to the power 9 the names of
the powers of 10 has been listed there is
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an interesting discussion made by bhaskara
at this point he says what is so special about
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this shakti means a certain potential so when
we declare something may be something it has
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been device by you but it has the potential
to convey some meaning to you very important
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meaning so he asked the question so what is
the potential that have been associating with
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this and what purposes he says so finally
what he is going to convey is so price
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something so you can just place it in that
place and thereby he conveys something so
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that has been something just been created
by you it service a very useful purpose in
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the day today transaction so that is kind
of discussion that represents here then he
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moves on to describe the fundamental operations
so we start with varga and he defines what
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does the term varga mean he says so one is
a certain geometrical representation so we
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have square
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and we have squaring scaring actually refers
to the operation and what is represented by
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this operation both are created here in this
thing 4 sided figure ; refers to square then
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he says so mathematically what does it represent
2 equal quantities so the product of 2 equal
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quantities is what is represented by there
and both are referred to as varga ok this
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then he ask in the commentary bhaskara processor
interesting question when you say so we
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can also think of a rhombus so rhombus is
also an object which has 4 sides equal why
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does the word varga convert rhombus so can
it also convert rhombus or not so this is
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the question he says see in fact this is the
very important statement in the sense that
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in order to understand the meaning of a particular
word so we have to go to the society to whatever
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sense it has been used by people he send the
world and therefore he says this particular
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shape rhombus has been never refer to be people
by this word and therefore it does not denote
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rhombus it has the potential only to convert
square then bhaskara also so all these are
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synonyms for square then i move on to describe
this algorithm which has been presented by
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aryabhatta to extract square root ok varga
refers to square vargamula refers to square
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root so the verse goes like this in fact yesterday
while i discuss to the
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aryabhatta systems of representing numbers
i said right so can refer to the square and
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when we think of the powers of 10 so 10 to
the power 0 1 is a square number then 10 to
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the power of 2 is a square number so similarly
in describing the operation which has to be
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done to extract the square root he clearly
states when we have a certain number which
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has been given to you you have to first of
all break that into 2 units so one is the
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varga part and the
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other is the so in understanding this verse
this has to be kept in mind so here refers
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to that which is ahh 10 to the power of 1
10 to the power of 3 and so on so this is
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called so the he says so the first work has
to be understood the vanga which is the square
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of some number subtracted so the process will
be very clear but i just try to make certain
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terminology clear before we show a certain
example so let us see this
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example now so here we have a number 55225
and this has to be written down like this
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putting varga avarga varga avarga varga avarga
so this is what he means by in the sloga varga
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avarga and so on and the procedure to be adopted
can be stated in 2 steps then he will move
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on to the example so in bhaskara commentary
what we find is so he says divide ok and he
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says so whenever you encounter a certain thing
so all that you need to do you
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have to do a certain division and whenever
we find this varga you move want to replace
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all that you need to do is means you have
to remove a certain square so we will see
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the algorithm and the basis of the algorithm
2 so in fact bhaskara very says a certain
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nume 54321 so this even place so this is odd
so all the hot places so they are referred
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to as varga so this is varga so all even spaces
are referred to us avarga so this is
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basically from the fact where the 10 the power
of 0 10 to the power of 2 10 to the power
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of 4 and so on so basically this has to be
divided into two so one is why is it done
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so basically a single digit so if you take
the square of it also can be only two digit
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therefore so depending upon the number of
digits and number which has been presented
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you first make a guess of how many digits
in the square root have yes so this is this
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classification
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separate into two so so the algorithm essentially
has 3 steps so what is states is starting
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from the least significant digit group the
digits of the number into two so the first
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thing that needs to be done is what i was
saying so some the least you just group them
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into two then from the remaining part the
most significant digit so which constitutes
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the so this is pretty evident so this is so
then this basically is so whatever be the
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most significant digit it
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can be one or it can be 2 so that is just
taken in the beginning and so which constitutes
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the subtract the maximum square that is possible
so having done that so long with the reminder
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so you bring down the next digit of the so
this is then once you remove the maximum square
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that is possible for remove for instance when
you have 5 to the maximum square that can
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be removed is 2 square so one will be remaining
and 4 will be remaining in
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the next step you have to start your operation
was 14 so along with the reminder bring down
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00:23:39,030 --> 00:23:44,720
next digit from the this has to be divided
by twice the in fact this has been stated
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00:23:44,720 --> 00:23:51,500
determine here at the first place this has
to be done at every states so this so at this
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00:23:51,500 --> 00:23:55,799
stage if you deal with the first two digits
or 1 digit so you have to jot down the whatever
197
00:23:55,799 --> 00:24:01,830
you get so suppose you have 5 by two so in
the next stage so when you do certain operation
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00:24:01,830 --> 00:24:07,190
you will add one more digit to this and that
should be considered as at that stage so things
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00:24:07,190 --> 00:24:10,940
will become clear when you look at the example
ahh basic operations 1 is divided by twice
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00:24:10,940 --> 00:24:16,009
that and then remove the square these are
the 2 operation with have to be repeated that
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00:24:16,009 --> 00:24:22,639
for the entire square root of the number so
now we go to the example so let us take this
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00:24:22,639 --> 00:24:24,210
number 55225 so we have group in this and
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00:24:24,210 --> 00:24:29,460
then we have this so the first things that
is to be done is remove the maximum square
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00:24:29,460 --> 00:24:39,130
root that is possible so 2 square can be removed
so you write 4 and then take the number 2
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00:24:39,130 --> 00:24:47,220
there so this is the so you just keep it somewhere
and so 1 is a reminder here then bring down
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00:24:47,220 --> 00:25:04,549
the next number from the plays what you have
is 15 so the operation the stated is so you
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00:25:04,549 --> 00:25:14,570
have to divide by twice the so two times 2
at this stage what you have is only 2 so 2
208
00:25:14,570 --> 00:25:22,799
times 2 is 4 so you have to divided by that
so what to get is 3 put 12 and the remainder
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00:25:22,799 --> 00:25:31,539
is 3 now the next digit has to be brought
down 32 is a number that is available so the
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00:25:31,539 --> 00:25:41,519
operation is over so the next operation that
is to be done is taken the number down so
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00:25:41,519 --> 00:25:47,789
you have to remove the square the square of
what square of the question that you obtained
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00:25:47,789 --> 00:25:52,360
in the previous place therefore you remove
9 so so what you have is 23 here and
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00:25:52,360 --> 00:25:58,509
23 were bringing down so the place is got
down so at this stage the that you have is
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00:25:58,509 --> 00:26:05,509
23 is 2 times 23 so you have to divide by
46 ok so what you get is 5 and then reminder
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00:26:05,509 --> 00:26:16,480
is 2 here so you then bring down at this stage
so what you got was 5 so so you have to remove
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00:26:16,480 --> 00:26:24,820
the square of these and 25 and the remainder
is 0 so this is basically the operationof
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00:26:24,820 --> 00:26:31,899
extracting square root at the same way aryabhatta
so let us take one more
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00:26:31,899 --> 00:26:38,820
example so we have to give into two so 41
then we have 94 we have 98 and what remains
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00:26:38,820 --> 00:26:46,960
is 2 so what can be extracted out is only
one square to remove that you write it 1 and
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00:26:46,960 --> 00:26:54,389
then the next step you bring down the number
so 19 and twice the at this stage so you have
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00:26:54,389 --> 00:27:01,520
to take 2*1/that so here you can actually
have the greater number pulled out but then
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00:27:01,520 --> 00:27:08,450
keeping in mind that the next step ahh you
do not get a negative number
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00:27:08,450 --> 00:27:13,570
okay so if you do that then you have to revert
that and then you will be able to do that
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00:27:13,570 --> 00:27:18,210
so here so what you get is 58 and so what
has to be removed here is 7 square to remove
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00:27:18,210 --> 00:27:22,289
49 then again you do the same operation twice
the 2 times 17 so you can see that so this
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00:27:22,289 --> 00:27:26,480
these are the 2 operations which have to be
repeated after describing the extraction of
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00:27:26,480 --> 00:27:32,690
square root he moves on to the square cube
cubing and ahh how to find cube
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00:27:32,690 --> 00:27:41,470
root so this is the next operation so understand
clearly lay down the procedure for extracting
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00:27:41,470 --> 00:27:49,509
cube root ok very clearly lay down procedure
so defining what is cube he says product so
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00:27:49,509 --> 00:28:04,250
you have quantity multiply by 3 times so that
gives what is called similarly so geometrically
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00:28:04,250 --> 00:28:11,809
what it represents so a cube what says is
an object which has 12 not size so it is the
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00:28:11,809 --> 00:28:17,899
kind of line ahh so both have been nicely
stated and so this is what
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00:28:17,899 --> 00:28:24,899
has been stated by bhaskara in fact they sometimes
basically a product is a product of 3 quantities
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00:28:24,899 --> 00:28:33,450
which are one in the same so this is what
is cube and what is interesting note is so
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00:28:33,450 --> 00:28:37,870
the aryabhatta definition of this cube scripting
out of the geometrical thing as quantity you
236
00:28:37,870 --> 00:28:43,100
protect the product price and you get this
so this is a sort of abstraction so beyond
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00:28:43,100 --> 00:28:49,679
what is this is a geometrical shape operation
has been again beautifully and concisely
238
00:28:49,679 --> 00:28:55,470
describe in one word so he says so this is
the procedure for extracting cube root of
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00:28:55,470 --> 00:29:01,410
a number see in case of square root so i said
you have to break the number given number
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00:29:01,410 --> 00:29:04,899
into 2 2 digits so you can usually guess that
in case you are extracting cube root you have
241
00:29:04,899 --> 00:29:12,169
to break them as units of 3 so these units
of 3 have been assign a certain nomenclature
242
00:29:12,169 --> 00:29:21,309
so there he call it as so here he uses 3 terms
and they are the name that he uses so this
243
00:29:21,309 --> 00:29:22,309
has to
244
00:29:22,309 --> 00:29:27,490
understand so from this second you have to
divide ok so what is to be done there he said
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00:29:27,490 --> 00:29:31,940
twice the here thrice ok so thrice of the
means the that you get at that stage ok then
246
00:29:31,940 --> 00:29:36,559
is be subtracted so what is to be subtracted
so 3 is number 3 and is previous number so
247
00:29:36,559 --> 00:29:40,592
means 3 times the previous number you have
to subtract so this has to be at the and when
248
00:29:40,592 --> 00:29:42,010
you come to the gana place so there from the
varga place
249
00:29:42,010 --> 00:29:47,309
you have to remove the varga ok so here at
the ghana place you have to remove the ghana
250
00:29:47,309 --> 00:29:51,450
the cube of something ok so all that will
be clear with example but the terminology
251
00:29:51,450 --> 00:29:55,720
has to be very clear when we read see this
understanding the algorithm will become quite
252
00:29:55,720 --> 00:30:02,110
evident the moment we understand that the
cube of a given number can of course of ok
253
00:30:02,110 --> 00:30:09,970
if you have three digit number 6 digits so
here it can be out most 3m and it can be it
254
00:30:09,970 --> 00:30:11,270
should be definitely
255
00:30:11,270 --> 00:30:18,440
greater than and less than or equal to 3m
so with this is mine so aryabhatta asked us
256
00:30:18,440 --> 00:30:22,840
to divide into groups of 3 and then carry
out the operation so i leave this this is
257
00:30:22,840 --> 00:30:27,659
basically translation the work and the essential
steps involved i just show this and then i
258
00:30:27,659 --> 00:30:31,480
go back to this previously slide here we have
this number 1771561 so the grouping i think
259
00:30:31,480 --> 00:30:36,429
i have to put a coma here so here ghana ghana1
ghana1 ghana ghana1 ghana2 and whatever remains
260
00:30:36,429 --> 00:30:42,139
here should be considered as ghana ok so group
of 3 start from the least significant and
261
00:30:42,139 --> 00:30:50,659
whatever remains with it is one two or three
that will be considered as ghana in the most
262
00:30:50,659 --> 00:30:56,110
significant place so in this example we have
a ghana with one number and the first thing
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00:30:56,110 --> 00:31:00,220
that has to be say all that he says he is
this possible to be repeated the process of
264
00:31:00,220 --> 00:31:05,179
his stage is with operation with have to do
operation other operation and this operation
265
00:31:05,179 --> 00:31:06,179
has to be
266
00:31:06,179 --> 00:31:11,721
related the number gets over so this is the
process we get a square also he said so with
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00:31:11,721 --> 00:31:15,640
the reference to do an operation there is
place you are doing this operation has been
268
00:31:15,640 --> 00:31:20,950
repeated and you will get the square root
so as an algorithm so what are the steps involved
269
00:31:20,950 --> 00:31:27,549
starting from the units place having grouped
the digits of the number into three from the
270
00:31:27,549 --> 00:31:35,590
remaining 1 2 or 3 the most significant digits
so the digit is called there he called
271
00:31:35,590 --> 00:31:40,389
period is called subtracting maximum soon
as possible this has to be done so this digit
272
00:31:40,389 --> 00:31:44,600
actually forms the most significant digit
of the cube root so then along with the reminder
273
00:31:44,600 --> 00:31:48,710
bring down the next digit from the so once
you do this operation this this number has
274
00:31:48,710 --> 00:31:52,720
to be brought down so this has to be divided
by thrice the square of the obtained so far
275
00:31:52,720 --> 00:31:56,070
in fact if you look at the verse ok so multiply
by 3 that for it
276
00:31:56,070 --> 00:32:00,710
has to be understood ok so the whatever you
have written divide thrice the square of the
277
00:32:00,710 --> 00:32:05,799
obtained so far so the portion forms the next
digit of the cube root ok so this is operation
278
00:32:05,799 --> 00:32:11,519
so whatever portion that we get here that
forms the next place to the cube so along
279
00:32:11,519 --> 00:32:15,480
in the remainder again the next digit has
to be brought down and at this stage so all
280
00:32:15,480 --> 00:32:20,659
that he says is is the square of this so whatever
we obtain 3 so 3 is 3 and
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00:32:20,659 --> 00:32:31,539
is the previous number ok so in the cube root
that you do whatever be the number the previous
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00:32:31,539 --> 00:32:40,990
number so second digit of the cube root and
previously have got one number so all of he
283
00:32:40,990 --> 00:32:48,490
says is 3 times the previous number and the
square of this ok that should be the ahh thing
284
00:32:48,490 --> 00:32:54,740
which has to be subtracted so this is the
prescription for so we so this
285
00:32:54,740 --> 00:32:58,009
the operation that has to be repeated so this
is all the ahh prescription and the process
286
00:32:58,009 --> 00:33:07,840
has to be repeated ok now let us look at this
example so we have this ghana place 1 so the
287
00:33:07,840 --> 00:33:17,380
maximum cube that can be removed is one cube
remove that we have 0 so you places 1 here
288
00:33:17,380 --> 00:33:23,779
and then the next digit the second that has
to be brought down so here the operation the
289
00:33:23,779 --> 00:33:28,970
square of that so this has to be divided when
you divide this 2
290
00:33:28,970 --> 00:33:36,000
whatever you get has to be taken as the next
digit in this place and the remainder is 1
291
00:33:36,000 --> 00:33:50,020
here to bring down this and this si and the
operation is so 3 times and number 1 so at
292
00:33:50,020 --> 00:34:03,279
this stage ahh 5 as remainder then you bring
down the ghana place at the moment you come
293
00:34:03,279 --> 00:34:16,440
to place you have to subtract the ghana ghana
of the previous digit so 2 cube has to be
294
00:34:16,440 --> 00:34:23,149
removed from here and you remove so you get
43 here then again repeat the same operation
295
00:34:23,149 --> 00:34:24,149
with
296
00:34:24,149 --> 00:34:31,089
ok 3 times 12 square ok so you get 1 and then
again 1 square of the last to determine 3
297
00:34:31,089 --> 00:34:39,730
is 12 here now ok at this stage when you got
2 the was 1 so when you move to 1 the is 12
298
00:34:39,730 --> 00:34:49,960
and the so it will be 2 now let us see what
is the rationale behind this procedure which
299
00:34:49,960 --> 00:34:56,550
has been given by aryabhatta this is straight
forward for us to see so consider a 3 digit
300
00:34:56,550 --> 00:35:00,760
number so this three digit number can be represented
as axsquare+bx+c ok
301
00:35:00,760 --> 00:35:10,690
x represents 10 fine the cube of this number
axsquare+bx+c will have these terms so we
302
00:35:10,690 --> 00:35:18,250
can group them out it is clear so that i have
done is cube of this and then i have thought
303
00:35:18,250 --> 00:35:26,319
of group to them ok as powers of x so the
maximum thing is going to be so a cube and
304
00:35:26,319 --> 00:35:34,500
the this to be multiplied by x to the power
of 6 here this is the largest term and coefficient
305
00:35:34,500 --> 00:35:44,520
is going to be a cube when we consider the
next x to the power of 5 so the coefficient
306
00:35:44,520 --> 00:35:45,730
will be 3 a square b so
307
00:35:45,730 --> 00:35:48,330
then you do x power 4 this will be the coefficient
x cube will be the coefficient and this is
308
00:35:48,330 --> 00:35:57,920
how the cube of this number can be written
down when you look at the operation which
309
00:35:57,920 --> 00:36:30,530
has been given by aryabhatta so all that he
said was see when you have the maximum digit
310
00:36:30,530 --> 00:36:41,300
you have to remove the maximum cube that can
be remove so you subtract this so this is
311
00:36:41,300 --> 00:36:48,450
what is operation cube has the maximum digit
it has to be done then in
312
00:36:48,450 --> 00:37:03,110
the next stage all that he said was you have
to divide see we know the cube of this number
313
00:37:03,110 --> 00:37:21,800
and what you want to find out is basically
abc so what the first state to determine a
314
00:37:21,800 --> 00:37:33,890
then you in order to determine b all that
you have to do is you have to divide by 3a
315
00:37:33,890 --> 00:37:46,990
square so there is a operation which he says
so you have to do that so you will basically
316
00:37:46,990 --> 00:37:56,589
get b at this stage so what needs to be done
if you have to remove 3 ab square in order
317
00:37:56,589 --> 00:38:03,550
to get this ok so this is basically the principle
behind and this process has to be completely
318
00:38:03,550 --> 00:38:05,250
repeated so at this
319
00:38:05,250 --> 00:38:19,740
stage see you have to get c so then you have
to do this 3 times ab square so this is the
320
00:38:19,740 --> 00:38:30,320
basic algebra which explains the process of
extraction of cube root as described by aryabhatta
321
00:38:30,320 --> 00:38:37,401
you can easily see here 3 times a square is
a first thing then see is one operation the
322
00:38:37,401 --> 00:38:47,510
other operation is subtracted so here divide
the third operation is again you have to understand
323
00:38:47,510 --> 00:38:50,790
the so 1 division and 2 subtraction so that
you what we can see here so we
324
00:38:50,790 --> 00:39:03,980
have this subtraction in the ghana place subtraction
in this place so this is repeated and once
325
00:39:03,980 --> 00:39:34,050
it is done you have be able to get
the cube root of the given number ok this
326
00:39:34,050 --> 00:39:44,920
has been repeated so at this stage since you
are applying the a and b 2 things have been
327
00:39:44,920 --> 00:40:38,390
obtained so a+b and is c do so this is the
operation which has been described from a
328
00:40:38,390 --> 00:40:55,520
different view point if you
look at the algorithm through understanding
329
00:40:55,520 --> 00:41:00,720
of the decimal place value
330
00:41:00,720 --> 00:41:23,740
system the other way it seems to
331
00:41:23,740 --> 00:41:46,589
be impossible for anybody to describe an operation
in so systematically by which you
332
00:41:46,589 --> 00:42:12,900
will be able to extract the square
root
333
00:42:12,900 --> 00:42:46,690
and it also sort of ahh indicated as to how
they have been able to do this out of algebraic
334
00:42:46,690 --> 00:43:08,040
manipulation so if aryabhatta has to given
the algorithm so he should have analyse so
335
00:43:08,040 --> 00:43:26,570
very clearly so the process which goes in
humming and the way of extracting 3 cube root
336
00:43:26,570 --> 00:44:07,650
is
the reverse process of
337
00:44:07,650 --> 00:44:08,650
it
338
00:44:08,650 --> 00:44:31,780
which is
339
00:44:31,780 --> 00:46:37,010
what has
been presented by aryabhatta in
340
00:46:37,010 --> 00:49:30,369
this beautiful world so more
341
00:49:30,369 --> 00:50:00,109
about aryabhatta in
342
00:50:00,109 --> 00:50:01,260
next lecture thank you