Negative Numbers Outline
A History
• Negative numbers were rejected as solutions of problems in early times (1, pg 259)
• The math of the Nine Chapters (chinese math book dating back to the Han Dynasty, 206BC
22AD) have matrices with negative numbers, the first appearance in history (2, pg33)
• Negative numbers were first accepted in a limited way in India around the 6th century, by
which time the Greek developments had long ceased (1, pg 185)
• They were eventually admitted in Hindu practical math through monetary problems, since
the idea of receiving and owing money was a simple and obvious one, that is a negative
solution could be interpreted as a debt (1, pg 260)
• Negative numbers first appeared when people began to solve equations (3, pg 81)
• Egypt and Mesopotamia scribes could solve equations, but didn't consider negative answers
solutions (3, pg 81)
• The Greeks ignored negative numbers completely because they dealt with geometry
(3, pg 81)
• Diophantus, a Greek writer and mathematician wrote a whole book about solving equations
and didn't consider negative numbers, saying the solution was absurd (3, pg 82)
• Brahmagupta recognized and worked with negative quantities to some extent as early as the
7th century, he treated them as debts, and even stated rules in computing with them (3, pg 82)
• Arabs knew how to deal with negative numbers, but didn't understand negatives themselves
(3, pg 83)
• After the Renaissance, Europeans, who learned from the Arabs to not use negatives,
continued to see them as absurd despite the advances made in astronomy, navigation,
physical science, warfare, and commerce (3, pg 83)
• Bhaskara in the 12th century stated a negative root of an equation, but discounted it (3, pg 82)
for which he indulged in some skepticism (2, pg 66)
• By the early 17th century, negative numbers became useful and hard to ignore (3, pg 83)
• Early in the 17th century, Descartes called negative solutions false, and solutions involving
square roots of negatives imaginary (3, pg 84)
• Our Cartesian plane is named after Descartes, but he didn't include a negative x or y axis
(3, pg 84)
• They were accepted by some in the middle of the 18th century (3, pg 85)
• De Morgan also saw them as absurd or inconsistent in 1831 (3, pg 85)
• As late as the early 19th century, some works still suggest that they do not have any real
existence (1, pg 260)
• Negative numbers didn't become accepted until the middle of the 19th century, about the time
of the American Civil War (3, pg 81)
• Abstract algebra and the structure of numbers then began to take hold with the works of
Gauss, Galois, Abel and others, and in the abstract setting the operational relationships
became essential. In other words, negative numbers were seen as the additive opposites of
the positive numbers (3, pg 86)
Negative Numbers, Page 1
Where Should They Go
• John Wallis claimed they were larger than infinity, because if 3/0 is infinite, changing the
denominator to a smaller number should be even larger. That is 3/(-1) = -3 is larger than
infinity (3, pg 84)
• In 1707 Isaac Newton in his book Universal Arithmetick said that a negative quantity is less
than nothing, but how could a quantity be less than nothing? (3, pg 84)
• Euler in 1770 said that they are less than nothing (3, pg 85)
1. Numbers: Their History and Meaning by Flegg
2. A Concise History of Mathematics by Struik
3. Math Through the Ages by Berlinghoff
4. http://www.mathpages.com/home/kmath298.htm
5. http://nti.educa.rcanaria.es/penelope/uk_confboye.htm#elements
Negative Numbers, Page 2