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'''Algebra''' is one of the broad areas of Mathematics. The Hindu name for the science of algebra is bījagaṇita. Bīja means "element" or | '''Algebra''' is one of the broad areas of Mathematics. The Hindu name for the science of algebra is bījagaṇita. Bīja means "element" or | ||
Brahmagupta (628) calls algebra as ''kuṭṭaka - gaṇita'' or ''kuṭṭaka''. ''Kuṭṭaka'' means pulveriser. Algebra is also called as ''avyakta - gaṇita'' or the science of calculation with unknowns (''avyakta'' means unknown) in contrast to the name ''vyakta - gaṇita'' the science of calculation with knowns (''vyakta'' means known'')'' for arithmetic including geometry and mensuration. | {{Infobox person | ||
| name = Algebra | |||
| image = Quadratic root.svg | |||
| alt = Algebra | |||
}} | |||
"analysis" and gaṇita means " the science of calculation". Bījagaṇita literally means "science of calculation with elements or the science of analytical calculation. | |||
[[Brahmagupta|Brahmagupta (628)]] calls algebra as ''kuṭṭaka - gaṇita'' or ''kuṭṭaka''. ''Kuṭṭaka'' means pulveriser. Algebra is also called as ''avyakta - gaṇita'' or the science of calculation with unknowns (''avyakta'' means unknown) in contrast to the name ''vyakta - gaṇita'' the science of calculation with knowns (''vyakta'' means known'')'' for arithmetic including geometry and mensuration. | |||
== Definition == | == Definition == | ||
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"Neither does analysis consist in symbols, nor are there different kinds of analyses; sagacity alone is analysis, for wide is imagination. "Analysis is certainly clear intelligence". "Or intelligence alone is analysis".In answer to the question, "if (unknown quantities) are to be discovered by intelligence alone what then is the need of analysis ?" he says, "Because intelligence is certainly the real analysis; symbols are its helps. The innate intelligence which has been expressed for the duller intellects by the ancient sages, who enlighten mathematicians as the sun irradiates the lotus, with the help of various symbols, has now obtained the name of algebra. | "Neither does analysis consist in symbols, nor are there different kinds of analyses; sagacity alone is analysis, for wide is imagination. "Analysis is certainly clear intelligence". "Or intelligence alone is analysis".In answer to the question, "if (unknown quantities) are to be discovered by intelligence alone what then is the need of analysis ?" he says, "Because intelligence is certainly the real analysis; symbols are its helps. The innate intelligence which has been expressed for the duller intellects by the ancient sages, who enlighten mathematicians as the sun irradiates the lotus, with the help of various symbols, has now obtained the name of algebra. | ||
[[File:Algebraic equation notation.svg|alt=Algebraic Equation|thumb|Algebraic Equation]] | |||
Thus, according to Bhāskara II, algebra may be defined as the science which treats of numbers expressed by means of symbols, and in which there is scope and primary need for intelligent artifices and ingenious devices. | Thus, according to Bhāskara II, algebra may be defined as the science which treats of numbers expressed by means of symbols, and in which there is scope and primary need for intelligent artifices and ingenious devices. | ||
Bījagaṇita means a 'seed'. The unknown quantities are like a seed and their values become | Bījagaṇita means a 'seed'. The unknown quantities are like a seed and their values become evident when the equations are solved. Since algebra deals with unknown quantities, it is called Bījagaṇita in Samskrit. Kṛṣṇa Daivajña, a renowned mathematician of the 16th Century CE wrote Bijapallava, a commentary on Bījagaṇita (1150 CE) of Bhāskara II. Kṛṣṇa Daivajña, explains the name Bījagaṇita as below | ||
अव्यक्तत्वादिदं बीजमित्युक्तं शास्त्रकर्तृभिः | | अव्यक्तत्वादिदं बीजमित्युक्तं शास्त्रकर्तृभिः | | ||
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== Origin == | == Origin == | ||
The | [[File:Hindu astronomer, 19th-century illustration.jpg|alt=Brahmagupta|thumb|Brahmagupta]] | ||
The origin of Hindu algebra can be definitely traced back to the period of the śulba (800-500 B.C.) and the Brāhmaṇa (c. 2000 B.C.). | |||
B.C.). | |||
"The Hindus were the first to make systematic use of the letters of the alphabet to denote unknowns. They were also the first to classify and make a detailed study of equations. Thus they may be said to have given birth to the modern science of algebra." <ref>Datta, 1938, Vol.2, Preface</ref> | "The Hindus were the first to make systematic use of the letters of the alphabet to denote unknowns. They were also the first to classify and make a detailed study of equations. Thus they may be said to have given birth to the modern science of algebra." <ref>Datta, 1938, Vol.2, Preface</ref> | ||
''<big>ś</big>ulbasūtrās'' mentions variable quantity. ''āryabhaṭīya'' of [[Aryabhata|''āryabhaṭa'']] mentions solution to linear and quadratic equations. Brahmagupta in his ''Brāhma- sphuṭa - siddhānta'' mentions the operations performed on unknowns using symbols. ''kuṭṭakādhyāya'' (chapter 18) explains the ''parikarma'' (calculations) with avyakta (or algebraic symbols). Hence Brahmagupta is treated as the father of algebra. The other texts on algebra includes Mahāsiddhānta of ''āryabhaṭa II ,siddhāntaśekhara ''of ''śripati, Bījagaṇita'' of ''Bhāskara II, bījagaṇitāvataṃsa'' of ''[[Development of Mathematics|nārāyaṇa paṇḍita.]]'' | |||
''Brahmagupta'' has given the rules for arithmetic operations with positive numbers, negative numbers and zero in ''kuṭṭakādhyāya'' of ''Brāhma- sphuṭa - siddhānta. Also equations with one unknown, equations with many unknowns, equations with products of unknowns and indeterminate equations of first and second order (kuṭṭaka and varga-prakṛti) are dealt by Brahmagupta.'' | |||
== Technical Terms == | == Technical Terms == | ||
=== Unknown Quantiy === | === Unknown Quantiy === | ||
The unknown quantity was called in the ''Sthānāṅga-sūtra'' (before 300 B.C.) ''yāvat - tāvat'' (as many as or so much as, meaning an arbitrary quantity). In the so-called ''Bakhshālī'' treatise, it was called ''yadṛcchā , vāñcā or kāmikā'' (any desired quantity) | The unknown quantity was called in the ''Sthānāṅga-sūtra'' (before 300 B.C.) ''yāvat - tāvat'' (as many as or so much as, meaning an arbitrary quantity). In the so-called ''Bakhshālī'' treatise, it was called ''yadṛcchā , vāñcā or kāmikā'' (any desired quantity). āryabhaṭa I (499) calls the unknown quantity as ''gulikā'' (shot). This term strongly leads one to suspect that the shot was probably then used to represent the unknown. From the beginning of the seventh century the Hindu algebraists are found to have more commonly employed the term ''avyakta'' (unknown). | ||
=== Equation === | === Equation === | ||
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== Symbols == | == Symbols == | ||
'''Symbols of Operation.''' There are no special symbols for the fundamental operations in the ''Bakhshālī'' work. Any particular operation intended is ordinarily indicated by placing the | '''Symbols of Operation.''' There are no special symbols for the fundamental operations in the ''Bakhshālī'' work. Any particular operation intended is ordinarily indicated by placing the shorthand abbreviation, the initial syllable of a Sanskrit word of that import, after, occasionally before, the quantity affected. Thus the operation of addition is indicated by ''yu'' (an abbreviation from ''yuta,'' meaning added), subtraction by + ''which is very probably from kṣa (abbreviated from kṣaya,'' diminished), multiplication by ''gu'' (from ''guṇa'' or ''guṇita,'' multiplied) and division by ''bhā'' (from ''bhāga or bhājita'' or'','' divided). | ||
Bhāskara II (1150) says, "Those (known and unknown numbers) which are negative should be written with a dot ''(bindu)'' over them." | Bhāskara II (1150) says, "Those (known and unknown numbers) which are negative should be written with a dot ''(bindu)'' over them." | ||
'''Symbols for Powers and Roots.''' The symbols for powers and roots are abbreviations of Sanskrit words | '''Symbols for Powers and Roots.''' The symbols for powers and roots are abbreviations of Sanskrit words which are placed after the number affected. Hence the. square is represented by ''va'' (from ''varga),'' cube by ''gha'' (from ''ghana),'' the fourth power by ''va-va'' (from ''varga-varga),'' the fifth power by ''va-gha-gh''ā (from ''varga-ghana-gh''ā''ta),'' the sixth power by ''gha-va'' (from ''ghana-varga),'' the seventh power by ''va-va-gha-gh''ā (from ''varga-varga-ghana-gh''ā''ta)'' and so on. The product of two or more unknown quantities is shown by writing ''bh''ā (from bhā''vita,'' product) after the unknowns with or without interposed dots; e.g., yā''va-k''ā''gha-bh''ā or ''y''ā''vak''ā''ghabh''ā means ''(y''ā'')<sup>2</sup> (k''ā'')<sup>3</sup>.'' In the ''Bakhshālī'' treatise the square-root of a quantity is shown by writing after it ''mū'' which is an abbreviation for ''mūla'' (root). | ||
For instance | |||
21 ''yu'' 4 ''mū'' 5 | |||
1 1 1 | |||
Means | |||
<math>\sqrt{21 + 4} = 5</math> | |||
and | and | ||
23 7<sup>+</sup> ''mū'' 4 | |||
1 1 1 | |||
Means | |||
<math>\sqrt{23 - 7} = 4 | |||
</math> | |||
In other treatises the symbol of the square-root is ''ka'' (from ''karaṇī,'' root or surd) which is usually placed before the quantity affected. | In other treatises the symbol of the square-root is ''ka'' (from ''karaṇī,'' root or surd) which is usually placed before the quantity affected. | ||
For example ''ka'' | For example ''ka'' 19 ka ''50 ka'' 57 ''ka 9''4 indicated as <math>\sqrt{19}+\sqrt{50}+\sqrt{57}+\sqrt{94}</math> | ||
'''Symbols for Unknowns''' | '''Symbols for Unknowns''' | ||
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!Symbol | !Symbol | ||
!Meaning | !Meaning | ||
! | !Reference | ||
|- | |- | ||
|''y''ā''vat-t''ā''vat'' | |''y''ā''vat-t''ā''vat'' | ||
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या | या | ||
|as many as | |as many as | ||
|''Sthānāṅgasūtra,'' | |''Sthānāṅgasūtra,'' | ||
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|''ya vā kā'' | |''ya vā kā'' | ||
य वा का | य वा का | ||
| | |desired | ||
quantity | quantity | ||
|''Bakhshālī Manuscript'' | |''Bakhshālī Manuscript'' | ||
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|Brahmagupta, Bhāskara II | |Brahmagupta, Bhāskara II | ||
|} | |} | ||
''Bakhshālī Manuscript'' mentions where there are five unknowns, the first letters of ordinals were used. that is pra from prathama (first), dvi from dvitīya (second), tṛ from tṛtīya (third), ca from caturtha (fourth) and paṃ from pañcama(fifth) to represent the unknowns. | |||
== Laws Of Signs == | == Laws Of Signs == | ||
Kauṭilya's arthaśāstra mentions negative quantities like debt(ṛṇa). Brahmagupta uses the words ''dhana'' and ''ṛṇa'' for denoting positive and negative numbers in ''Brāhma- sphuṭa - siddhānta''. Nowadays Integers include positive numbers, negative numbers and zero<ref>{{Cite book|title=A Primer to Bhāratīya Gaṇitam , Bhāratīya-Gaṇita-Praveśa- Part-1|publisher=Samskrit Promotion Foundation|year=2021|isbn=978-81-951757-2-7}}</ref> . | |||
=== Addition === | === Addition === | ||
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व्यस्तं तदन्तरं स्यादृणं धनं धनमृणं भवति ॥<ref>Brahma-sphuta-siddhanta, ch.18, vs.31 p.309</ref> | व्यस्तं तदन्तरं स्यादृणं धनं धनमृणं भवति ॥<ref>Brahma-sphuta-siddhanta, ch.18, vs.31 p.309</ref> | ||
Brahmagupta | Brahmagupta Says: "From the greater should be subtracted the smaller; (the final result is) positive, if positive from positive. and negative, if negative from negative. If, however, the greater is subtracted from the less, that difference is reversed (in sign). negative ,becomes positive and positive becomes negative. When positive is to be subtracted from negative or negative from positive then they must be added together. | ||
=== Multiplication === | === Multiplication === | ||
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भक्तमृणेन धनमृणं धनेन हृतमृणमृणं भवति ॥<ref>Brahma-sphuta-siddhanta (ch.18, vs.34, p.310)</ref> | भक्तमृणेन धनमृणं धनेन हृतमृणमृणं भवति ॥<ref>Brahma-sphuta-siddhanta (ch.18, vs.34, p.310)</ref> | ||
Brahmagupta | Brahmagupta says: "Positive divided by positive or negative divided by negative becomes positive. But positive divided by negative is negative and negative divided by positive remains negative. | ||
=== Evolution and Involution === | === Evolution and Involution === | ||
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== Negative Quantities == | == Negative Quantities == | ||
Regarding a negative quantity , moving in one direction is considered as positive, moving in the opposite direction is considered as negative. | |||
Kṛṣṇa Daivajña shows positive and negative directions along a line. If east is considered as positive direction, then, west should be considered as negative. T | |||
Kṛṣṇa Daivajña shows positive and negative directions along a line. If east is | |||
Kṛṣṇa Daivajña also | Kṛṣṇa Daivajña also talks about these opposites of negative and positive in terms of kāla (time) and vastu (object). With regard to time, if the future denotes positive, the opposite of it, the past will be negative. If we borrow something, we are indebted to pay it back. It is called ''ṛṇa'' (negative). The opposite of it is ''dhana'' (positive) where we own the object or we are bound to receive something. In common terminology, the two words ''dhana'' and ṛṇa are used for wealth and debt respectively. The idea of negative numbers goes back to Arthaśāstra. | ||
All these concepts are summarized by Kṛṣṇa Daivajña in his commentary as follows: | All these concepts are summarized by Kṛṣṇa Daivajña in his commentary as follows: | ||
ऋणत्वमिह त्रिधा तावदस्ति देशतः कालतः वस्तुतश्चेति ... तच्च वैपरीत्यमेव। ... तत्रैकरेखा स्थिता द्वितीया दिक विपरीता दिगित्युच्यते । यथा पूर्वविपरीता पश्चिमा दिक् । यथा उत्तरदिग्विपरीता दक्षिणा दिगित्यादि । तथा च पूर्वापरदेशयोर्मध्ये एकतरस्य धनत्वे कल्पितं तं प्रति तदितरस्य ऋणत्वम्।<ref> | ऋणत्वमिह त्रिधा तावदस्ति देशतः कालतः वस्तुतश्चेति ... तच्च वैपरीत्यमेव। ... तत्रैकरेखा स्थिता द्वितीया दिक विपरीता दिगित्युच्यते । यथा पूर्वविपरीता पश्चिमा दिक् । यथा उत्तरदिग्विपरीता दक्षिणा दिगित्यादि । तथा च पूर्वापरदेशयोर्मध्ये एकतरस्य धनत्वे कल्पितं तं प्रति तदितरस्य ऋणत्वम्।<ref>Bījapallava on Bījagaṇita p.13</ref> | ||
“Negativity is of three kinds - according to place, time and things. It is in short the contrary. Just as west is the contrary direction to east and south to north. Thus between two places situated in the east and west, If one is taken to be positive the other is relatively negative." | “Negativity is of three kinds - according to place, time and things. It is in short the contrary. Just as west is the contrary direction to east and south to north. Thus between two places situated in the east and west, If one is taken to be positive the other is relatively negative." | ||
== Fundamental Operations == | == Fundamental Operations == | ||
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=== Squaring === | === Squaring === | ||
The rule for squaring of an algebraic expression is | The rule for squaring of an algebraic expression is | ||
(a+b)² =a²+b²+2ab | (a+b)² =a²+b²+2ab | ||
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be known terms, proceed with the remainder in the same way after taking the square-root of the knowns." | be known terms, proceed with the remainder in the same way after taking the square-root of the knowns." | ||
== See Also == | |||
* [[बीजगणित]] | |||
== External Links == | |||
* [http://www.ms.uky.edu/~sohum/ma330/files/brahmagupta_algebra.pdf Brahmagupta's Algebra - Mathematics] | |||
* [https://rarebooksocietyofindia.org/book_archive/196174216674_10152963932466675.pdf Bija Ganita or The algebra of the Hindus] | |||
* [https://rarebooksocietyofindia.org/book_archive/196174216674_10152963932466675.pdf Algebra] | |||
== Further Reading == | |||
* Bhāskara (II.), Edward Strachey. Bija Ganita: Or The Algebra Of The Hindus... ISBN-13 978-1249957041. | |||
== References == | == References == | ||
<references /> | |||
[[Category:Articles using infobox templates with no data rows]] | |||
[[Category:Articles with hCards]] | |||
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
[[Category:Organic Articles English]] | |||
[[Category:Pages using duplicate arguments in template calls]] | |||
Latest revision as of 09:35, 26 December 2022
Algebra is one of the broad areas of Mathematics. The Hindu name for the science of algebra is bījagaṇita. Bīja means "element" or
Algebra | |
|---|---|
 |
"analysis" and gaṇita means " the science of calculation". Bījagaṇita literally means "science of calculation with elements or the science of analytical calculation.
Brahmagupta (628) calls algebra as kuṭṭaka - gaṇita or kuṭṭaka. Kuṭṭaka means pulveriser. Algebra is also called as avyakta - gaṇita or the science of calculation with unknowns (avyakta means unknown) in contrast to the name vyakta - gaṇita the science of calculation with knowns (vyakta means known) for arithmetic including geometry and mensuration.
Definition
Bhāskara II (1150) has defined Algebra as "Analysis (bīja) is certainly the innate intellect assisted by the various symbols (varṇa), which, for the instruction of duller intellects, has been expounded by the ancient sages who enlighten mathematicians as the sun irradiates the lotus; that has now taken the name algebra (bījagaṇita)".
That algebraic analysis' requires keen intelligence and sagacity has been observed by him on more than one occasion.
"Neither does analysis consist in symbols, nor are there different kinds of analyses; sagacity alone is analysis, for wide is imagination. "Analysis is certainly clear intelligence". "Or intelligence alone is analysis".In answer to the question, "if (unknown quantities) are to be discovered by intelligence alone what then is the need of analysis ?" he says, "Because intelligence is certainly the real analysis; symbols are its helps. The innate intelligence which has been expressed for the duller intellects by the ancient sages, who enlighten mathematicians as the sun irradiates the lotus, with the help of various symbols, has now obtained the name of algebra.
Thus, according to Bhāskara II, algebra may be defined as the science which treats of numbers expressed by means of symbols, and in which there is scope and primary need for intelligent artifices and ingenious devices.
Bījagaṇita means a 'seed'. The unknown quantities are like a seed and their values become evident when the equations are solved. Since algebra deals with unknown quantities, it is called Bījagaṇita in Samskrit. Kṛṣṇa Daivajña, a renowned mathematician of the 16th Century CE wrote Bijapallava, a commentary on Bījagaṇita (1150 CE) of Bhāskara II. Kṛṣṇa Daivajña, explains the name Bījagaṇita as below
अव्यक्तत्वादिदं बीजमित्युक्तं शास्त्रकर्तृभिः |
"Since this (quantity) is unknown, it was called bīja by makers of sciences,"
Origin
The origin of Hindu algebra can be definitely traced back to the period of the śulba (800-500 B.C.) and the Brāhmaṇa (c. 2000 B.C.).
"The Hindus were the first to make systematic use of the letters of the alphabet to denote unknowns. They were also the first to classify and make a detailed study of equations. Thus they may be said to have given birth to the modern science of algebra." [1]
śulbasūtrās mentions variable quantity. āryabhaṭīya of āryabhaṭa mentions solution to linear and quadratic equations. Brahmagupta in his Brāhma- sphuṭa - siddhānta mentions the operations performed on unknowns using symbols. kuṭṭakādhyāya (chapter 18) explains the parikarma (calculations) with avyakta (or algebraic symbols). Hence Brahmagupta is treated as the father of algebra. The other texts on algebra includes Mahāsiddhānta of āryabhaṭa II ,siddhāntaśekhara of śripati, Bījagaṇita of Bhāskara II, bījagaṇitāvataṃsa of nārāyaṇa paṇḍita.
Brahmagupta has given the rules for arithmetic operations with positive numbers, negative numbers and zero in kuṭṭakādhyāya of Brāhma- sphuṭa - siddhānta. Also equations with one unknown, equations with many unknowns, equations with products of unknowns and indeterminate equations of first and second order (kuṭṭaka and varga-prakṛti) are dealt by Brahmagupta.
Technical Terms
Unknown Quantiy
The unknown quantity was called in the Sthānāṅga-sūtra (before 300 B.C.) yāvat - tāvat (as many as or so much as, meaning an arbitrary quantity). In the so-called Bakhshālī treatise, it was called yadṛcchā , vāñcā or kāmikā (any desired quantity). āryabhaṭa I (499) calls the unknown quantity as gulikā (shot). This term strongly leads one to suspect that the shot was probably then used to represent the unknown. From the beginning of the seventh century the Hindu algebraists are found to have more commonly employed the term avyakta (unknown).
Equation
The equation is called by Brahmagupta (628) sama-karaṇa or samī- karaṇa (making equal) or more simply sama (equation). pṛthūdakasvāmī (860) employs also the term sāmya (equality or equation); and Sripati (1039) sadṛśī- karaṇa (making similar). Narayana (1350) uses the terms samī- karaṇa , sāmya and samatva (equality). An equation has always two pakṣa (side).
Absolute Term
In the Bakhshālī treatise the absolute term is called dṛśya (visible). In later Hindu algebras it has been replaced by a closely allied term rūpa (appearance), though it continued to be employed in treatises on arithmetic. Thus the true significance of the Hindu name for the absolute term in an algebraic equation is obvious. It represents the visible or known portion of the equation while its other part is practically invisible or unknown.
Power
The oldest Hindu terms for the power of a quantity, known or unknown, are found in the Uttarādhyayana-sūtra (c. 300 B.C. or earlier). In it the second power is called varga (square), the third power ghana ( cube), the fourth power varga-varga (square-square), the sixth power ghana-varga (cube-square), and the twelfth power ghana-varga-varga (cube-square-square), using the multiplicative instead of the additive principle. In this work we do not find any method for indicating odd powers higher than the third. In later times, the fifth power is called varga-ghana-ghāta(product of cube and square, ghāta = product), the seventh power varga-varga-ghana-ghāta (product of square-square and cube) and so on. Brahmagupta's system of expressing powers higher than the fourth is scientifically better. He calls the fifth power pañca-gata (literally, raised to the fifth), the sixth power ṣaḍ-gata (raised to the sixth) ; similarly the term for any power is coined by adding the suffix gata to the name of the number indicating that power. Bhāskara II has sometimes followed it consistently for the powers one and upwards. In the Anuyogadvāra-sūtra, a work written before the commencement of the Christian Era, we find certain interesting terms for higher powers, integral as well as fractional, particularly successive squares (varga) and square-roots (varga-mūla). According to it the term prathama-varga (first square) of a quantity, say a2 means a; dvitīyavarga (second square) = (a2)2 = a4 ; tṛtīya -varga (third square) = ((a2)2 )2 = a8 and so on. In general, nth varga of a = a2x2x2x ……. to n terms =a2ⁿ. Similarly, prathama-varga-mūla (first square-root) means √a ; dvitīya-varga-mūla (second square-root) =√ (√a) = a1/4 ; and, in general, nth varga-mūla of a = a1/2ⁿ
Again we find the term tṛtīya-varga-mūla-ghana (cube of the third square-root) for (a1/23)3 = a3/8•
The term varga for "square" has an interesting origin in a purely concrete concept. The Sanskrit word varga literally means "rows," or "troops" (of similar things). Its application as a mathematical term originated in the graphical representation of a square, which was divided into as many varga or troops of small squares, as the side contained units of some measure.
Coefficient
In Hindu algebra there is no systematic use of any special term for the coefficient. Ordinarily, the power of the unknown is mentioned when the reference is to the coefficient of that power. In explanation of similar use by Brahmagupta his commentator pṛthūdakasvāmī writes "the number (aṅka) which is' the coefficient of the square of the unknown is called the 'square' and the number which forms the coefficient of the ( simple) unknown is called 'the unknown quantity. However, occasional use of a technical term is also met with. Brahmagupta once calls the coefficient saṃkhyā (number) and on several other occasions guṇaka, or guṇākara (multiplier). pṛthūdakasvāmī (860) calls it aṅka (number) or prakṛti (multiplier). These terms reappear in the works of Sripati (1039)5 and Bhāskara II (1150). The former also used rūpa for the same purpose.
Symbols
Symbols of Operation. There are no special symbols for the fundamental operations in the Bakhshālī work. Any particular operation intended is ordinarily indicated by placing the shorthand abbreviation, the initial syllable of a Sanskrit word of that import, after, occasionally before, the quantity affected. Thus the operation of addition is indicated by yu (an abbreviation from yuta, meaning added), subtraction by + which is very probably from kṣa (abbreviated from kṣaya, diminished), multiplication by gu (from guṇa or guṇita, multiplied) and division by bhā (from bhāga or bhājita or, divided).
Bhāskara II (1150) says, "Those (known and unknown numbers) which are negative should be written with a dot (bindu) over them."
Symbols for Powers and Roots. The symbols for powers and roots are abbreviations of Sanskrit words which are placed after the number affected. Hence the. square is represented by va (from varga), cube by gha (from ghana), the fourth power by va-va (from varga-varga), the fifth power by va-gha-ghā (from varga-ghana-ghāta), the sixth power by gha-va (from ghana-varga), the seventh power by va-va-gha-ghā (from varga-varga-ghana-ghāta) and so on. The product of two or more unknown quantities is shown by writing bhā (from bhāvita, product) after the unknowns with or without interposed dots; e.g., yāva-kāgha-bhā or yāvakāghabhā means (yā)2 (kā)3. In the Bakhshālī treatise the square-root of a quantity is shown by writing after it mū which is an abbreviation for mūla (root).
For instance
21 yu 4 mū 5
1 1 1
Means
and
23 7+ mū 4
1 1 1
Means
In other treatises the symbol of the square-root is ka (from karaṇī, root or surd) which is usually placed before the quantity affected.
For example ka 19 ka 50 ka 57 ka 94 indicated as
Symbols for Unknowns
Bhāskara II (1150) observes, "Here (in algebra) the initial letters of (the names of) knowns and unknowns should be written for implying them." It has been stated before that at one time the unknown quantity was called yāvat-tāvat (as many as, so much as). In later times this name, or its abbreviation yā is used for the unknown.
यावत्तावत् कालको नीलकोऽन्यो वर्णः पीतो लोहितश्चैतदाद्याः।
अव्यक्तानां कल्पिता मानसंज्ञास्तत्संख्यानं कर्तुमाचार्यवर्यैः ॥[2]
"The great ācāryās assumed symbols to represent unknowns from the initial letters of yāvat-tāvat and colours such as kālaka (black), nīlaka (blue), pīta (yellow), lohita (red) etc."
Bhāskara II (1150) says: "yāvat-tāvat (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."
"In those examples where occur two, three or more unknown quantities, colours such as yāvat-tāvat, etc., should be assumed for them. As assumed by the previous teachers, they are: yāvat-tāvat (so much as), kālaka (black), nīlaka (blue), pītaka (yellow), lohitaka (red), harītaka (green), śvetaka (white), citraka (variegated), kapilaka (tawny) , piṅgalaka (reddish-brown), dhūmraka (smoke-coloured), pātalaka (pink), śavalaka (spotted), śyāmalaka (blackish), mecaka (dark blue) etc. Or the letters of alphabets beginning with ka, should be taken as the measures of the unknowns in order to prevent confusion.
Thus symbols like were used to represent unknown quantities. In today's context , we see the letters x, y, z etc. being used to represent the unknown quantities. The following table gives the different names and symbols used to mean the unknown quantities in the early works on Algebra.
| Term | Symbol | Meaning | Reference |
|---|---|---|---|
| yāvat-tāvat | yā
या |
as many as | Sthānāṅgasūtra,
Bhāskara I, Bhāskara II |
| yadṛcchā , vāñcha or kāmikā | ya vā kā
य वा का |
desired
quantity |
Bakhshālī Manuscript |
| gulikā | gu
गु |
ball | āryabhaṭa |
| kālaka, nīlaka, pīta, lohita (red) | kā nī pī lo
का नी पी लो |
black, blue,
yellow, red |
Brahmagupta, Bhāskara II |
Bakhshālī Manuscript mentions where there are five unknowns, the first letters of ordinals were used. that is pra from prathama (first), dvi from dvitīya (second), tṛ from tṛtīya (third), ca from caturtha (fourth) and paṃ from pañcama(fifth) to represent the unknowns.
Laws Of Signs
Kauṭilya's arthaśāstra mentions negative quantities like debt(ṛṇa). Brahmagupta uses the words dhana and ṛṇa for denoting positive and negative numbers in Brāhma- sphuṭa - siddhānta. Nowadays Integers include positive numbers, negative numbers and zero[3] .
Addition
धनयोर्धनमृणमृणयोर्धनर्णयोरन्तरं समैक्यं खम् ।
ऋणमैक्यं च धनमृणधनशून्ययोः शून्ययोः शून्यम् ॥[4]
Brahmagupta (62.8) says:
"The sum of two positive numbers is positive. Sum of two negative numbers is negative. Sum of a positive and a negative number is their difference. If positive and negative numbers are equal, their sum is zero. The sum of zero and negative is negative. Sum of a positive number and zero is positive. Sum of two zeroes is zero."
Subtraction
ऊनमधिकाद्विशोध्यं धनं धनादृणमृणादधिकमूनम् ।
व्यस्तं तदन्तरं स्यादृणं धनं धनमृणं भवति ॥[5]
Brahmagupta Says: "From the greater should be subtracted the smaller; (the final result is) positive, if positive from positive. and negative, if negative from negative. If, however, the greater is subtracted from the less, that difference is reversed (in sign). negative ,becomes positive and positive becomes negative. When positive is to be subtracted from negative or negative from positive then they must be added together.
Multiplication
ऋणमृणधनयोर्घातो धनमृणयोर्धनवधो धनं भवति ।
शून्यर्णयो: खधनयो: खशून्ययोर्वा वधः शून्यम् ॥[6]
Brahmagupta says:"The product of a positive and a negative number is negative; product of two negatives is positive; product of positive multiplied by positive is positive. The product of zero and negative, or of zero and positive is zero. Product of two zeroes is zero.
Division
धनभक्तं धनमृणहृतमृणं धनं भवति खं खभक्तं खम्।
भक्तमृणेन धनमृणं धनेन हृतमृणमृणं भवति ॥[7]
Brahmagupta says: "Positive divided by positive or negative divided by negative becomes positive. But positive divided by negative is negative and negative divided by positive remains negative.
Evolution and Involution
Brahmagupta says:
. "The square of a positive or a negative number is positive . The (sign of the) root is the same as was that from which the square was derived."
Bhāskara II: "The square of a positive and a negative number is positive; the square-root of a positive number is positive as well as negative. There is no square-root of a negative number, because it is non-square."
Negative Quantities
Regarding a negative quantity , moving in one direction is considered as positive, moving in the opposite direction is considered as negative.
Kṛṣṇa Daivajña shows positive and negative directions along a line. If east is considered as positive direction, then, west should be considered as negative. T
Kṛṣṇa Daivajña also talks about these opposites of negative and positive in terms of kāla (time) and vastu (object). With regard to time, if the future denotes positive, the opposite of it, the past will be negative. If we borrow something, we are indebted to pay it back. It is called ṛṇa (negative). The opposite of it is dhana (positive) where we own the object or we are bound to receive something. In common terminology, the two words dhana and ṛṇa are used for wealth and debt respectively. The idea of negative numbers goes back to Arthaśāstra.
All these concepts are summarized by Kṛṣṇa Daivajña in his commentary as follows:
ऋणत्वमिह त्रिधा तावदस्ति देशतः कालतः वस्तुतश्चेति ... तच्च वैपरीत्यमेव। ... तत्रैकरेखा स्थिता द्वितीया दिक विपरीता दिगित्युच्यते । यथा पूर्वविपरीता पश्चिमा दिक् । यथा उत्तरदिग्विपरीता दक्षिणा दिगित्यादि । तथा च पूर्वापरदेशयोर्मध्ये एकतरस्य धनत्वे कल्पितं तं प्रति तदितरस्य ऋणत्वम्।[8]
“Negativity is of three kinds - according to place, time and things. It is in short the contrary. Just as west is the contrary direction to east and south to north. Thus between two places situated in the east and west, If one is taken to be positive the other is relatively negative."
Fundamental Operations
Number of Operations
The Number of fundamental operations in algebra is recognised by all Hindu algebraists to be six, namely " addition, subtraction, multiplication,
division, squaring and the extraction of the square-root. So the cubing and the extraction of the cube-root which are included amongst the fundamental operations of arithmetic, are excluded from algebra.
But the formula
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a + b)3 = a3 + 3ab(a+b) + b3,
is found to have been given, as stated before, in almost all the Hindu treatises on arithmetic beginning with that of Brahmagupta (628).
Addition and Subtraction
Brahmagupta says: Of the unknowns, their squares, cubes, fourth powers, fifth powers, sixth powers, etc., addition and subtraction are (performed) of the like; of the unlike (they mean simply their) statement severally.
Bhāskara II:
"Addition and subtraction are performed of those of the same species (jāti) amongst unknowns; of different species they mean their separate statement."
Multiplication
Brahmagupta says: The product of two like unknowns is a square; the product of three or more like unknowns is a power of that designation. The multiplication of unknowns of unlike species is the same as the mutual product of symbols; it is called bhāvita (product or factum).
Division
Bhāskara II states: By whatever unknowns and knowns, the divisor is multiplied (severally) and subtracted from the dividend
successively so that no residue is left, they constitute the quotients at the successive stages.
Squaring
The rule for squaring of an algebraic expression is
(a+b)² =a²+b²+2ab
or in its general form
(a+b+c+d+ ... )2=a2+b2+c2+d2+ ..+2Σab.
Square-root
For finding the square-root of an algebraic expression Bhāskara II gives the following rule:
"Find the square-root, of the unknown quantities which are squares; then deduct from the remaining terms twice the products of those roots two and two; if there
be known terms, proceed with the remainder in the same way after taking the square-root of the knowns."
See Also
External Links
Further Reading
- Bhāskara (II.), Edward Strachey. Bija Ganita: Or The Algebra Of The Hindus... ISBN-13 978-1249957041.
References
- ↑ Datta, 1938, Vol.2, Preface
- ↑ Bījagaṇita, ch. Avyakta-kalpanā, vs.5, p.7
- ↑ A Primer to Bhāratīya Gaṇitam , Bhāratīya-Gaṇita-Praveśa- Part-1. Samskrit Promotion Foundation. 2021. ISBN 978-81-951757-2-7.
- ↑ Brahma-sphuţa-siddhānta (ch.18, vs.30, p.309)
- ↑ Brahma-sphuta-siddhanta, ch.18, vs.31 p.309
- ↑ Brahma-sphuţa-siddhānta (ch.18, vs.33, p.310)
- ↑ Brahma-sphuta-siddhanta (ch.18, vs.34, p.310)
- ↑ Bījapallava on Bījagaṇita p.13
