Infinitesimals through Indian polynomials

Created: 2025-10-07 Tue 16:01

Introduction

Polynomials and non-Archimedean arithmetic

  • We have seen that the arithmetic of polynomials is non-Archimedean
  • Unlike \(\mathbb{R}\) which is Archimedean
  • non-Archimedean arithmetic has both infinities and infinitesimals,
  • It does not have any limits.

Polynomial arithmetic traditionally existed in India

  • From ancient times.
  • Brahamagupta in his chp, 18 of Brāhma-Sphuṭa-Siddhānta (ब्राह्म-स्फुट-सिद्धांत)
  • uses the term avyakta gaṇita (अव्यक्त गणित)
  • or "the ganita of unexpressed numbers"
  • for numbers represented by polynomials,
  • involving an "unknown" \(x\), such as \(2x+3\).
  • I believe avyakt ganita started long before Brahamgupta
  • who only gave the first coherent account of it
  • and solved quadratic equations.
  • According to Dutta and Singh (B. Dutta and S. N. Singh. History of Hindu Mathematics: A Source Book. Asia Publishing House, vol. II, p. 6, 1935)
  • the need to solve polynomial equations arose in the context of the problem of constructing Vedic altars.
  • E.g. to construct a square with side \(c\), and the same area as a rectangle of a given side \(a\), requires the solution of $ax = c².$

यावत तावत

  • The term for the unknown \(x\) was yavat tavat (यावत तावत् ).
  • (यावत = as much as, तावत = so much).
  • Eng trans. does not make good sense
  • Better to say यावत = जितना,
  • तावत = उतना.
  • As in नद्यां यावत् जलम् अस्ति, तावत् जलं सरोवरे नास्ति.
  • = "जितना जल नदी में है, उतना जल सरोवर में नहीं."
  • “As much water there is in the river, so much is not there in the lake”
  • which sounds natural in Hindi but artificial in English.
  • This term yavat tavat for \(x\) is BETTER than using the term "unknown",
  • because unknown may also mean what is forever unknowable,
  • as in "what happens at the far edges of the cosmos is unknown [and may be forever unknowable]".
  • However, yavat tavat means जितना होगा उतना
  • "as much as it will be, so much",
  • making clear that the value of \(x\) is currently unknown
  • but will be known eventually, i.e, it is "unknown but will be known".
  • It is regrettable that none of the Sanskrit translations makes this elementary point clear.

Jaina contribution

  • Brahmagupta is NOT the first to use the term yavat tavat.
  • This term was used in the Sthānāṅga-sūtra (before - 300 CE.)
  • In the Bakhshali MS it was called yadṛcchā, vāñchā or kāmika (any desired quantity).
  • That is to say, algebra began long before Brahmagupta and Bhaskara.
  • Hence, also wrong to use the term "Hindu algebra" for two reasons.
  • 1. Because there is no Christian algebra or Muslim algebra,
  • because algebra does not relate to religious beliefs and it is foolish to use terms which suggest such a link.
  • 2. Because Jains who were not Hindu too were involved from the earliest times,
  • and we don’t want to create a fight between Hindus and Jains.
  • (The Brits always wanted to create religious quarrels,
  • e.g. between Hindus and Muslims, to help them to “divide and rule)”

Method of false position

  • So, it is not about the term yavat tavat etc.
  • but about the method, today called the method of False Position
  • used e.g. in Bakshali to solve equations.
  • E.g. To solve \(ax + b =p\),
  • assume any value \(g\) and evaluate \(ag + b =p'\) .
  • Then the solution is \(x = \frac{p-p'}{a} + g\)

Bijaganita

  • Translation: "What was said earlier [Lilavati] was vyakt ganita [expressed arithmetic]
  • of which avyakt ganita [unexpressed arithmetic] is the seed [bija];
  • often problems [such as the solution of quadratic equations in Lilavati]
  • cannot be solved without avyakt ganita [… therefore, I will now explain the bija methods]"

Several variables

  • Indians used polynomial in several variables,
  • instead of \(x, y, z\), etc, after yavat tavat these were denoted by varna-s = colors.
  • As explained by Bhaskara II these were colors such as kalaka, nilak, lohita, pita
  • or just the first letters य or या, क, नी, लो etc.

Translation of Beejaganita 21

  • "Yāvattāvat (so much as), kālaka (black), nīlaka (blue), pīta (yellow), lohita (red)
  • and other colours have been taken by the venerable professors as notations for the measures of the unknowns,
  • for the purpose of calculating with them. […] (The maxim), ‘colours such as yāvattāvat, etc., should be assumed for the unknowns,’
  • gives (only) one method of implying (them).
  • Here, denoting them by names, the equations may be formed by the intelligent (calculator)".

Others

  • According to the Amarakosh of the celebrated Sanskrit lexicographer Amarasiṃha (f. 400 A.D.)
  • In the case of more unknowns, it is usual to denote the first by yāvattāvat
  • and the remaining ones by colours or the alphabets with which those color-names begin.
  • Similar statements are made by others before Bhaskara II
  • Pṛthūdakasvāmī (860) in his commentary on the Brāhmasphuṭasiddhānta of Brahmagupta (628) says:

-"In an example in which there are two or more unknown quantities, colours such as yāvattāvat, etc., should be assumed for their values".

  • Śrīpati (1039) in the Siddhāntaśekhara: "Yāvattāvat and colours such as kālaka (black), nīlaka (blue), etc., should be assumed for the unknowns".

Coefficients and the constant term

  • The constant term in a polynomial was denoted by rupa.
  • The coefficients of the unknowns were denoted by anka, samkhya, gunaka (multiplier) etc.

Powers

  • Powers were denoted by varga (square), ghana (cube), varga-varga (4th power), ghana-varga (5th power) etc.
  • Brahmagupta 18-41 has a more systematic terminology, पञ्चगत (5th power), षड्गत (6th power) etc.
  • where गत corresponds to the modern Hindi term for exponent घात (which term was used by Brahmagupta for multiplication).

Fractional powers

  • Fractional powers were also used as in प्रथम वर्ग मूल (first square root, of, say \(a\)) = \(\sqrt a\)
  • द्वितीय वर्ग मूल = \(\sqrt {\sqrt a} = a^{\frac{1}{4}}\)
  • तृतीय वर्ग मूल = \(a^{\frac{1}{8}}\)
  • तृतीय वर्ग मूल घन = cube of the third square root = \[ a^{\frac{3}{8}}\]

Algebra

  • The word algebra used today for avyakta ganita or Beejaganita
  • comes from al Khwarizmi’s book al jabr wa’al Muqabala (full title: al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wa’al-Muqābalah )
  • which translated Brhamagupta.
  • What does this title mean?
  • Western translators who translated al Khwarizmi did a bad job:
  • they could not correctly translate the 2 terms jabr and Muqabala.
  • The typical translation of the title as given in Wikipedia etc. is
  • The Concise Book of Calculation by Restoration and Balancing.

Bad translation

  • The term jabr actually means forceful or by force, as in jabrdasti
  • or jabariyas= (determinists).
  • The term muqabala means contest as in this Bollywood song
  • about a dance contest between Vyjantimala (classic Bhartanatyam)
  • and Helen (free form) who played a vamp and did cabaret dances for Bollywood.
  • So, a better translation of "al jabra waal muqabala" is:
  • (al jabr) forcible (re)-solution
  • by means of a contest (muqabala)
  • between two sides (paksha-s) of an algebraic equation (समीकरण = equalization).
  • Of course, al Khwarizmi too did a bad job of translating Indian ganita.
  • For example, he has no negative numbers
  • which necessarily arise in the solution of quadratic equations.

Linear and quadratic equations

  • Various type of equation were solved
  • Linear equation in one variable (एकवर्ण समीकरण)
  • simultaneous linear equations in 2 or more variables
  • Quadratic equations

Linear equations

  • Goes back to Sthananga sutra 747 and Bakhshali MS.
  • As stated by Brahmagupta 18.43 (also trans. and Eng. trans.)
  • That is, the solution of the linear polynomial equation \(ax+c=bx+d\) is
  • \[x= \frac{d-c}{a-b}\].

Simultaneous equations

  • For simultaneous equations of the form

\[̱ \begin{align}x+y&=a\\ x-y&=b\end{align}\]

Quadratic equations

Positive and negative polynomials and zero

  • Brahmagupta in his Brāhma Sphuṭa Siddhānta (18-30-35)
  • has a number of verses on positive, negative entities, and zero.
  • West has wrongly treated this as the first account of zero and negative numbers.
  • Foolish to believe what the West says
  • when they had so many difficulties in understanding zero and negative numbers.

Recall that place value-system (स्थान-मान प्रणाली) existed since Veda-s

  • Numbers till parārdha \(10^{12}\) found in Yajurveda 17.2
  • Numbers till tallakshana \(10^{53}\) found in

Lalita vistara sutta.

  • compared to numbers limited to a myriad (10⁴) in Western pebble arthmetic
  • But how do you write even 108 in place-value system without zero?

Europeans failed to understand zero

Suspicion of zero

  • Adding zero at the end can inflate a contract
  • (not possible with Roman numerals: at most IIII can be added at the end)
  • Hence Florence passed a law against zero in 1299.

Fibonacci failed to understand negative numbers

  • a Florentine trader understood the efficiency of Indian arithmetic
  • hence its great usefulness for commerce
  • Alas, he failed to understand negative numbers (like AL Khwarizmi).
  • Compare Fibonacci Liber Abaci toc with Mahavira's toc

This rejection of negative numbers till Augustus De Morgan 19th c.

  • A very influential Western mathematician of the 19th c.
  • Professor of math at University College London.
  • E.g.1 Algebra and Calculus [p. xi]
  • E.g.2 Study of Difficulties in math [p. 72]
  • A dunce, but not alone, part of an influential group
  • of Western mathematicians who rejected negative numbers.
  • Why would you want to rely on these mathematicians?
  • or the Wikipedia which cites only them to promote the Western narrative.
  • and chat bots trained on that data
  • As a result of such colonial influence even the Hindi translation of those verses is wrong

Brahamgupta's verses about avyakt ganita

  • NOT just about vyakt ganita.
  • negative numbers and zero existed in vyakt ganita from long before.
  • Check the original Sanskrit
  • Is there any mention of anka as stated in the Hindi trans.?
  • Stock English translations are even more horrible.

Infinitesimals and sum of infinite geometric series

  • But the sum of the INFINITE geometric series was first given by Nīlakanṭh
  • You learn that $a + ar + ar² + … = \frac {a}{1-r} $, \(r <1\)
  • Instead of a common multiplicand \(r\) take a common divisor \(d = \frac{1}{r}\)
  • the n the above sum becomes \(\frac {a}{1-r} = \frac {a} {1-\frac{1}{d}}= \frac {ad}{d-1}\), \(d > 1\).

How was the sum obtained?

  • Simple polynomial arithmetic tells us
  • \((1+x+x²+ ... +xⁿ)(1-x)= (1-x^{n+1})\)
  • Hence, the sum of the geometric series \[(1+x+x²+ ... +xⁿ) = \frac {(1-x^{n+1})}{1-x}\]
  • With non-Archimedean arithmetic if \(n\) is infinite
  • and $x < 1 $, then \(x^{n+1}\) is infinitesimal, and can be neglected
  • so that the infinite sum is just \(\frac{1}{1-x}\).
  • This shows how infinitesimals and non-Archimedean arithmetic enable us to sum an infinite series.
  • This is similar to limits by order-counting as described in Cultural Foundations of Mathematics
  • \(\frac{xⁿ}{xᵐ}\), with \(n

Zeroism

  • The critical step in the above derivation is that an infinitesimal is zeroed: because mathematics is not exact,
  • so that something insignificant may be neglected or discarded or zeroed.
  • This is in line with the use of terms like सविशेष (with an avasesh), सानित्य (स अनित्य = impermanent), आसन्न (near value) etc, sicne the sulba sutra-s.
  • Ganita does have exactitude.
  • Note that an infinitesimal is zeroed at the end of a calculation.
  • Why not at the beginning?
  • This is similar to the situation \(1+\epsilon = \epsilon\) with floats.
  • \(\epsilon\) itself is not zero; it is only neglected relative to 1.
  • At the beginning of a calculation there is nothing to compare the quantity \(\epsilon\) with.

Rational functions were explicitly used in India

  • Note that order-counting is easily done for polynomials \(p(x)\)
  • $ p = O (xⁿ)$
  • where \(xⁿ\) is the leading term of \(p(x)\)
  • Similarly for a rational function \(r(x)= \frac{p(x)}{q(x)\), if $ p = O (xⁿ)$, and $ q = O (xᵐ)$, \(r= O (x^{n-m})\).

Kha-hāra

  • Note the peculiarity that Brahmagupta refers to division by śūnya as "cheda"
  • Likewise, Bhaskara Bijaganita 18 refers to division by zero as kha-hāra
  • The meanings are not quite clear, and
  • I have discussed this in various ways.
  • Since infinitesimals are zeroed at the end of a calculation
  • one possibility is that the term sunya could be ambiguously used for both infinitesimal and zero.
  • Indeed, long ago (Raju, C. K. ‘Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhāṣā’. Philosophy East and West 51, no. 3 (2001): 325–62. http://ckraju.net/papers/Hawaii.pdf.

)

  • I suggested that this way \(\frac{0}{0}=0\) is also implicitly used in current mathematics
  • especially in the Lebesgue integral, as in current statistics
  • Another way is to suppose that infinitesimals were regarded as non-representables
  • as in non-normal floats
  • There is a third way.
  • Regardless of what has been said by ancient Indian mathematicians
  • and regardless of the results of standard analysis
  • we can decide on our own what is the most practical and feasible way to follow
  • We have seen that there is no need for fluxions
  • and calculus based on real analysis does not work for all situations in physics
  • Therefore, there is no need to preserve the results of standard analysis
  • by first constructing real numbers, then hyper real numbers etc.
  • On the other hand, there is no need to go literally by everything written in ancient Indian texts
  • Perhaps our ancestors were right, but things have changed
  • For example, the ancient planetary model no longer works accurately
  • so we should change it to suit current circumstances, or use a better model.
  • Likewise, for all applications today
  • Aryabhata's numerical method of solving differential equations works very well
  • Also, non-Archimedean arithmetic of polynomials
  • combined with Zeroism works fine to some infinite series.
  • Therefore, this is the way we will choose to do calculus
  • for its practical value and ease.
  • Historical controversies about what BrahmaGupta and Bhaskar said and and will go on.