Indian Astronomer and Mathmatican Brahmagupta

BrahmaguptaMathematics

Algebra

Brahmagupta gave the solution of the a general linear Equation in chapter eighteen of Brahmasphutasiddhant,

The difference between rupas, when inverted and divided by the difference of the unknowns, is the unknown in    the equation. The rupas are [subtracted on the side] below that from which the square and the unknown are to be subtracted.

which is a solution for the equation equivalent to , where rupas refers to the constants c and e. He further gave two equivalent solutions to the general quadratic equation . Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. [The result is] the middle [number].
Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.

which are, respectively, solutions for the equation equivalent to,

x = \frac{\sqrt{4ac+b^2}-b}{2a}

and

x = \frac{\sqrt{ac+\tfrac{b^2}{4}}-\tfrac{b}{2}}{a}

He went on to solve systems of simultaneous indeterminate equations stating that the desired variable must first be isolated, and then the equation must be divided by the desired variable’s coefficient. In particular, he recommended using “the pulverizer” to solve equations with multiple unknowns.

Subtract the colors different from the first color. [The remainder] divided by the first [color’s coefficient] is the measure of the first. [Terms] two by two [are] considered [when reduced to] similar divisors, [and so on] repeatedly. If there are many [colors], the pulverizer [is to be used].

Like the algebra of Diaphanous, the algebra of Brahmagupta was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms.The extent of Greek influence on this syncopation, if any, is not known and it is possible that both Greek and Indian syncopation may be derived from a common Babylonian source.

to be continue……

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