After the decline of the Roman Empire (not that the Romans did anything for maths!) India became the temporary centre of mathematical research. The most important contributions of the Hindus in the second half of the first millenium were the decimal place system, the introduction of zero and negative numbers, and the development of algebra. Whereas Diophantus' first step in the solution of a linear equation was to remove the negative terms, the Hindus worked with negative numbers from about 600 A.D.
A number was turned into the corresponding negative quantity by placing a dot over it. They also had a method for representing positive and negative numbers pictorially by line segments in different directions, corresponding to our representation using a number line.
In their treatment of equations in several unknowns, the Hindus also achieved some advance on Diophantus, in that they actually worked with several unknowns using different colours to distinguish them. Thus the second unknown was called "the black one", the third "The blue one", etc.
Since they allowed negative numbers in their solution of quadratic equations, they could combine the various cases considered by Diophantus into one rule, and had a method of solution similar to our formula for quadratics today. The Hindus were the first to show an awareness of the fact that roots occur in pairs, and occasionally even admitted negative roots as solutions.
The Arabs took over the preparatory work done by the Greeks and Hindus in algebra. Their most important algebraist was al-Khowarizmi (9th century - his name is commemorated in the word "algorithm"). His major work is entitled "Al-jabr wa'lmugabalah" (restoration and balancing) and from the first word in this title we now have the word "algebra". However his algebra was a rhetorical algebra which, unlike the work of Diophantus, did not use symbols for particular arithmetical operations.
Before the Renaissance, all algebra was motivated by geometry and any methods for solving equations were derived or proved geometrically. The geometric proofs given by al-Khowarizmi for solving quadratics are interesting. For the case
x + px = q (p > 0, q> 0)
with solution x = -
the geometric derivation appears as shown in the diagrams.
The total area of the large square can be expressed in two forms:
1. A = (x + 2.) = (x + )
2. A = x + 4(px) + 4.
= x + px +
= q +
(x + ) = q +
or x + =
or x = -
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