Invention of Decimals and Negative Numbers

Date 595-665

Locale India, China

The development of a decimal place-value system made numbers easier to use, while acceptance of negative numbers aided with the development of algebra and new physical applications. The best evidence available points to India as the locale for the most significant steps in the process.

Key Figures

• Āryabhaṭa the Elder (c. 476-c. 550), Indian mathematician and astronomer
• Brahmagupta (c. 598-c. 660), Indian mathematician and astronomer
• Mahāvīra (c. 800-c. 870), Jaina mathematician
• Bhāskara (1114-c. 1185), Indian mathematician and astronomer
• al-Khwārizmī (c.780-c. 850), Indian mathematician

Summary of Event

The awareness of the “number” concept and its applications is fundamental to civilization and the building of knowledge. Indeed, many ancient cultures around the world developed the ability to count, measure time and space, and make arithmetical and geometric calculations for astronomy and other scientific endeavors. The various numeral systems that resulted generally denoted numbers by words or by a large set of symbols. Only positive numbers were considered.

Place-value systems—meaning that each “digit” in a number represented a multiple of the base—existed in Babylonia at least in part around 2000 b.c.e., in China by 200 b.c.e., and in the Maya Empire between 200 and 665 c.e. Sometime between 200 b.c.e. and 600 c.e., however, Indian mathematicians and scribes began writing numbers in true place-value notation with symbols for the numerals 1 through 9, which had evolved from the middle of the third century b.c.e. Writers gradually discarded the separate symbols they had for 10, 100, 1000, . . . ; 20, 30, 40, . . . 90; and 200, 300, 400, . . . 900. For example, Āryabhaṭa the Elder wrote a mathematics and astronomy textbook called Āryabhaṭīya (499; The Aryabhatiya, 1927) that contained numbers in place-value form with nine symbols (but no zero). A donation charter of Dadda III of Sankheda in the Bharukachcha region prepared in 595 is the oldest known dated Indian document containing a number in decimal place-value notation including zero.

Indeed, a symbol for zero is necessary for a fully decimal-positional system. Empty spaces in numbers may have been marked in ancient Egypt, Babylonia, and Greece. The Maya certainly used zero as a placeholder in their base-20 system by 665. In India, a dot as a zero to mark an empty place appeared in the Bakhshali manuscript, which may date to the 600’s or earlier. Other Indian texts used ten symbols in a decimal place-value system to facilitate such tasks as multiplication. The word kha was sometimes used instead of a zero symbol, and the empty circle was widely adopted late in the ninth century.

Unlike Maya numerals, which were confined to that civilization, the Indian system quickly spread into other regions of the world. Inscriptions that date to 683 and 684 and employ zero as a placeholder have been found in Cambodia and in Sumatra, Indonesia. Indian astronomers used their numerals in the service of the Chinese emperor by 718. Arab scholars and merchants learned of the nine-sign Indian system in the 600’s and 700’. All ten digits had reached Baghdad by 773, and they were used for positional notation in Spain by the 800’.

However, the symbols used to represent the numbers evolved separately in the western and eastern regions of the Arab Empire, with the symbols in the west (North Africa and Spain) remaining more like the original Indian versions by 1000. These symbols were standardized into today’s form with the advent of printing in the 1400’. Many European scholars were introduced to the decimal place-value system through a book on the Indian symbols written in 825 by al-Khwārizmī, which was anonymously revised and translated into Latin in the 1100’s as Algoritmi de numero Indorum (al-Khwārizmī on the Indian art of reckoning; “Thus Spake al-Khwarizmi,” 1990). Some European Christians were already familiar with Indian number symbols, though; for example, they have been found in the Codex Vigilanus, which was copied by a Spanish monk in 976.

Negative numbers most likely first appeared in China. The anonymous work Jiuzhang suanshu (nine chapters on the mathematical art), which dates approximately to the second century, provides correct rules for adding and subtracting with both negative and positive numbers. The concept of negative numbers was apparently transmitted to India in the second century, where mathematicians developed true fluency in handling negatives, including the ability to multiply and divide these numbers. These Indian advancements were then transmitted back to China by the 1300’. For instance, Brahmagupta introduced negative numbers to an Indian audience in 628 through the astronomy text Brahmasphuṭasiddhānta (the opening of the universe). His arithmetical rules of operation were updated by Mahāvīra in Ganita sara sangraha (850; compendium of the essence of mathematics). In the twelfth century, the six books by Bhāskara represented the peak of contemporary mathematical knowledge. He improved notation by placing a dot over a number to denote that it was negative. He accepted negative solutions and encouraged others to accept them as well, providing several word problems to test the reader’s calculating skills.

Many of these works were also notable for their authors’s efforts to treat zero as an abstract number and to understand its properties. Brahmagupta and Bhāskara agreed that any number minus itself was zero and that any number multiplied by zero was zero. They disagreed on the result when dividing by zero. Brahmagupta said the result when dividing zero by zero was zero. Bhāskara realized that Brahmagupta was incorrect, but he concluded that (a.0)/0 is a in his work on mathematics, Līlāvatī (c. 1100’; the beautiful). In a later book on algebra, Bījaganita (c. 1100’; seed counting or root extraction), he suggested that a divided by zero yielded infinity. This would force zero multiplied by infinity to equal every number a, or to prove that all numbers are equal. Bhāskara did not attempt to resolve this issue or to admit that dividing by zero is impossible.

Significance

Although the decimal place-value system facilitates arithmetical computation, it was not easily accepted as it moved outward from India. The dissemination of Indian numeral symbols was necessarily slowed by the complex paths of transmission that roughly followed medieval trade routes. Additionally, even though writers such as al-Uqlīdisī trumpeted the utility of decimal numbers in Kitāb al-fuḥūl fī al-ḥisāb al-Hindī (952-953; The Arithmetic of al-Uqlidisi, 1978), artisans and merchants often saw no compelling reason to give up their existing numerical practices, such as finger reckoning. Indian number symbols also sometimes mixed with existing symbol sets as they entered new cultures. Finally, it took time for mathematicians to understand and adopt ten-character decimal symbols (rather than nine) that employed zero first as a placeholder and then as an abstract number in its own right.

Negative numbers also aroused the foundational concerns, definitional difficulties, and philosophical baggage of the number zero. Although writers such as al-Khwārizmī did not recognize negative numbers or zero as algebraic coefficients, this stumbling block was perhaps especially prevalent in Europe, where the rules for decimal and negative numbers in Leonardo of Pisa’s Liber abaci (English translation, 2002), were widely read but not always taken up immediately. In fact, European mathematicians into the eighteenth century questioned the validity of negative numbers and often made computational errors when they did work with these numbers. Such influential Renaissance and early modern mathematicians as Regiomontanus, Gerolamo Cardano, and François Viète went so far as to discard negative solutions. Nevertheless, these numbers simultaneously enabled the development of modern algebra. In the end, the decimal and negative numbers that arrived in Europe from India via Islam revolutionized and algebraized mathematics. They became the basis of the European number system and were key components of the new mathematical discipline—including analytical geometry, mechanics, and differential and integral calculus—that emerged in the early modern period.

Bibliography

Calinger, Ronald. A Contextual History of Mathematics. Upper Saddle River, N.J.: Prentice Hall, 1999. History of mathematics with significant discussions of the development of arithmetic and numbers.

Gupta, R. C. “Spread and Triumph of Indian Numerals.” Indian Journal of History of Science 18, no. 1 (1983): 23-38. Classic work on the subject with many examples of early uses of the symbols outside India, although Gupta’s claim that the numerals entered Europe by 500 is wrong.

Joseph, George Gheverghese. The Crest of the Peacock: The Non-European Roots of Mathematics. London: Tauris, 1991. Standard reference on the history of non-Western mathematics.

Kaplan, Robert. The Nothing That Is: A Natural History of Zero. New York: Oxford University Press, 2000. Entertaining account that can by enjoyed even by readers who like numbers but not mathematics.

Martzloff, Jean-Claude. History of Chinese Mathematics. Translated by Stephen S. Wilson. Berlin: Springer, 1987. Comprehensive introduction to Chinese mathematics.

Pycior, Helena M. Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra Through the Commentaries on Newton’s Universal Arithmetick. New York: Cambridge University Press, 1997. Details how struggles with the concept of negative numbers continued through early modern Europe.