Jain Mathematics : Numbers and units of space and time

Note : This article is written for those who find it difficult to read Hindi. For those who can read Hindi , it is recommended to read the original material from “ Samyak Gyan Chandrika by Pandit Todarmalji ”. Here I have tried to simplify the language, however the original Hindi words are retained as it is since there are no equivalents.

The relationship between numbers is represented by 14 series in Jainism. The very first is the Sarva Dhara which is nothing but serial sequence of numbers as 1,2,3,……and so on. However the important difference is that the last number is keval gyan, which we have seen as the highest infinity and no number can exceed that. It may also be noted that this number is believed to be even and divisible by 4. The other series is sequence of even numbers and odd numbers and so on. But the most interesting is  the series of square of 2, number 2 being the min samkhyat ( Numeral).

The Series of Square of 2 ( Dwiroop varg dhara)

The series starts with square of 2 being first number and the subsequent numbers are square of the previous number. This can be seen in the table below:

Position	1	2	3	4	5	6
Number	4	16	256	65536	4294967296	1844-----
	2^21	2^22	2^23	2^24	2^25	2^26

The table shows the first six numbers of this series. Each position in this series is called Varga Shalaka. For example the number 65536 has varga shalaka of 4 since that is its position in the series. This number is given the name Panathhi since it begins with number 65. The fifth position is a 10 digit number which is called Badaal since it starts with number 42. The next number in the series is 20 digit number called Ikathhi since it starts with number 18. These names itself tell us how important these numbers were  in ancient times since they were given specific names. Another word which needs to be learnt at this stage is called “Ardhachheda” which is Log₂(Number)  , in other words   2 ^Ardhachheda  = Number

For example if Number is 32 then Ardhachheda =  5  since 2⁵ =32

If Number =65536 then ardhachheda= 16   since  Ardhachheda= Log₂( Number)  = Log₂(2^2⁴)= 2⁴= 16

Another relationship which can be seen is that Ardhachheda=  2^{Varga shalaka}

We can also derive that Varga shalaka = Log₂(Log₂Number)= ardhachheda of ardhachheda

It can be easily seen that although they did not call this  mathematics as Logarithms, the principle  was just that. Another interesting thing any student of computer science would have noticed is the remarkable use of  series of square of 2, since that forms the basics of binary system. In Jainism also the decimal numbers were not so much prominent as the binary numbers to base of 2.

With this introduction we will proceed with our series of square of 2.

It can be seen that the number grows at a very fast pace in this series. The sixth position itself give the number as 2^2⁶ = 2⁶⁴ . This number is prominent in Jainism since 2⁶⁴-1 is the total number of non repetitive letters possible with 64 letters of vocabulary hence that is the total extent of sruta gyana. Now if we proceed further in this series , we find that virtually all the important quanitities are part of this series. This is denoted by indicating its position in the series of square of 2 , as follows:

1)      Position – several samkhyat position ( where samkhyat represents numerable)

Number = varga shalaka of Min ParitaSamkhyat   ( where Min ParitaSamkhyat is min innumerable). Note that the number has not reached Min Paritasamkhyat but its varga shalaka which is far smaller number that Min ParitaSamkhyat ( MPS).

2)      Position – several samkhyat position further

Number = ardhachheda of MPS

3)      Position – several samkhyat position further

Number = square root of MPS

4)      Position -  one position later

Number =  MPS

5)      Position – several samkhyat position after 4)

Number = Min Yuktasamkhyat = One Avali

Where avail is a unit of measurement of time.

6)      Position – Next position after 5)

Numbet = square of Avali = Prataravali = Min Asamkhyatasamkhyat

7)      Position – several innumerable steps after 6)

Number = varga shalaka of AddhaPalya 

AddhaPalya is a unit of measurement of time

8)      Position – several innumerable steps after 7)

Number = ardhachheda of addhapalya

9)      Position- several innumerable steps after 8)

Number = square root of AddhaPalya

10)   Position- one step after 9)

Number = AddhaPalya

11)   Position – several innumerable steps after 10)

Number = Soochyangul

Where soochyangul is the number of spatial units(Pradesh) in one Angul space.

12)   Position – next step after 11)

Number=  Pratangul  ( square of soochangul)

13)   Position – several innumerable steps after 12)

Number= Cubic root of Jagatshreni

Where Jagatshreni is the height of Lokakash in spatial units of Pradesh.

14)   Position – several innumerable steps later

Number = Varga shalaka of Min Paritanant

Number = Ardhachheda of Min Paritanant

Number = Square root of Min Paritanant

Number = Min Paritanant

Here four steps are given which occur in sequence each after several innumerable steps except the fourth which is immediate next after square root of Min Paritanant.

15)   Position – several innumerable steps later

Number = Min Yuktanant

16)   Position – next step after 15)

Number = Min Anantanant

17)   Position – several Infinite steps after 16)

Number =  Varga shalaka of Jivas

Number = Ardhachheda of Jivas

Number = Square root of Jivas

These three occur after several infinite steps after each other. Jivas is the number of all  life forms in the universe.

18)   Position – Next step after third part of 17)

Number = Jivas

19)   Position – several infinite steps after 17)

Number = Number of Permaanu ( atoms) in the universe

This shows the relationship between number of life forms to the matter.

20)   Position – Several infinite steps after 19)

Number = Number of time units from past to future in units of Samay.

21)   Position – several infinite steps later

Number = Height  of the Akash in spatial units of Pradesh.

Here the Akash implies both Lokakash and Alokakash together. While Lokakash is bound space in which all jivas and matter is present. Alokakash is inifinite space surrounding the Lokakash wherein no jivas or matter or time are  present.

22)   Position -  Next step after 21)

Number = Number of space units in Pratarakash

Pratarakash is square of the Akash spatial elements in  height form.

23)   Position – After infinite steps

Number = Number of avibhaag pratichhedas of Agurulaghutva quality in Dharma and Adharma dravyas

Here abibhaag pratichheda represents the smallest indivisible unit of the quality of agurulaghutva . Dharma and Adharma are two primary dravya or elements of the universe. Agurulaghutva is one of the qualities possessed by all the six prime  dravyas or elements of the universe.

24)   Position – After infinite steps

Number = Number of avibhaag pratichhedas of the agurulaghutva quality of the single Jiva which increases in six orders of magnitude namely: infinitesimal increase, innumerable part increase, numerable part increase, numerable times increase, innumerable times increase, infinite times increase. The decrease also occurs in a similar six order manner.

25)   Position – After infinite steps

Number = Number of avibhaag pratichhedas of the min infinitesimal  single letter knowledge of the smallest Nigod Jiva.

26)   Position – several infinite steps later

Number = Number of avibhaag pratichhedas of the quality of Kshayik labdhi of asamyat samyak drishti Tiryanch (animal) Jiva.

27)   Position – several infinite steps after 26)

Number =   varga shalaka of Keval Gyan

Keval Gyan is the knowledge units of omniscient.

28)    Position – several infinite steps later

Number = ardhachheda of keval gyan

29)   Position – several infinite steps later

Number = Eighth square  root of the keval gyan

30)   Position - + 1,2,3,4,5,6,7 steps later

Number = 7^th, 6^th, 5^th, 4^th, 3^rd, 2^nd and 1^st square root of keval gyan

31)   Position – Next step after 30)

Number = Avbhaag Pratichhedas of the Keval Gyana

Thus we see that the series of the square of 2 contains quite important numbers , namely number of Jivas, Number of permaanu , number of akash Pradesh, number of time units , as well as Keval Gyan. Thus in  a single series we get to know the relationship of one with the other.

Series of Cube of 2 ( Dwiroop Ghan dhara)

The previous series started with square of 2, this series starts with cube of 2 and the subsequent numbers are square of the previous ones.  The first four positions are shown below:

Position	1	2	3	4
Number	8	64	4096
	23	43	163	2563

The series can be worked out similar to the previous of the square of 2. It may be noted that this series will not overlap with the previous one. It means that the same number will not appear in both series. On the other hand it is complementary to the previous with some important differences which are highlighted below:

1.       While series of square of 2 has cube root of Jagat sreni or the height of lokakash as one of the elements  {see 14) in previous description}, here Jagat sreni is the constituent.

2.       While previous series had Number of Jivas as the elements, here cube of same occurs.

3.       In this series the cube of entire akash i.e. lokakash and alokakash together is a constituent.

4.       The last number in this series is cube of the second square root of Keval gyan. If we try to derive the next number then we find that it becomes cube of the square root of Keval Gyan which will be higher than kevan gyan . Hence such a number is not possible.

Series of cube of cube of 2 ( Dwiroop Ghanaghan dhara)

            This is the last of the 14 series which are used to describe numbers and their relationship. This series is that of cube of the cube of 2, and the subsequent numbers are square of the previous ones.

Position	1	2
Number	512	262188
	(23)^3	((23)^3)^2

As can be seen above the series grows at a very fast pace. The important numbers here are :

1.       The number of Pradesh  or units of space of Lokakash.

2.       Several numbers pertaining to the Nigod and single sensed JIvas  and their life span.

3.       Here the last number is the fourth square root of Keval Gyan

With these 14 series the relationship of most of the important quantities related to numbers , space, duration and even the bhava i.e the number of  thoughts of the mind are related to each other and one can understand the magnitude of those numbers.

Units of Space and Time

Units of space : Pradesh   The smallest element in the universe is permaanu or atom. However it is not same as the atom described by our science. In fact it is infinite times smaller than atom because it is the single particle by which all the atoms  and matter has been made. In that sense it is similar to the God particle  described today. In Jain philosophy such a particle is cause of all the material seen. The permaanu is indivisible smallest particle of matter which occupies a single unit of space which is called Pradesh. It is also described as the distance travelled by one permaanu at slowest speed in one samay where samay is the smallest unit of time. It is important to note that slowest speed is mentioned since at highest speed the permaanu can travel from one end of the universe to the other.

Now permaanus join with one another and create skandh or molecules. This joining also follows specific laws and is quite scientific. Interestingly the matter is named Pudgala which itself means one who has property of associating and disassociating. This is precisely what matter is wherein the permaanus keep adding or subtracting at any given time. Even infinite permaanus form the skandh which will still not be visible. The smallest is called avasannasann skandh. Without going into details we quote the value 8¹¹ times of this skandh will be equal to one maize seed. 8 of these equal to one angul i.e. one finger. This is also known as Soochyangul or Vyavhar angul. However this is another angul called Prananangul which is 500 times that of previous one. This corresponds to the finger of Bharat Chakravarti king in the fourth  cycle of Avasarpini kaal prevailing now. At that time the body of humans was largest equal to 525 Dhanush. His angul is called Praman angul since all measurements with respect to the seas, oceans and space are related to that absolute measurement. The table below now shows  the relationship of the space units further.

500 Utsedh Angul	1 Praman Angul
6 specified angul	1 paad
2 paad	1 vitasti
2 Vitasti	1 hand
2 hands	1 rikku
2 rikkus, 4 hands	1 dand or dhanush
2000 dand/ dhanush	1 kos
4 kos	1 yojan
500 vyavhar yojan	1 praman yojan

       This praman yojan is the basic unit by means of which the islands and seas are measured. For example the diameter of Jambu dweep on which we reside has a diameter of 1 lakh yojan.

Innumerable soochanguls together equal to one Jagat sreni i.e the length of lokakash in which all jivas and matter reside. The space is called Akash which is divided into two parts namely lokaksh and alokakash . In lokakash all the other 5 dravyas or the elements of the universe are found namely Jiva, ajiva, kaal, dharmastikaya and adharmastikaya. The volume of lokakash is equal to cube of Jagat sreni also known as ghan loka. The length of Jagat sreni is also known as 7 Raju where 343 cube raju is the volume of the lokakash.

Unit of Time : Samay  As per Jain scripture the lokakash is packed with units of time known as kaal which is known as different dravya or one of the six constituents of the universe. The lokakash is supposed to be comprised of innumerable number ( not infinite) of pradesh units while alokakash comprises of infinite space units surrounding the lokakash.  While kaal element is packed through out the universe, it by itself is inert but is the cause of modification in the jiva and matter at every instant. Hence its function is that of a catalytic agent. From a practical point of view the time taken by a single permaanu to cross one pradesh of space is called one samay which is the smallest unit of time. This is indivisible unit of time far far smaller than the unit of time which we call “second”.

Innumerable samay units constitute one Avali  which is still smaller than second. This innumerable number is Min Yuktasamkhyat which we have encountered in the article on infinity ( Jan 09/2017)

The following table represents some of the other time units

Min Yuktasamkhyat	1 Avali
Numerable Avali	1 breath = 2880/3773 seconds
7 breaths	1 stok
7 stok	1 love
38 ½ love	1Ghadi =24 Minutes
1 Ghadi= 48 minutes	1 Muhurta
30 Muhurta	24 hours = 1 day
30 days	1 month
12 months	1 year

However in Jainism the time measurement needed to be far far higher than years. Even millions of years are not sufficient to define age of a typical Heaven or Hell resident. Hence another type of unit is used which is called Upama Maan  i.e. the unit defined by Simile or Anology.

Upama Maan Imagine a well having the diameter of 1 Praman Yojan and a depth of 1 Praman Yojan. This filled to the brim with the tips of the hair of newborn sheep ( which is born in superior Bhoga Bhumi , where the hair is thinnest). The number of hair in  this well is called one Palya or Palyopam. This number can be computed to be equal to a 45 digit number. Now take out one hair at an interval of 100 years each. The period in which all the hair are taken out, measured in number of samay units is equal to the number of samay in one Vyavhar Palyopam.

                Now multiplying this with number of samay of innumerable years give rise to number of samay of Uddhar Palyopam. Further multiplying with another number of samay of innumberable years give rise to number of samay of Addha Palyopam. 10 Koda Kodi of Addha palya is equal to one Addha Sagar which is a major unit of measurement of life span in heaven or hell, period of Karmas and so on. One koda kodi represents one karod karod or 10⁵x10⁵. Incidentally  the analogy of Sagar is quite apt because this one sagar period would be the time taken to empty the water of Lavan Sea with surrounds our Jambu dweep taken  at a specific rate.

                Further the place like Bharat Kshetra where we reside undergoes time cycle of upward or downward trend. Currently we are undergoing downward trend known as Avasarpini Kaal. This duration is total of 10 Koda Kodi Sagar in which we positioned at  about 2500 years of the fifth period known as Dushma kaal. This cycle would terminate at about 39500 years from now and the Utsarpini kaal of 10 koda kodi sagar would start. Thus 20 koda kodi sagar constitute one kalpa kaal.

Conclusion  The purpose of such units of time and space demonstrates the limitless of time and space  as told by the omniscient. It also conveys to us the insignificance of our life span compared to the millions and millions of years in which we are migrating from one form to another without reaching any type of destination. If we take a look at the presentation of Panch Paravartan , it would be more clear to us. Typically the Jivas  have a maximum life span of 2000 sagar in which they take birth as  two sensed or above  life form , after which invariably they revert to one sensed life form in which they may spent unlimited years. Hence the urgency for self realization  is felt more strongly if we keep in mind this fact.