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.

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.).

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 yadrccha, vancha or kamika (any desired quantity).This term was originally connected with the Rule of False Position. Aryabhata I (499) calls the unknown quantity as gulika (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-karana or sami-karana (making equal) or more simply sama (equation). Prthudakasvami (860) employs also the term samya (equality or equation); and Sripati (1039) sadrsi-karana (making similar). Narayana (1350) uses the terms sami-karana, samya and samatva (equality). An equation has always two paksa (side).

Absolute Term

In the Bakhshali treatise the absolute term is called drsya (visible). In later Hindu algebras it has been replaced by a closely allied term rupa (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 Uttaradhyayana-sutra (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-ghata (product of cube and square, ghata = product), the seventh power varga-varga-ghana-ghata (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 panca-gata (literally, raised to the fifth), the sixth power sad-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. Bhaskara II has sometimes followed it consistently for the powers one and upwards. In the Anuyogadvara-sutra, 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-mula). According to it the term prathama-varga (first square) of a quantity, say a² means a; dvitiyavarga (second square) = (a²)² = a⁴ ; trtiya-varga (third square) = ((a²)² )² = a⁸ and so on. In general, nth varga of a = a^{2x2x2x ……. to n terms} =a^2ⁿ. Similarly, prathama-varga-mula (first square-root) means √a ; dvitrya-varga-mula (second square-root) =√ (√a) = a^1/4 ; and, in general, nth varga-mula of a = a^1/2ⁿ

Again we find the term trtiya-varga-mula-ghana (cube of the third square-root) for (a^1/23)3 = a^3/8•

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 Prthudakasvami writes "the number (anka) 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 samkhya (number) and on several other occasions gunaka, or gunakara (multiplier). Prthudakasvaml (860) calls it anka (number) or prakrti (multiplier). These terms reappear in the works of Sripati (1039)5 and Bhaskara II (1150). The former also used rupa for the same purpose.

Laws Of Signs

Addition

Brahmagupta (62.8) says: "The sum of two positive numbers is positive, of two negative numbers is negative; of a positive and a negative number is their difference.'

Subtraction

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

Brahmagupta says:"The product of a positive and a negative (number) is negative; of two· negatives is positive; positive multiplied by positive is positive.

Division

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

The square of a positive and a negative number is positive. The square-root of a positive number will be positive and also negative. It has been proved that a negative number, being non-square, has no square root.

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 (62.8).

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.

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 bhavita (product or factum).

Division

Bhaskara 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 Bhaskara 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."

References

Citations

1.History of Hindu Mathematics by Bibhutibhusan Datta and Avadesh Narayan Singh