The road to negativity is paved with the shattered preconceptions of some of maths greatest minds, and the implementation of more abstract thinking within the world of mathematics over a fairly short time frame. In this day and age we take for granted the fact that negative numbers have become so common place that they are taught at an elementary school level. Now imagine a time where the greatest minds in the field struggled with not only this concept, but the very idea of zero having not only an existence or an identity. In its infancy, numbers began as tool for counting and tracking physical quantities; one cow, two cats, three apples, etc. to have zero was to have nothing, to have a negative was “Ridiculous” or “Absurd”! The overall shift in paradigm this requires is fascinating, showing the evolution in conceptual thinking over the years impacted by societal changes.

It was not until around 1600 BC that the Babylonians gave zero a purpose. Up until this point, zero was interpreted as nothing by many people. However, due to the means of recording numerals within this culture, changes had to be made. “They had begun to recognize…the absence of quantity…as a quantity in its own right” (Berlinghoff, page 82). The Babylonians utilized a base 60 numeral system recorded using clay tablets carved by hand, with spaces being used as a means of separating 60s place from the 1s place, resulting in values that were inconsistent or unknown without prior contextual knowledge. Eventually, closer to 4^th century BC, the Babylonians began to use their equivalent of a period/end of sentence symbol as to indicate skipping of a place in larger values. It was the primitivity and inconsistency of the hand tools available at the time that forced their minds to expand its definition of how numbers would be used and manipulated moving forward.

Fast forward several centuries and you find the now common place base ten system in place worldwide, and one of the greatest shifts in mathematical thinking of all time. The concept that even the absence of a quantity can be considered a quantity itself, was revolutionary at the time. With numbers created for counting and tracking, to make the leap from this sole purpose to identifying them as ideas or symbols which can be modified, requires a stretch of the imagination. By the 9^th century this concept was expanded, to include the idea that the numbers existed and could be manipulated, regardless if there was anything to count or track. “Then, and only then, does it make sense to treat zero as a number (Berlinghoff, page 82) It was this recognition of Zero as a number that paved the road for Algebra, even if this wasn’t commonly accepted as a root of equations by 16^th and 17^th century mathematicians. By the 18^th century zero had gone from just a place holder to an invaluable tool within the world of algebra, reaching critical importance during the 19^th century.

This is about the time that negative numbers began gaining traction as a viable concept. While the ancient Egyptians and Mesopotamians could solve equations that were in the realm of negativity, they appeared as coefficients--rather than as an answer. A non-positive solution was nonsense, or at the least seen as an indication that the question has been worded incorrectly. That is not to say what happens when taking of square root of a negative number. In this specific case, “-1 can be neither positive nor negative” according to the speaker in Berlinghoff’s textbook (page 98). Even as far back as the 7^th century, the negatives were given rules per Indian mathematician, Brahmagupta. He regarded the use of negative numbers to be important when dealing with debts and developed rules for how these number could be manipulated; however, this did not have a direct influence on European development. The 9^th century brought two books written by Muhammed ibn Musa al-Khwarizimi which brought early understanding of negative numbers to Europe. By the early 17^th century, the dynamic was beginning to change. Up until this point the idea of a negative numbers was considered “absurd”, and when negatives appeared as the solutions to equations they were discounted as fiction. Even with the great minds such as Descartes and Newton actively working to discount the concepts throughout the 17^th and 18^th century, the importance and influence of negative numbers could not be ignored for long. It was during mid-18^th century that the negative numbers finally reached their place amongst the core concepts of number manipulation