Negative numbers got a boost from Leonard of Pisa who lived from around 1170 to 1250. Nowadays, better known as Fibonacci, son of Bonaccio, he published his Latin language book Liber Abaci in 1202. The book uses negative numbers and describes rules for adding and multiplying negative numbers.
Michael Stifelwas a German Monk who lived from 1487to 1567. He produced rules for arithmetic with negative numbers. At the same time he described them as ‘numeri absurdi’, as did Girolamo Cardano (1501-1576). In his Latin language work on algebra, Ars Magna, he described a solution for finding the roots of cubic equations. Cardano avoided negative square roots. They made even less sense to him than negative numbers.
Bombelli (1526 – 1572) set up the multiplication rules for the square root of negative numbers that include the rule that √-1 x √-1 = -1. His rules allowed cubic equations such as y3 = 15y + 4 to be resolved using Cardano’s solution.
John Napier (1550-1617) was the inventor of logarithms. He called negative numbers ‘defectivi’.
Rene Descartes (1596-1650), the inventor of analytic geometry, called negative solutions to equations ‘false roots’. Along with Fermat (1607 – 1665), Descartes was the creator of analytic geometry. This is also known as co-ordinate geometry with the use of x and y-axes. The Cartesian coordinates used in co-ordinate geometry are named in Descartes’ honour. He did not have any negative numbers on his x and y-axes. Co-ordinate geometry linked algebra and geometry for the first time. It lead to our current day acceptance of the number line with its negative numbers.
Isaac Newton (1642 – 1726) accepted negative numbers and treated them as being analogous to debts and shortfalls. He did not consider the square roots of negative numbers to be numbers. He did accept them as the solutions to some polynomial equations but saw the solutions as having no application in the real world.
John Wallis (1660 – 1703) worked on infinite sums and products which are of great use in applied mathematics. He remained less certain about the status of negative numbers, since in his view it was impossible for a quantity to be ‘Less than Nothing, or any number fewer than None’. He used geometric methods to work on the square roots of negative numbers.
Adrien-Quentin Buée(1745 -1825) wanted √-1 to be considered ‘a purely geometric operation. It is a sign of perpendicularity.’
Caspar Wessel (1745 – 1818) described lines by their length and direction giving us the concept of vectors.
In 1806 Argand described -1 as a rotation of the point 1 by 180o. He concluded from this that half of this rotation would result in the point 1 being rotated to √-1.
Gauss (1777 – 1855) and Hamilton (1805 -1865) gave us our current understanding of the square roots of negative numbers through their work on complex numbers. Gauss blamed the terminology for many of the problems of comprehension and suggested a different terminology based on geometry. He said ‘If +1, −1, √-1 had been described verbally not as ‘positive, negative, imaginary’ (or [the latter] even as ‘impossible’) but, for example as ‘direct, inverse, lateral’ instead, there would have been no cause to refer to any such darkness’.
William Rowan Hamilton (1805 – 1865) maintained that any complex number could be described using a pair (a, b) of real numbers on the Cartesian plane. To implement this in algebra, he introduced 2 new complex numbers j and k. In the same way as i represents a rotation of 90o around the x-axis, j represents a rotation of 90o around the y-axis and k represents a rotation of 90o around the z-axis. This lead to his famous equation i2 = j2 = k2 = ijk = -1.
In conclusion, I believe it is fair to say that negative numbers have not had overwhelming support from the great mathematicians. At best, mathematicians tolerate negative numbers as necessary to make mathematics work. This is because they are not fully fit for purpose and it is time to develop an updated system of mathematics.
This is a comment for History Part 2