Algebra

Algebra is one of the broad areas of Mathematics. The Hindu name for the science of algebra is bījagaṇita. Bīja means "element" or "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 manifest 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. He explains the name Bījagaṇita as follows;

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."

The idea of a variable quantity was known in  ś ulbasūtrās. Solution of linear and quadratic equations is discussed in āryabhaṭīya of āryabhaṭa. It was Brahmagupta who  gave systematic discussion of algebra as a branch of mathematics. In his Brāhma- sphuṭa - siddhānta, he explicitly discusses the operations performed on unknowns using symbols. these parikarma (calculations) with avyakta (or algebraic symbols) are explained in kuṭṭakādhyāya (chapter 18). Hence Brahmagupta is considered as the father of algebra. This was followed by several later texts. Mahāsiddhānta of āryabhaṭa II ,siddhāntaśekhara of śripati, Bījagaṇita of Bhāskara II, bījagaṇitāvataṃsa of  nārāyaṇa paṇḍita.

In kuṭṭakādhyāya of Brāhma- sphuṭa - siddhānta, Brahmagupta  gave the precise rules for arithmetic operations with positive numbers, negative numbers and zero. He also deals with 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).

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).This term was originally connected with the Rule of False Position. 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 tachygraphic 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 of those imports and are placed after the number

affected. Thus 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 indicated 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 indicated by writing after it mū which is an abbreviation for mūla (root).

For instance

11 yu 5 mū  4

1        1       1

Means √(11+5) =  4

and

11   7+   mū   2

1    1             1

Means √(11 -7) =  2

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 9   ka 450 ka 75   ka  54 means √9   + √450 + √ 75 + √54 $$\sqrt{9}+\sqrt{450}+\sqrt{75}+\sqrt{54}$$

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 ॥

"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. In Bakhshālī Manuscript when the problem involved as many as five unknowns, the first letters of ordinals were used. thus pra from prathama (first), dvi from dvitīya (second), tṛ from tṛtīya (third), ca from caturtha (fourth) and paṃ from pañcama(fifth) have been used to represent the unknowns.

Laws Of Signs
The idea of negative quantities was known in India from ancient times. Kauṭilya's arthaśāstra talks of negative quantities like debt(ṛṇa). In Mathematics, this is discussed for the first time in Brāhma- sphuṭa - siddhānta of Brahmagupta. Brahmagupta uses the words dhana and ṛṇa for denoting positive and negative quantities. Nowdays positive numbers, negative numbers and zero are collectively called integers.

Addition
धनयोर्धनमृणमृणयोर्धनर्णयोरन्तरं समैक्यं खम् ।

ऋणमैक्यं च धनमृणधनशून्ययोः शून्ययोः शून्यम् ॥

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 ॥

Brahmagupta writes: "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 ॥

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 ॥

Brahmagupta states: "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
The rules stated above by Brahmagupta are based upon a proper understanding of the nature of negative quantity.

What is a negative quantity in the real world ? If moving in one direction is taken as positive, moving in the opposite direction is negative. Kṛṣṇa Daivajña shows positive and negative directions along a line. If east is taken as positive direction, then, west should be negative. This is illustrated in the figure below with zero at the centre of the positive and negative numbers. This was the representation of negative as opposed to positive in terms of deśa (direction).

Kṛṣṇa Daivajña also mentions about these opposites of negative and positive ms of kāla (time) and vastu (object). With regard to time, if the future denotes positive, the opposite of it, the past will be negative. What is positivity and negativity with reference to objects ? If you borrow something, you are indebted to pay it back. It is called ṛṇa (negative). The opposite of it is dhana (positive) where you own the object or you are bound to receive something. In common parlance, the two words dhana and ṛṇa are used for wealth and debt respectively. The use of these words for denoting positivity and negativity can be well appreciated. In this sense, 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।

“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."

and Algebraic Equations

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 the same as the treatises on arithmetic,

(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."