Part 14 - Twenty-one kinds of numbers

As noted on p. xxiii, unity is outside the sphere of calculation. Numbers fit for calculation (gananasankhya) begin with 2, and go up to the highest possible infinity. They are classified under three groups: (1) sankhyata (numerable), (2) asankhyata (innumerable³) and (3) ananta (infinite). The first group has three subdivisions viz., jaghanya (lowest), madhyama' (intermediate) and utkrsta (highest). The second group has three main divisions viz., (1) paritta, (2) yukta and (3) asankhyata, each of which is again of three types known as (1) jaghanya, (2) madhyama and (3) utkrsta.. Thus in all, the second group

4-6 This is only a rough rendering. 7 This is also styled as ajaghanyotkrsta. 5 gani0

has 9 divisions. The third group, too, has the same number of divisions; for, firstly ananta has three divisons viz., (1) paritta, (2) yikta and (3) ananta, and secondly each of these three has three subdivisions viz., (1) jaghanya, (2) madhyama and (3) utkrsta. All these 3+9+9 i. e. 21 classes of numbers can be hence represented as under :Gananasankhya Sankhyata Asankhyata Ananta (1) Jaghayna, (1) Paritta-J.M.U.1 (1) Paritta-J.M.U. (2) Yukta-J.M.U. (2) Madhyama, (2) Yukta-J.M.U. (3) Utkrsta (3) Asankhyata-J.M.U. (3) Ananta.-J.M.U. The number 2 is the jaghanya-sankhyata. The number 3 and the following up to one preceeding the utkrsta-sankhyata come under the class known as madhyama-sankhyata. Utkrstasankhyata is explained by means of an example as under :Suppose we have four palyas each of the size of the Jambudvipa whose diameter is 100,000 yojanas, whose circumference is 316, 227 yojanas, 3 gavyutis, 128 dhanusyas, 13 angulas and a little more, whose depth is 1000 yojanas, which has a jagati 8 yojanas in height and a vedika, two yojanas in height. I Out of these four palyas named as anavasthita, salaka, pratisalaka and mahasalaka let us fill the first with white mustard seeds, and then start throwing one seed out of them in Jambudvipa, another in Lavanasamudra and SO on in the successive dvipas and samudras of the Jaina cosmography. When all the seeds are exhausted, let us construct another palya having its diameter equal to that of the dvipa or samudra where the last seed was thrown. This palya, too, must be of the same depth and height as the anavasthita palya. Let this newly constructed palya be also named as anavasthita. Let us fill this with seeds as before and start once more throwing a seed in dvipas and samudras till this palya becomes empty. Let us throw one sced in salaka this 1 J, M and U stand for jaghanya, madhyama and utkrsta respectively.

time with a view to note that anavasthita became empty. Once more we should now construct a new palya having depth and height as before but having its diameter equal to the dvipa or samudra where the last seed was thrown. Let us fill this palya again named as anavasthita and start throwing seeds as before. When this gets emptied, a seed is to be thrown in salaka. When this process is repeated for a number of times, it will so happen that salaka will be completely full. At this stage we should again construct a new palya as before and fill it up with seeds. Then we should commence throwing seeds from salaka till it gets emptied. This time we should throw one seed in pratisalaka and start throwing seeds from anavasthita. When this process is repeated several times, salaka will become full. Then this palya should be emptied as before and to mark that stage one seed must be thrown in pratisalaka. In course of time, this process when repeated, will fill up pratisalaka. We should then start throwing seeds from it till it becomes. empty and to note that stage, we should throw one seed in mahapratisalaka. Let us then start emptying salaka which has been already filled up with seeds by this time. When it becomes empty, a seed is to be thrown in pratisalaka, and the process of emptying anavasthita, throwing one seed in salaka, constructing a new anavasthita etc. is to be repeated till all the four palyas including the anavasthita finally constructed get filled up with seeds. On this stage being reached, we should make a heap of seeds of these four palyas and add to it all the seeds thrown in various dvipas and samudras. When this work is over, let us count the number of the seeds. When one is deducted from the number thus obtained, the remaining number is spoken of as utkrsta-sankhyata. This utkrsta-sankhyata number of the early Jainas may be compared with what is called "Alef-zero" in modern Mathematics. This number is explained in "The theory of functions of a real variable and the theory of Fourier's Series" by E. W. Hobson Sc. D., F. R. S. (A. D. 1907, p. 154), as under:- 1 According to Hemacandra Suri's commentary (p. 236) on Anuyogadara one seed was thrown even earlier. What is stated here is, however, in accor dance with Lokaprakasa (I, 140).

"The cardinal number of the aggregate of all the finite integers I, 2, 3,.........n, is called Alef-zero, and is denoted by 2; thus = {1}. The number x. is identical with the number which has been previously denoted by a". 2. By adding unity to the utkrsta-sankhyata (the highest numerable), jaghanya-paritta-asankhyata (the lowest nearly innumerable) is obtained. Then follow the intermediate numbers which form the class known as madhyama-paritta-asankhyata until utkrsta-paritta-asankhyata (the highest nearly innumerable) is reached. Which is this 'highest nearly innumerable'? The answer is as under:Jaghanya-paritta-asankhyata multiplied by itself not only once but jaghanya-paritta-asankhyata times,³ leads to a number called jaghanya-yukta-asankhyata. This number diminished by one goes by the name of utkrsta-paritta-asankhyata. Numbers between jaghanya-yukta-asankhyata and utkrstayukta-asankhyata form the class known as madhyama-yuktaasankhyata. Jaghanya-yukta-asankhyata multiplied by itself jaghanyayukta-asankhyata times gives us a number styled as jaghanyaasankhyata-asankhyata. This number diminished by one is utkysta-yukta-asankhyata. Jaghanya-asankhyata-asankhyata when multiplied by itself jaghanya-asankhyata-asankhyata times gives rise to jaghanyaparitta-ananta. This number diminished by one is utkrstaasankhyata-asankhyata. This Jaghanya-baritta-ananta multiplied by itself jaghanyaparitta-ananta times comes to jaghanya-yukta-ananta. number diminished by one is utkysta-paritta-ananta. Jaghanya-yukta-ananta multiplied by itself jaghanya-yukta. ananta times leads to jaghanya-ananta-ananta. This number dimished by one is utkrsta-yukta-ananta. 1. "A cardinal number is characteristic of a class of equivalent aggregates". It is so defined on p. 8 in "The Theory of functions of a real variable and the theory of Fourier's series" (p. 8) above referred to. 2 "The cardinal number . is greater than all the finite cardinal numbers and it is less than any other transfinite cardinal number" (Ibid., p. 155). 3 This is called abhyasa of jaghanya-paritta-asankhyata; for, abhyasa means a number raised to itself e. g. the abhyasa of x is xx. 4 This is equal to the number of samayas in one avali.

All numbers beyond jaghanya-ananta-ananta come under the class known as madhyama-ananta-ananta; for, according to the canonical works there is nothing like utkrsta-ananta-ananta. The Karma-granthas agree up to the definition of jaghanyayukta-asankhyata. Thereafter they differ as under:Jaghanya-yukta-asankhyata multiplied by itself and then diminished by unity gives utkrsta-yukta-asankhyata. The addition of one to this number gives jaghanya-asankhyata-asankhyata. Find out the square of this jaghanya-asankhya-asankhyata, then its square and again its square. Add to this number 10 particular asankhyatas. This resulting number is to be squared and then this process is to be repeated twice. The result thus arrived at, is jaghanya-paritta-ananta. This number diminished by unity is utkrsta-asankhyata-asankhyata. The abhyasa of jaghanya-paritta-ananta is equal to jaghanyayukta-ananta. This number which corresponds to the number of the abhavyas, when diminished by unity equals utkrstaparitta-ananta. The square of jaghanya-yukta-ananta comes to jaghanyaananta-ananta. This number diminished by one is utkrstayukta-ananta. Find out the square of the jaghanya-ananta-ananta, then find out its square and then find out the square of this resulting number. Add to this number finally obtained, six particular anantas. The number thus got is to be squared. This resulting number is also to be squared. Repeat this process once more. Then the number arrived at, gives us utkrsta. ananta-ananta, when the ananta paryayas of kevala-jnana and those of kevala-darsana are added to it. 1 This is in short, the eighth power of jaghanya-asankhyata asankhyata. 2 They are: (i-iv) The pradesas of (a) lokakasa, (b) of dharmastikaya, (c) of adharmastikaya, and (d) of a soul, (v-vi) adhyavasayasthanas of sthitibandha and anubhaga, (vii) indivisible parts of mental, vocal and physical yogas, (viii) the samayas of kalacakra, (ix) pratyeka jivas and (x) the bodies of the anantakayas. Here, everywhere 'number' is understood. 3 They are:-(i) the number of the vanaspatikayas, (ii) the number of the nigodas, (ii) the number of the liberated, (iv) the number of the paramanus, (v) the number of the samayas of the time (past, present and future) and (vi) the number of the pradesas of alokakasa.

Dr. Datta has treated this subject in his article "The Jaina School of Mathematics" (pp. 141-142) as under:- "Consider a certain trough which is of the size of the Jambudvipa whose diameter is 100,000 yojanas, and whose circumference is 316, 227 yojanas, 3 gavyuti, 128 dhanus, 13 angula and a little more. Fill it up with white mustard seeds counting them one after another. Continue in this way to fill up with mustard seeds other troughs of the sizes of the various lands and seas of the Jain cosmography. Still it is difficult to reach the highest number amongst the numerables. So the highest numerable number of the early Jainas corresponds to what is called Alef-zero in modern mathematics. For numbers beyond that Anuyoga-dvara-sutra further proceeds: By adding unity to the higest 'numerable', the lowest 'nearly innumerable' is obtained. After that are the intermediate numbers until the highest 'nearly innumerable' is reached. Which is the highest 'nearly innumerable"? The lowest 'nearly innumerable' number multiplied by the lowest 'nearly innumerable' number and then diminished by unity will give the highest 'nearly innume. rable' number. Or the lowest 'truly innumerable' number diminished by unity gives the highest 'nearly innumerable' number. Which is the lowest 'truly innumerable'? The lowest 'truly innumerable' is obtained by multiplying the lowest 'nearly innumerable' number by itself; or by adding unity to the 'highest nearly innumerable' number. This number is also equivalent to Avali. After that are the intermediate numbers until the highest 'truly innumerable' number is reached. Which is this highest 'truly innumerable' number? It is the lowest 'truly innumerable' number multiplied by the Avali and then diminished by unity; or the lowest 'innumerably innumerable' number decreased by unity. Which is the lowest 'innumerably innumerable' number? It is the lowest 'truly innumerable' multiplied by Avali or the highest 'truly innumerable' number increased by unity. After that, are the intermediate numbers until the highest 'innumerably 1 See "The Bulletin of the Calcutta Mathematical Society" vol. XXI, No. 2, 1929.

innumerable' is reached. Which is the highest 'innumerably innumerable' number? It is the lowest 'innumerably innumerable' number multiplied by itself and then diminished by unity, or the lowest 'nearly infinite' number diminished by unity. Which is the lowest 'nearly infinite' number? The lowest 'innumerably innumerable' number multiplied by itself or the highest 'innumerably innumerable' increased by unity. After that are the intermediate numbers until the highest 'nearly infinite' is reached. Which is this highest 'nearly infinite' number? The lowest 'nearly infinite' number multiplied by itself and the product decreased by unity; or the lowest 'truly infinite' decreased by unity. Which is the lowest 'truly infinite' nnmber? The lowest 'nearly infinite number' multiplied by itself, or the highest 'nearly infinite' increased by unity. It is also called the Abhavisiddhi. After that are the intermediates until the highest 'truly infinite' is obtained. Which is the highest 'truly infinite' number? The lowest 'truly infinite' number multiplied by the Abhavisiddhi and diminished by unity or the lowest 'infinitely infinite' number diminished by unity. Which is the lowest 'infinitely infinite' number? It is the lowest 'truly infinite' number multiplied by the Abhavisiddhi number, or the highest 'truly infinite' added by unity. After that are intermediate numbers. Such are the numbers of calculation." He further observes: "It will be easily recognised that the above classification can be represented by the following series: 2...N (N+1)...{ (N+1)²-1 } 1 (N+1)²...(N+1)*-1 } 1 (N+1){ (N+1)-1} 1 (N+1)³... (N+1)-1} | 32 (N+i)...{ (N+i)3 } \(N+i)3... where N denotes the highest numerable number as defined before...... The series contains as recorded in the work the extreme numbers of each class and the different classes have been separated by a vertical line. It will be noticed that in the classification of numbers stated above there is an attempt to define numbers beyond Alef-zero....The fact that an attempt was made in India to define such numbers as early as the first century before the

Christian era, speaks highly of the speculative faculties of the ancient Jaina Mathematicians."1

1 For a treatment of numbers according to the Digambara sources, the reader is referred to the "Jaina Gem Dictionary" (pp. 140-148).