# The controversy over negative numbers – 04/20/2021 – Marcelo Viana

The first written mention of negative numbers goes back to the “Nine Chapters of Mathematical Art” published in China around the year 200. In the centuries that followed, the Chinese, Indians, and Arabs learned to perform operations with these numbers. But there was also no consensus: Bhaskara (1114–1185) said that negative solutions for the quadratic equation are not valid because “people do not approve of negative solutions”.

It was worse in the west. In the mid-18th century, the Englishman Francis Maseres (1731-1824) was still arguing that negative numbers “obscure the entire theory of equations and complicate things that are by their very nature perfectly obvious and simple”.

The French Nicolas Chuquet was the first European to use negatives as exponents in the second half of the 15th century. But like many others he called them numeri absurdi (absurd numbers). The Franciscan Luca Pacioli (1445-1517) used negative numbers to represent debts in his work “Summa” published in 1494, which created the two-entry book model.

Another Italian, Rafael Bombelli (1526–1572), wrote the operating rules learned in school (“less often less is more”) in his 1572 published “Algebra”. He used m. (“Minus”) to represent negative and p. (“Plus”) to represent positively. The – e + signs we use today became popular over the next century.

René Descartes’ (1596–1650) position was ambivalent: he viewed negative solutions for solutions as “wrong”, but he knew how to convert negative solutions into positive ones, and this led him to accept the negative numbers.

The Englishman John Wallis (1616-1703) had strange ideas: he disagreed that negative was less than nothing, but he thought it was more than infinite. Ironically, he was the first to give a clear interpretation of the negative numbers by the line where the positive ones mark the distance to one side of zero and the negative ones to the other.

Gottfried Leibniz (1646–1716) agreed with the objections to the Numbers Absurdi, but argued that they can continue to be used as long as they provide correct results. This pragmatism was already adopted by Cardano, as we will comment later.

In 1765 Leonhard Euler (1707–1783) began his complete introduction to algebra with operations on positive and negative numbers and returned to the idea of debt to explain it. However, the controversy over negatives was not pacified until the 19th century with the formalization of arithmetic. At that point the expansion of the idea of number had gone much further.

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