# Where do negative numbers come from? – 13/04/2021 – Marcelo Viana

When I wrote about the solution of the cubic equation by the Italian Renaissance mathematicians Scipione del Ferro, Niccolò Tartaglia, and Gerolamo Cardano, I mentioned that one of the difficulties was that the negative numbers had not yet been discovered at this point. A reader wrote to me questioning the adjective: In his opinion, mathematics was created by humanity and not discovered.

This is a fascinating philosophical discussion that is far from being resolved: there are good arguments for both positions. Most mathematicians, including myself, believe that mathematical ideas exist in the structure of the universe and it is our job to discover them. In this sense, mathematics is also an experimental science.

The German Leopold Kronecker (1823-1891) was an exception. In an 1860 speech, he stated that “integers were created by the Lord God, everything else is the creation of men”. But it was an extreme case: He also criticized his colleague Ferdinand von Lindemann (1852–1939) for saying that π is a transcendent number: “Why investigate such questions when there aren’t even irrational numbers?”

Natural numbers (positive integers) have always been linked to the counting operation. Another important operation, measurement, resulted in rational numbers given by the fractions of natural numbers. In ancient Greece, the advance of geometry, especially the Pythagorean theorem, required a further expansion of the idea of number to include so-called irrational numbers.

Pause to regret that the names of the number sets are so clumsy: the fractions are neither more nor less “rational” than the other numbers. Unfortunately, the terminology disaster didn’t end there, as we’ll see later.

Like their Mesopotamian and Egyptian predecessors, the Greeks only thought of positive numbers. Diophantus (200–284 BC) considered negative solutions to equations “absurd”.

The “Nine Chapters of Mathematical Art”, published in China around the year 200, contain the first documented use of negative numbers: They are represented with red symbols as in Excel! A hundred years later, Chinese mathematicians added and subtracted negative numbers. From the 7th century onwards, Indians and Arabs also knew the rules for multiplication and division.

Perhaps because of the Greek influence, the West was left behind.

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