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The conflict between geometry and algebra

The ancient Greeks did not really address the problem of negative numbers, because their mathematics was founded on geometrical ideas. Lengths, areas and volumes resulting from geometrical constructions necessarily all had to be positive. Their proofs consisted of logical arguments based on the idea of magnitude. Magnitudes were represented by a line or an area, and not by a number (like 4.3 metres or 26.5 cubic centimetres). In this way, they could deal with ²awkward² numbers like square roots by representing them as a line. For example, you can draw the diagonal of a square without having to measure it.
About 300 CE, the Alexandrian mathematician Diophantus (200-c.284 CE) wrote his Arithmetica, a collection of problems where he developed a series of symbols to represent the ²unknown² in a problem and powers of numbers. He dealt with what we now call linear and quadratic equations. In one problem Diophantus wrote the equivalent of 4 = 4x + 20 which would give a negative result, and he called this result ²absurd².
In the 9^th century in Baghdad Al-Khwarizmi (c.780-c.850 CE) presented six standard forms for linear or quadratic equations and produced solutions using algebraic methods and geometrical diagrams. In his algebraic methods he acknowledged that he derived ideas from the work of Brahmagupta and therefore was happy with the notion of negative numbers. However, his geometrical models (based on the work of Greek mathematicians) persuaded him that negative results were meaningless (how can you have a negative square?). In a separate treatise on the laws of inheritance, Al-Khwarizmi represents negative quantities as debts.
In the 10^th century Abul-Wafa (940-998 CE) used negative numbers to represent a debt in his work on ²what is necessary from the science of arithmetic for scribes and businessmen². This seems to be the only place where negative numbers have been found in medieval Arabic mathematics. Abul-Wafa gives a general rule and gives a special case where subtraction of 5 from 3 gives a "debt" of 2. He then multiples this by 10 to obtain a "debt" of 20, which when added to a ²fortune² of 35 gives 15.
In the 12^th century Al-Samawal (1130-1180) produced an algebra where he stated that:

· if we subtract a positive number from an ²empty power², the same negative number remains,

· if we subtract the negative number from an ²empty power², the same positive number remains,

· the product of a negative number by a positive number is negative, and by a negative number is positive.

Negative numbers did not begin to appear in Europe until the 15^th century when scholars began to study and translate the ancient texts that had been recovered from Islamic and Byzantine sources. This began a process of building on ideas that had gone before, and the major spur to the development in mathematics was the problem of solving quadratic and cubic equations.

As we have seen, practical applications of mathematics often motivate new ideas and the negative number concept was kept alive as a useful device by the Franciscan friar Luca Pacioli (1445-1517) in his Summa published in 1494, where he is credited with inventing double entry book-keeping.

TEXT 6

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