Monday 10 October 2022

#15 Negative ...from old math terms notes

Negative numbers, and the equivalent word for negative were introduce by Brahmagupta, a Hindu mathematician around 600 AD. The Latin root of today's word is negare, to deny. The negative numbers, in this sense, denying or invalidating an equivalent positive quantity.

The negative numbers were themselves denied for a long part of mathematical history, and only slowly came to be accepted. The first record of the operational rules for what we today call positive and negative numbers came from the pen of Diophantus (around 250 AD) who referred to them as "forthcomings" and "wantings". His work may have been drawn from proposition five in Euclid's Book II of the Elements in which Euclid demonstrates with geometric figures what we would write in modern algebra as (a+b)(a-b)+b2 = a2. This, of course, is easily recast as the more common identity (a+b)(a-b)= a2 - b2. Diophantus would accept negatives only as a way of diminishing a greater quantity, but did not accept them as independent quantities and would not accept a solution that was negative. Al-Khwarizmi (850 AD), whose writings brought Arabic numerals to the west, used a similar approach with negatives allowed in-process but not as a final result.

Descartes, around 1636, used the French fausse, false, for negative solutions. Thomas Harriot had described negative roots as the solution to an alternate form of the equation with the signs of the odd powers changed. Today his idea would be expressed by saying that the appearance of -c as a root of f(x) was only to be understood to mean that c is a root of f(-x).

In Mathematics: The Loss of Certainty, by Morris Kline includes the following argument against negative numbers by Antoine Arnauld (1612-1694), mathematician, theologian, and friend of Blaise Pascal; "Arnauld questioned that -1:1 = 1:-1 because, he said, -1 is less than +1; hence, how could a smaller be to a greater as a greater is to a smaller?"

Franz Lemmermeyer wrote in a posting to the Historia-Matematica newsgroup that Gleanings from the History of the Negative Number by PGJ Vrendenduin suggests that a number line with both positive and negative numbers could be found in the work of Wallis (1657). Another posting to the same list quoted Kline's "Mathematical Thought from Ancient to Modern Times":

"Though Wallis was advanced for his times and accepted negative numbers, he thought they were larger than infinity but not less than zero. In his 'Arithmetica Infinitorum' (1665), he argued that since the ratio a/0, when a is positive, is infinite, then, when the denominator is changed to a negative number, as in a/b with b negative, the ratio must be greater than infinity."

Even as late as 1831, De Morgan would still write that one "must recollect that the signs + and - are not quantities, but directions to add and subtract."

According to a post from Laura Laurencich, the Incas had a method of indicating both positive and negative numbers on their quipus as documented by the Jesuit Priest Blas Valeria in 1618.