## Boost for Negative Numbers.

The history continues with a boost for negative numbers from Leonardo of Pisa. He 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.

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 y^{3} = 15y + 4 to be resolved using Cardano’s solution.

William Rowan Hamilton (1805 – 1865) stated 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 90^{o} around the x-axis, ‘j’ represents a rotation of 90^{o} around the y-axis and ‘k’ represents a rotation of 90^{o} around the z-axis. This lead to his famous equation i^{2} = j^{2} = k^{2} = ijk = -1.

However, the history of negative numbers provides us with examples of many mathematicians who did not fully support negative numbers.

## The Critics.

Michael Stifel, a German Monk, lived from 1487 to 1567. He produced rules for arithmetic with negative numbers. Nevertheless, he described them as ‘*numeri absurdi*’, as did Girolamo Cardano (1501-1576). In his Latin language work on algebra, *Ars Magna*, Cardano described a solution for finding the roots of cubic equations. He avoided negative square roots. They made even less sense to him than negative numbers.

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 that 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) saw √-1 as ‘*a purely geometric operation. It is a sign of perpendicularity.*’

## Conclusion.

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*’.

In 1806 Argand described -1 as a rotation of the point 1 by 180^{o}. He concluded from this that half of this rotation would result in the rotation of a point at 1 to √-1.

I believe it is fair to say that history shows 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.

They are not alone. Accountants and Physicists appear to try and avoid negative numbers.