bhAskarAchArya’s vision of infinity in arithmetic

Infinity as a number has always been a point of great commotion for mathematicians throughout the history. The Romans didn’t have a symbol of zero in their arithmetic. Whether they avoided zero to avoid division by zero is a matter for a different post. In this post, however, we will take a look at bhAskarAchArya‘s vision of infinity.

utpAdakam yat pravadanti buddheradhishTitam satpurusheNa sAMkhyAH |
vyaktasya krutsnasya tadekabIjamavyaktamISam gaNitam cha vande ||

उत्पादकम् यत् प्रवदन्ति बुद्धेरधिष्टितम् सत्पुरुषेण संख्याः |
व्यक्तस्य कृत्स्नस्य तदेकबीजमव्यक्तमीशम् गणितम् च वन्दे ||

This is the first shloka from bhAskarAchArya‘s 2nd part of siddhAnta shiromaNi known as “bIja gaNita“[1]. Apart from the obviousness of the shloka, one can only wonder what bhAskarAchArya was experiencing when penning that verse. Note the sheer reverence to the subject of the text – “gaNita“! “bIja” literally means seed. bhAskarAchArya uses the shabdaavyaktabIjam” and explains why we must study it in the following shloka.

pUrvam prOktam vyaktamavyaktabIjam prAyaH praSnAno vinA avyaktayuktyA |
jnAtum SakyA mandadhibhinitAntam yasyAttasmAt avachmi bIjakriYam cha ||

पूर्वं प्रोक्तं व्यक्तम्व्यक्तबीजम् प्रायः प्रश्नानो विना-अव्यक्तयुक्त्या |
ज्ञातुं शक्यामन्दबुद्धिभिनितान्तम् यस्यात्तस्मात् अवच्मि बीजक्रियाम् च ||

In leelAvati, bhAskarAchArya doesn’t talk about complex concepts. He uses simple rational number arithmetic and most of the examples he used are from day to day life. The true mathematical perspective of bhAskarAcharya seems to emerge from bIja gaNita. As he explains in the opening shlokas, in bIja gaNita, he goes into details of that which “avyakta” i.e., unapparent. He explains that he is formulating bIja gaNita because the avyakta cannot be understood or known even by paNditas without proper tools to question and answer.

Scholars like Colebrooke and others seem to conclude that “examples quoted by bhAskarAchArya were problems for leisure”. A serious study of original texts seems to indicate that Colebrooke’s conclusion may have been prejudiced (we stress “may” because Colebrooke had the intellectual honesty to credit Hindu mathematicians for some of the most fundamental discoveries in Mathematics[2]). There is, however, lot more to bhAskarAchArya‘s work than simple math.

It would be an injustice to bhAskarAchArya to take only maths out of siddhanta SiromaNi and ignore the great statements he made like the one in first shloka. To bhAsharAchArya, avyaktabIjam is the manifestation of the avyaktamISam. Therefore, he says, this analysis is necessary to know the unapparent “ISa“. Unfortunately, even a scholar like Colbrooke seems to translate “avyaktabIjam” as “unknown quantity” but doesn’t see beyond what is apparent. The intellectual depth of bhAskarAchArya‘s work comes out as he moves further into uncomfortable areas in arithmetic.

In one of the shlokas, bhAskarAchArya calls the result of division by 0 as “khahara“. Had he left it there, Colebrooke would have been right but the author seems to suggest something more. bhAskarAchArya says the result of division by 0 is the “ananta”. He says that like “achyuta” (i.e., Vishnu) from whom the whole world (including bhUtagaNa) appears at the time of creation, adding or deleting anything from the quantity khahara doesn’t affect the latter i.e., remains as is.

What bhAskarAchArya called as “khahara“, modern world calls as “the infinite”. The original shloka is as follows:

asmin vikAraH khahare na rASAvapi pravishTeshtatrapi nihsruteshu |
bahushvapi syAllayasrushtikAle-anante-achyute bhUtagaNeshu yadvat ||

अस्मिन् विकारः खहरे न राशावपि प्रविष्तेश्वपि निः स्रुतेषु |
बहुष्वपि स्याल्लय स्रुष्तिकाले अनन्ते अच्युते भूतगणेषु यद्वत् ||

To give a sense of the times of bhAskarAchArya to the reader, bhAskarAchArya was a 12th century scholar and during this time, the English were in periods of anarchy, episodes of usurpation, wars and witch hunts. The quality of thought during the same period in India sustained by mathematicians like bhAskarAchArya most certainly is worth our time and devotion.


[1] “Bijaganita – Elements of Algebra with Sanskrit commentary of Sri Jiva Natha Jha” by Pandit Sri Achyutananda Jha

[2] “Translation of Lilavati and Bijaganita” by H T Colebrooke


Here is a question to the interested reader: Where can you find original manuscripts of bhAskarAchArya‘s siddhAnta SiromaNi?

The answer is : Wellcome Library, London. Refer here Imagine. These manuscripts were our wealth. British took them to England. Now they would like to sell a digitized copy to us. You might say “they preserved it and we would have not”. Think again why that argument is flawed and fundamentally so.


One comment

  1. Interesting article Vivek 🙂

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