1 00:00:18,230 --> 00:00:26,890 so far we looked at the mathematical concepts that were available in some of the most ancient 2 00:00:26,890 --> 00:00:35,769 text so vedas and sulvasutras the next 3-4 talks we will be focusing our attention on 3 00:00:35,769 --> 00:00:48,390 a single text called aryabhatiya of aryabhatta aryabhatta who was in the later part of the 4 00:00:48,390 --> 00:00:57,980 5th century and his text aryabhatiya is one the most seminal text on indian astronomy 5 00:00:57,980 --> 00:01:02,489 and mathematics so we will start with the period 6 00:01:02,489 --> 00:01:11,360 of aryabhatta and then we will have a brief description on the work aryabhatiya aryabhatiya 7 00:01:11,360 --> 00:01:18,080 as i was mentioning so while discussing this decimal place value system as we compose in 8 00:01:18,080 --> 00:01:24,890 a very first time and therefore without it will be extremely difficult for us to understand 9 00:01:24,890 --> 00:01:33,300 the contents so the earliest commentary that is available for this is one by bhaskara so 10 00:01:33,300 --> 00:01:43,380 bhaskara lived in the early part of 7 century and it is a very profound commentary on aryabhatiya 11 00:01:43,380 --> 00:01:44,380 we 12 00:01:44,380 --> 00:01:50,890 will see a brief description of where bhaskara lived and what was his period and then we 13 00:01:50,890 --> 00:01:58,709 will move on to this text percy so aryabhatta first introduces the notational places then 14 00:01:58,709 --> 00:02:07,380 he start discover describing about the fundamental operations square squaring finding out the 15 00:02:07,380 --> 00:02:14,069 square root cube root and so on then he moves on to various other topics the text aryabhattiya 16 00:02:14,069 --> 00:02:21,840 as such is composed in four parts we will discuss all that in great detail but 17 00:02:21,840 --> 00:02:33,129 before i move on to that i just wanted to say about the very name aryabhatta so aryabhatta 18 00:02:33,129 --> 00:02:40,709 so this is how one can derive the word aryabhatta so bhata normally means a soldier or guardian 19 00:02:40,709 --> 00:02:48,420 and so on so the term arya refers to a noble person in fact very interesting definition 20 00:02:48,420 --> 00:03:01,310 has been presented in one of the text called shabdakalpadruma so that have noted down this 21 00:03:01,310 --> 00:03:12,409 is a very beautiful definition so the one who performs what is to be performed by 22 00:03:12,409 --> 00:03:17,780 thing so this is not sufficient and you also process in the negative form so this what 23 00:03:17,780 --> 00:03:26,659 is called have to be done but i will also engage myself in a serious activities that 24 00:03:26,659 --> 00:03:36,379 is not acceptable and therefore that is acceptable in the society at that particular point of 25 00:03:36,379 --> 00:03:47,230 time so the one who so the aryabhatta cab be derived in 2 ways so the one who protects 26 00:03:47,230 --> 00:03:55,930 that and one who save guardian and so on and so forth any way so term can be derived and 27 00:03:55,930 --> 00:04:00,370 the very name 28 00:04:00,370 --> 00:04:07,659 of the text is based on the name of the author so sometimes it just happens the text could 29 00:04:07,659 --> 00:04:17,040 be completely different so here it is called so that which belongs to aryabhatta or that 30 00:04:17,040 --> 00:04:25,480 which has been composed by aryabhatta so that is why it is called the name aryabhata appears 31 00:04:25,480 --> 00:04:32,560 twice in the text so which is not quite common in many of the text in the indian literature 32 00:04:32,560 --> 00:04:38,380 but here write in the beginning of deepika pada which is the first section so he 33 00:04:38,380 --> 00:04:47,030 mention his name and again once more in the next section called ganitha pada the time 34 00:04:47,030 --> 00:04:54,791 of aryabhatta is also pretty clear to us that is no ambiguity because there is a verse in 35 00:04:54,791 --> 00:05:17,290 iii section goes like this 36 00:05:17,290 --> 00:05:29,580 i was 23-years-old at a certain point of time so that is mention in the first part of 37 00:05:29,580 --> 00:05:40,030 the solga so so when 3600 years of the i was 23-year-old so when we go back and then find 38 00:05:40,030 --> 00:05:45,330 out so what was the beginning of so it just happens that these 499 ad 39 00:05:45,330 --> 00:05:53,940 so people are opinion that this verse has been composed in and therefore aryabhatta 40 00:05:53,940 --> 00:06:00,130 was 23-year-old and he compose this work so this actually clearly tells that aryabhatta 41 00:06:00,130 --> 00:06:12,710 was born in 4676 ad coming to aryabhatiya as i was mentioning so this is the text which 42 00:06:12,710 --> 00:06:21,910 is very clearly datable and it was composed in 499 and as far as we know this is the earliest 43 00:06:21,910 --> 00:06:28,400 datable text that is available in full form for us so they have been in earlier words 44 00:06:28,400 --> 00:06:29,750 in fact aryabhatta himself 45 00:06:29,750 --> 00:06:44,530 towards the end the work he says he says i entered into the ocean of knowledge with the 46 00:06:44,530 --> 00:06:49,710 intellect as board and i lifted up gym so which means i mean he is trying to point out 47 00:06:49,710 --> 00:06:54,280 that there has been earlier literature so from which he has carried out some important 48 00:06:54,280 --> 00:07:01,780 things and presented in this work so this aryabhatiya is available for us in complete 49 00:07:01,780 --> 00:07:08,640 form and the earlier works which were available for aryabhatta have been shot of compiled 50 00:07:08,640 --> 00:07:10,520 by varahamihira in his 51 00:07:10,520 --> 00:07:17,970 text called panchasiddhantika will come little later more or less contemporary so the systematically 52 00:07:17,970 --> 00:07:27,300 discuss the procedures for planetary computation and it has been composed in a very cryptic 53 00:07:27,300 --> 00:07:32,150 style and therefore some people prefer though it is in the form of verses it is in the form 54 00:07:32,150 --> 00:07:43,970 of aryametre so the text is many times refer to as sutra itself and there are 4 padas as 55 00:07:43,970 --> 00:07:49,710 i said 4 sections deepikapada 13 verses ganitapada 33 verses kalakriyapada 5 verses 56 00:07:49,710 --> 00:07:58,090 golapada 50 verses in all we find 121 versus composed in arya meter which consists of the 57 00:07:58,090 --> 00:08:06,120 text aryabhatiya adn in fact yesterday you might have that srinivas was mentioning that 58 00:08:06,120 --> 00:08:16,170 so is also called as 108 so this 108 says it just drop out the 13 verses so it just 59 00:08:16,170 --> 00:08:25,310 happens to be 108 so we will see it little more in greater detail later why this 108 60 00:08:25,310 --> 00:08:31,340 so to give you an idea of what are the contents of this 4 chapter so gotikapada so basically 61 00:08:31,340 --> 00:08:32,940 he llays 62 00:08:32,940 --> 00:08:39,279 out all the parameters that are essential for doing astronomical computations in the 63 00:08:39,279 --> 00:08:46,670 to be described in the later part of the work then ganitapada as a name itself implies discuss 64 00:08:46,670 --> 00:08:51,730 all the necessary mathematics that required for doing this planetary competition starts 65 00:08:51,730 --> 00:08:58,290 with square square root cube cube root and then it moves on to discuss the solution for 66 00:08:58,290 --> 00:09:02,499 indeterminate equations of first order and which is called which will be discuss the 67 00:09:02,499 --> 00:09:04,000 great length ahh by 68 00:09:04,000 --> 00:09:12,629 professor sriram and also myself with you later in kalakriyapada we primary find the 69 00:09:12,629 --> 00:09:19,540 geometrical picture and implies by the computational procedure it has been discussed in great detail 70 00:09:19,540 --> 00:09:24,959 and in golapada the variety of topics which are discuss so it is in fact almost occupy 71 00:09:24,959 --> 00:09:31,449 50% of 50 verses in golapada so it discuss various details regarding the shape of the 72 00:09:31,449 --> 00:09:37,709 earth so the source of light and planets calculation of eclipse visibility of planets and so forth 73 00:09:37,709 --> 00:09:45,060 so these are the various topics which are discussed in golapada before i proceed into 74 00:09:45,060 --> 00:09:46,060 the text 75 00:09:46,060 --> 00:09:52,529 aryabhatiya i will say a couple of words on bhaskara commentary because i will be more 76 00:09:52,529 --> 00:09:59,040 or less dealing both of them together so i am not going to take a bhaskara separately 77 00:09:59,040 --> 00:10:09,350 so we will see a brief note on bhaskar bhaskar as actually return three major works one is 78 00:10:09,350 --> 00:10:16,480 aryabhatta bhartiya as i was mentioning the other 2 are sort of independent works but 79 00:10:16,480 --> 00:10:24,930 bhaskara describes them as aryabhatta means so it a primarily exposition on what has been 80 00:10:24,930 --> 00:10:25,930 described in 81 00:10:25,930 --> 00:10:33,189 aryabhatta that is how we describe those who are called that these 3 words put together 82 00:10:33,189 --> 00:10:38,839 so a lot of light on the kind of mathematical knowledge as well as the astronomical theories 83 00:10:38,839 --> 00:10:47,949 which were present around that period so bhaskara's time as we estimated to the 629 ad and his 84 00:10:47,949 --> 00:10:55,389 work actually displays a great amount of scholarship and it is really in intellectual peace to 85 00:10:55,389 --> 00:11:00,980 be his bhasya so i had an occasion to go through in great detail in connection with this 86 00:11:00,980 --> 00:11:14,589 course in fact in one place he says so he refers to a certain number and you can take 87 00:11:14,589 --> 00:11:44,879 this as an exercise now yesterday we discussed 0 and then agni rama so 3 rama and therefore 88 00:11:44,879 --> 00:12:06,190 3 and and so on you can see this so based on the last four digits ahh 1000 times itself 89 00:12:06,190 --> 00:12:22,510 is 4320000 years so based on some ahh analysis for it has been shown by that this corresponds 90 00:12:22,510 --> 00:12:30,649 to 629 ad bhaskara was just about 140 years after aryabhatta very close 91 00:12:30,649 --> 00:12:39,300 to the period of composition of aryabhatiya then there are references in this work as 92 00:12:39,300 --> 00:12:47,420 well as the work of brahmagupta that aryabhatta had many so one of the most famous one was 93 00:12:47,420 --> 00:12:55,249 refers to him in various places and also brahmaputra refers to them ok this was an introduction 94 00:12:55,249 --> 00:13:03,870 now we move on to the text aryabhatiya itself so the first section gitikapada consists of 95 00:13:03,870 --> 00:13:08,579 13 versus as i was mentioning so verse 1 is indicatory work and 96 00:13:08,579 --> 00:13:33,649 it goes like so we find a very clear statement where aryabhatta himself says aryabhatta states 97 00:13:33,649 --> 00:13:43,209 so what does he state so so this gitikapada so which essentially present numbers certain 98 00:13:43,209 --> 00:13:49,540 parameters so is considered to be out of the text in some set so the details of person 99 00:13:49,540 --> 00:13:55,309 so the numbers because which are required and the kind of find aa sin table etc are 100 00:13:55,309 --> 00:14:01,899 all essential for doing computation but they need not be integrated with the text perse 101 00:14:01,899 --> 00:14:02,899 and you want to 102 00:14:02,899 --> 00:14:08,949 understand astronomy so it is in this sense i mean that it has been segregated out and 103 00:14:08,949 --> 00:14:26,629 the last verse goes like this 104 00:14:26,629 --> 00:14:33,339 this is called the so leaving out the first verse and the last verse and the kind of colours 105 00:14:33,339 --> 00:14:42,890 with all that she says so once a person knows this so it is a kind of but what is the point 106 00:14:42,890 --> 00:14:50,879 that i want to convey here was this so this very clearly tell that the basic text of aryabhatiya 107 00:14:50,879 --> 00:14:55,069 is leaving out gitikapada and just consists of 108 verses and 108 00:14:55,069 --> 00:14:59,300 there is also a palace with the end in fact one find another invitation at the beginning 109 00:14:59,300 --> 00:15:06,040 of ganitapada so ganesha see invocation basically marks the beginning of the text so we find 110 00:15:06,040 --> 00:15:11,660 2 invocation so one is completely separated out and then again in ganitapada i will find 111 00:15:11,660 --> 00:15:17,949 invocation so we have basically 3 part of aryabhatiya ganita and gola what we are going 112 00:15:17,949 --> 00:15:40,889 to discuss here is only ganitapada ok the ganitapada commences with this verse 113 00:15:40,889 --> 00:15:47,679 basically offering my veneration my veneration to what you can easily guess so 114 00:15:47,679 --> 00:16:08,019 these are the names for the planets so leave it out who starts from refers to moon then 115 00:16:08,019 --> 00:16:16,420 group of stars constellations so to all the celestial bodies and then i start ahh this 116 00:16:16,420 --> 00:16:29,389 work aryabhatta means states states what so aryabhatta does not playing that i am going 117 00:16:29,389 --> 00:16:40,740 to say completely everything out of my head so it is not that so all that he says is so 118 00:16:40,740 --> 00:16:48,370 the knowledge which was highly rewired in place called and going to narrate here 119 00:16:48,370 --> 00:17:06,730 so commenting up on these word in bihar and earlier it was called place of great learning 120 00:17:06,730 --> 00:17:13,730 where even nalanda university was existing at a point of time aryabhatta says that i 121 00:17:13,730 --> 00:17:19,079 actually narrate the knowledge which was highly review in this place and so further bhaskara 122 00:17:19,079 --> 00:17:22,510 says is being hurt so what is hurt in fact here we find reference to all the five which 123 00:17:22,510 --> 00:17:23,940 has been compiled in ah so he says means bramha so is 124 00:17:23,940 --> 00:17:29,010 brahma siddhanta so we having other so by the people and basing on aryabhatta has presented 125 00:17:29,010 --> 00:17:32,789 his work after the invocatory verse the next work basically presents the ahh notational 126 00:17:32,789 --> 00:17:37,490 places so he says so the names of the various places in fact this is extremely essential 127 00:17:37,490 --> 00:17:41,970 for even deciphering the number which has been given by aryabhatta so he basically tells 128 00:17:41,970 --> 00:17:44,270 the means so 10 to the power 9 the names of the powers of 10 has been listed there is 129 00:17:44,270 --> 00:17:46,679 an interesting discussion made by bhaskara at this point he says what is so special about 130 00:17:46,679 --> 00:17:49,590 this shakti means a certain potential so when we declare something may be something it has 131 00:17:49,590 --> 00:17:51,830 been device by you but it has the potential to convey some meaning to you very important 132 00:17:51,830 --> 00:17:54,429 meaning so he asked the question so what is the potential that have been associating with 133 00:17:54,429 --> 00:17:59,690 this and what purposes he says so finally what he is going to convey is so price 134 00:17:59,690 --> 00:18:03,590 something so you can just place it in that place and thereby he conveys something so 135 00:18:03,590 --> 00:18:07,200 that has been something just been created by you it service a very useful purpose in 136 00:18:07,200 --> 00:18:12,809 the day today transaction so that is kind of discussion that represents here then he 137 00:18:12,809 --> 00:18:19,309 moves on to describe the fundamental operations so we start with varga and he defines what 138 00:18:19,309 --> 00:18:24,540 does the term varga mean he says so one is a certain geometrical representation so we 139 00:18:24,540 --> 00:18:25,540 have square 140 00:18:25,540 --> 00:18:29,900 and we have squaring scaring actually refers to the operation and what is represented by 141 00:18:29,900 --> 00:18:34,330 this operation both are created here in this thing 4 sided figure ; refers to square then 142 00:18:34,330 --> 00:18:40,380 he says so mathematically what does it represent 2 equal quantities so the product of 2 equal 143 00:18:40,380 --> 00:18:45,010 quantities is what is represented by there and both are referred to as varga ok this 144 00:18:45,010 --> 00:18:46,740 then he ask in the commentary bhaskara processor interesting question when you say so we 145 00:18:46,740 --> 00:18:52,260 can also think of a rhombus so rhombus is also an object which has 4 sides equal why 146 00:18:52,260 --> 00:18:57,700 does the word varga convert rhombus so can it also convert rhombus or not so this is 147 00:18:57,700 --> 00:19:03,700 the question he says see in fact this is the very important statement in the sense that 148 00:19:03,700 --> 00:19:10,890 in order to understand the meaning of a particular word so we have to go to the society to whatever 149 00:19:10,890 --> 00:19:18,620 sense it has been used by people he send the world and therefore he says this particular 150 00:19:18,620 --> 00:19:21,071 shape rhombus has been never refer to be people by this word and therefore it does not denote 151 00:19:21,071 --> 00:19:26,160 rhombus it has the potential only to convert square then bhaskara also so all these are 152 00:19:26,160 --> 00:19:29,360 synonyms for square then i move on to describe this algorithm which has been presented by 153 00:19:29,360 --> 00:19:31,820 aryabhatta to extract square root ok varga refers to square vargamula refers to square 154 00:19:31,820 --> 00:19:35,820 root so the verse goes like this in fact yesterday while i discuss to the 155 00:19:35,820 --> 00:19:37,779 aryabhatta systems of representing numbers i said right so can refer to the square and 156 00:19:37,779 --> 00:19:44,279 when we think of the powers of 10 so 10 to the power 0 1 is a square number then 10 to 157 00:19:44,279 --> 00:19:50,420 the power of 2 is a square number so similarly in describing the operation which has to be 158 00:19:50,420 --> 00:19:55,840 done to extract the square root he clearly states when we have a certain number which 159 00:19:55,840 --> 00:20:01,899 has been given to you you have to first of all break that into 2 units so one is the 160 00:20:01,899 --> 00:20:02,899 varga part and the 161 00:20:02,899 --> 00:20:11,049 other is the so in understanding this verse this has to be kept in mind so here refers 162 00:20:11,049 --> 00:20:18,659 to that which is ahh 10 to the power of 1 10 to the power of 3 and so on so this is 163 00:20:18,659 --> 00:20:25,200 called so the he says so the first work has to be understood the vanga which is the square 164 00:20:25,200 --> 00:20:30,659 of some number subtracted so the process will be very clear but i just try to make certain 165 00:20:30,659 --> 00:20:34,909 terminology clear before we show a certain example so let us see this 166 00:20:34,909 --> 00:20:40,399 example now so here we have a number 55225 and this has to be written down like this 167 00:20:40,399 --> 00:20:45,490 putting varga avarga varga avarga varga avarga so this is what he means by in the sloga varga 168 00:20:45,490 --> 00:20:54,929 avarga and so on and the procedure to be adopted can be stated in 2 steps then he will move 169 00:20:54,929 --> 00:21:01,220 on to the example so in bhaskara commentary what we find is so he says divide ok and he 170 00:21:01,220 --> 00:21:03,610 says so whenever you encounter a certain thing so all that you need to do you 171 00:21:03,610 --> 00:21:07,679 have to do a certain division and whenever we find this varga you move want to replace 172 00:21:07,679 --> 00:21:18,049 all that you need to do is means you have to remove a certain square so we will see 173 00:21:18,049 --> 00:21:23,070 the algorithm and the basis of the algorithm 2 so in fact bhaskara very says a certain 174 00:21:23,070 --> 00:21:31,950 nume 54321 so this even place so this is odd so all the hot places so they are referred 175 00:21:31,950 --> 00:21:41,830 to as varga so this is varga so all even spaces are referred to us avarga so this is 176 00:21:41,830 --> 00:21:50,039 basically from the fact where the 10 the power of 0 10 to the power of 2 10 to the power 177 00:21:50,039 --> 00:21:55,539 of 4 and so on so basically this has to be divided into two so one is why is it done 178 00:21:55,539 --> 00:22:01,500 so basically a single digit so if you take the square of it also can be only two digit 179 00:22:01,500 --> 00:22:05,881 therefore so depending upon the number of digits and number which has been presented 180 00:22:05,881 --> 00:22:12,780 you first make a guess of how many digits in the square root have yes so this is this 181 00:22:12,780 --> 00:22:13,780 classification 182 00:22:13,780 --> 00:22:20,779 separate into two so so the algorithm essentially has 3 steps so what is states is starting 183 00:22:20,779 --> 00:22:27,730 from the least significant digit group the digits of the number into two so the first 184 00:22:27,730 --> 00:22:33,960 thing that needs to be done is what i was saying so some the least you just group them 185 00:22:33,960 --> 00:22:40,100 into two then from the remaining part the most significant digit so which constitutes 186 00:22:40,100 --> 00:22:49,789 the so this is pretty evident so this is so then this basically is so whatever be the 187 00:22:49,789 --> 00:22:50,980 most significant digit it 188 00:22:50,980 --> 00:22:57,399 can be one or it can be 2 so that is just taken in the beginning and so which constitutes 189 00:22:57,399 --> 00:23:06,610 the subtract the maximum square that is possible so having done that so long with the reminder 190 00:23:06,610 --> 00:23:14,240 so you bring down the next digit of the so this is then once you remove the maximum square 191 00:23:14,240 --> 00:23:24,820 that is possible for remove for instance when you have 5 to the maximum square that can 192 00:23:24,820 --> 00:23:35,370 be removed is 2 square so one will be remaining and 4 will be remaining in 193 00:23:35,370 --> 00:23:39,030 the next step you have to start your operation was 14 so along with the reminder bring down 194 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 195 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 196 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 198 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 199 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 200 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 201 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 202 00:24:22,639 --> 00:24:24,210 number 55225 so we have group in this and 203 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 204 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 2 205 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 206 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 207 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 209 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 210 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 211 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 212 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 213 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 214 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 215 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 216 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 217 00:26:24,820 --> 00:26:31,899 extracting square root at the same way aryabhatta so let us take one more 218 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 219 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 220 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 221 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 222 00:27:01,520 --> 00:27:08,450 keeping in mind that the next step ahh you do not get a negative number 223 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 224 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 225 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 226 00:27:22,289 --> 00:27:26,480 these are the 2 operations which have to be repeated after describing the extraction of 227 00:27:26,480 --> 00:27:32,690 square root he moves on to the square cube cubing and ahh how to find cube 228 00:27:32,690 --> 00:27:41,470 root so this is the next operation so understand clearly lay down the procedure for extracting 229 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 230 00:27:49,509 --> 00:28:04,250 you have quantity multiply by 3 times so that gives what is called similarly so geometrically 231 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 232 00:28:11,809 --> 00:28:17,899 kind of line ahh so both have been nicely stated and so this is what 233 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 234 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 235 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 237 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 239 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 240 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 245 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 263 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 267 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 281 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 282 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