History of Negative Numbers


c. 570 - 500 B.C.E., Greece: The Pythagoreans thought of number as "a multitude of units". Thus one was not a number for them. There are no indications of negative numbers in their work. [K, 50], [S, 257]

Fourth Century B.C.E., Greece: Aristotle made the distinction between number (i.e., natural numbers) and magnitude ("that which is divisible into divisibles that are infinitely divisible"), but gave no indications of the concept of negative number or magnitude. [K, 56], [S, 257]

c. 300 B.C.E., Greece: Books VII, VIII, and IX of Euclid's Elements concern the elementary theory of numbers. Euclid continues Aristotle's distinction between number and magnitude, but there are still no indications of negative numbers. [K, 84], [S, 257]

 c. 100 B.C.E.-50 C.E., China: In The Nine Chapters on the Mathematical Art (Jiuzhang Suanshu), negative numbers were used in the chapter on solving systems of simultaneous equations. Red rods were used to denote positive coefficients, black to denote negative ones. Rules for signed numbers were given. [K, 19] (For more information on the history of mathematics in China, see Mathematics in China, http://aleph0.clarku.edu/~djoyce/mathhist/china.html)

Third century C.E., Greece: The first indication of negative numbers in a western work was in Diophantus' Arithmetica, in which he referred to the equation which in modern notation would be represented as 4x + 20 = 0 as absurd, since it would give the solution x = -4. . He also said, "a number to be subtracted, multiplied by a number to be subtracted, gives a number to be added." So, for example, he could deal with expressions such as 9in modern notation) x - 1 times x - 2. However, Diophantus gave indications that he had no conception of the abstract notion of negative number [S, 258] [C, 61]

Seventh Century C.E., India: Negative numbers were used to represent debts when positive numbers represented assets. Indian mathematician/astronomer Brahmagupta used negative numbers to unify Diophantus' treatment of quadratic equations from three cases (ax2 + bx = c, bx + c = ax2, and ax2 + c = bx) to the single case we are familiar with today. [C, 94]. He gave rules for operations with negative numbers. [K, 226]

Ninth Century C.E, Middle East.: Although the Arabs were familiar with negative numbers from the work of Indian mathematicians, they rejected them. [Kl, 192]Muhammad ibn Musa al-Khwarizmi's textbook Al-jabr wa'l- muqabala (from which we get the word "algebra") did not use negative numbers or negative coefficients [P]. Thus his discussion of quadratic equations dealt with six different types of equations, rather than the one general form we use. [K, 245]

Twelfth Century, India: Bhaskara gives negative roots for quadratic equations, but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots." [C. 93]

Thirteenth Century, China: Negative numbers were indicated by drawing a diagonal stroke through the right-most nonzero digit of a negative number. [S, 259]

Thirteenth Century, Italy:  Smith [S, 258] asserts that Fibonacci included no mention of negative numbers in his book Liber Abaci, but in a later volume, Flos, interpreted a negative solution in a problem as a loss. However, Mark Dominus (private communication) has pointed out that in Sigler's English translation of Liber Abaci there are some problems that do involve negative solutions, which are interpreted as debits. [Si, 226-227,320-322 , 484-486]

Fifteenth Century, Europe: Chuquet was the first to use negative numbers in a European work. He used them as exponents, writing, for example, for what we would write as -12x-2. [K, 350]. However, he referred to them as "absurd numbers." [Kl, 252]

Sixteenth Century, Europe:

Seventeenth Century, Europe:

Eighteenth Century, Europe:

Nineteenth Century, Europe:


[C] Cajori, Florian, History of Mathematics, 5th ed. Chelsea, New York, 1991

[K] Katz, Victor, A History of Mathematics: An Introduction, 2nd edition, Addison-Wesley, Reading, Mass., 1998

[Kl] Kline, Morris, Mathematical Thought form Ancient to Modern Times, Oxford University Press, New York, 1972

 [P] Parshall, Karen Hunger, The Art Of Algebra From Al-Khwarizmi To Viète: A Study In The Natural Selection Of Ideas, expanded version of an article in History of Science, Vol.26, No. 72, pp. 129 - 164, online at http://www.lib.virginia.edu/science/parshall/algebra.html

[Si] Sigler, L.E., Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation,  Springer, 2002.

 [S] Smith, David Eugene, History of Mathematics, v. II, Dover, New York, 1953

This page was prepared by Martha K. Smith, February 19, 2001