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It was not until the 19th century when British mathematicians likeDe Morgan, Peacock, and others, began to investigate the 'laws ofarithmetic' in terms of logical definitions that the problem ofnegative numbers was finally sorted out.However, there were references to negative numbers farearlier.In 200 BCE the Chinese number rod system (see note1 below)represented positive numbers in Red and Negative numbers in black.An article describing this system can be found here. These wereused for commercial and tax calculations where the black cancelledout the red. The amount sold was positive (because of receivingmoney) and the amount spent in purchasing something was negative(because of paying out); so a money balance was positive, and adeficit negative. The concept also appeared in Astronomy where the ideas of'strong' and 'weak' were used for approximating a number from aboveor below.

For example approaching 5 from above means for example,starting with 5.2 you can find better approximations 5.1, 5.05,5.025. Thus 5.025 was called a 'strong' approximation and a numberlike 4.9 'weak'. So 'strong' numbers were called positive and'weak' numbers negativeIn India, negative numbersdid not appear until about 620 CE in the work of Brahmagupta (598 -670) who used the ideas of 'fortunes' and 'debts' for positive andnegative. By this time a system based on place-value wasestablished in India, with zero being used in the Indian numbersytem. Brahmagupta used a special sign for negatives and stated therules for dealing with positive and negative quantities asfollows. The ancient Greeks didnot really address the problem of negative numbers, because theirmathematics was founded on geometrical ideas. Lengths, areas, andvolumes resulting from geometrical constructions necessarily allhad to be positive.

Their proofs consisted of logical argumentsbased on the idea of magnitude. Magnitudes were represented by aline or an area, and not by a number (like 4.3 metres or 26.5 cubiccentimetres). In this way they could deal with 'awkward' numberslike square roots by representing them as a line. For example, youcan draw the diagonal of a square without having to measure it (seenote 2 below).About 300 CE, the Alexandrian mathematician Diophantus (200 - c.284CE) wrote his Arithmetica, a collection of problems where he developed a series of symbolsto represent the 'unknown' in a problem, and powers of numbers. Hedealt with what we now call linear and quadratic equations.

In oneproblem Diophantus wrote the equivalent of 4 = 4x + 20 which wouldgive a negative result, and he called this result 'absurd'.In the 9th century in Baghdad Al - Khwarizmi (c.780 - c.850CE) presented six standard forms for linear or quadratic equationsand produced solutions using algebraic methods and geometricaldiagrams. In his algebraic methodshe acknowledged that he derivedideas from the work of Brahmagupta and therefore was happy with thenotion of negative numbers.

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However, his geometrical models (basedon the work of Greek mathematicians) persuaded him that negativeresults were meaningless (how can you have a negative square?). Ina separate treatise on the laws of inheritance, Al-Khwarizmirepresents negative quantities as debts.In the 10th century Abul -Wafa (940-998 CE) used negative numbersto represent a debt in his work on 'what is necessary from thescience of arithmetic for scribes and businessmen'? This seems tobe the only place where negative numbers have been found inmedieval Arabic mathematics.

Abul-Wafa gives a general rule andgives 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, whichwhen added to a 'fortune' of 35 gives 15.In the 12th century Al - Samawal (1130 - 1180) had produced analgebra where he stated that:. if we subtract a positive number from an 'empty power', thesame negative number remains,. if we subtract the negative number from an 'empty power', thesame positive number remains,. the product of a negative number by a positive number isnegative, and by a negative number is positive.