Negative numbers have been the source of wide controversy in the historical development of mathematics. Even as late
as the 18th century, some mathematicians argued that equations with negative solutions suggested that a false assumption
had been made. The fact that negative numbers could not represent physical quantities was the main reason people were
reluctant to accept negative numbers. The Greeks, for example, studied geometry where numbers were used to represent
length and distance. The Greeks understood that formulas might involve negative quantities, but since geometric
measurements were always positive, equations that yielded negative numbers were considered "absurd."
The Chinese are credited with being the first known culture to recognize and use negative numbers. The book Nine
Chapters on the Mathematical Art, written around the year 200 AD, documents some of the development of mathematics
in ancient China. Chapter 8 of this book describes how red rods are used to represent positive numbers and black rods
are used to represent negative numbers. Later, Chinese mathematician Liu Hui wrote negative numbers in a slanted position
to set them apart from positive numbers. Subsequent Chinese works, such as the 1299 book Introduction to Mathematical
Studies, fully document the rules for performing mathematical operations on positive and negative numbers.
The Indian mathematician Brahmagupta is known to have used negative numbers starting around 630 AD. During this time,
positive numbers were used to represent assets and negative numbers were used to represent debt. Brahmagupta described
the rules for the four basic operations on signed numbers-addition, subtraction, multiplication and division. He also
pointed out that the square root of a positive number could be positive or negative. Brahmagupta is credited with other
important mathematical contributions to the areas of arithmetic, algebra and geometry. For example, Brahmagupta's
treatment of quadratic equations and the quadratic formula, which can result in negative and irrational solutions, is of
The first evidence of the acceptance of negative numbers in Europe can be traced back to the year 1545. In this year,
Italian mathematician Girolamo Cardano wrote a book titled The Great Art. Like Brahmagupta, Cardano clearly stated
the rules for the use of negative numbers. Despite his understanding of computation and calculations with negative
numbers, Cardano was reluctant to give negative numbers practical acceptance. In fact, he called negative solutions
"fictitious" whereas positive solutions were deemed "true." French mathematician Rene Descartes held a similar view to
Cardano. He even described methods to transform negative solutions to those that were positive. In 1759, English
mathematician Baron Francis Maseres went so far as to publish a work describing how negative numbers could be avoided.
These mathematicians fully understood the rules ofnegative numbers but perhaps not their application to real-life.
By the late 1700's, the algebraic use of negative numbers had become widely accepted.