From Middle Horizon cord-keeping to the rise of Inka khipus in the central Andes

Antiquity ◽  
2014 ◽  
Vol 88 (339) ◽  
pp. 205-221 ◽  
Author(s):  
Gary Urton
Keyword(s):  
Number System ◽  
Central Andes ◽  
Decimal Place ◽  
Place Value ◽  
Middle Horizon ◽  
Value System ◽  
Intermediate Period ◽  
Andean South America ◽  
Ams Dating ◽  
Descendant Communities

Recording devices formed of knotted cords, known askhipus, are a well-known feature of imperial administration among the Inka of Andean South America. The origins and antecedents of this recording system are, however, much less clearly documented. Important insights into that ancestry are offered by a group of eight khipus dating from the later part of the Middle Horizon period (AD 600–1000), probably used by the pre-Inka Wari culture of the central Andes. This article reports the AMS dating of four of these early khipus. A feature of the Middle Horizon khipus is the clustering of knots in groups of five, suggesting that they were produced by a people with a base five number system. Later, Inka khipus were organised instead around a decimal place-value system. Hence the Inka appear to have encountered the base five khipus among Wari descendant communities late in the Middle Horizon or early in the Late Intermediate period (AD 1000–1450), subsequently adapting them to a decimal system.

The Indian Gift

2016 ◽  
Author(s):  
Joseph Mazur
Keyword(s):  
Number System ◽  
Decimal Place ◽  
Place Value ◽  
Ancient India ◽  
Finger Counting ◽  
History Of ◽  
Indian Mathematics

This chapter discusses the legacy of Indian mathematics. With very few archaeological clues, the origins of the Indian numbers must rely on a small wealth of writing that survives almost exclusively in the form of stone inscriptions. Some of those stone epigraphs used decimal place-value numerals, providing some evidence that ancient India was familiar with a kind of place-value numerical system. Some letter combinations of the Sanskrit words for numbers probably contributed suggestive shapes early in the morphographic history of our current script. The chapter first considers the Brahmi number system before turning to modern Hindu-Arabic numerals. It also examines how the Western system of numerals with zero came to be by focusing on finger counting, the dust boards, and the abacus.

scholarly journals On the Origin of Symbolic Mathematics and Its Significance for Wittgenstein’s Thought

2015 ◽  
pp. 7 ◽  
Author(s):  
Sören Stenlund
Keyword(s):  
Sixteenth Century ◽  
Decimal Place ◽  
Place Value ◽  
Late Nineteenth Century ◽  
Value System ◽  
Early Modernity ◽  
Modern Physics ◽  
Ontological Sense ◽  
Modern Mathematics ◽  
Algebraic Symbolism

The main topic of this essay is symbolic mathematics or the method of symbolic construction, which I trace to the end of the sixteenth century when Franciscus Vieta invented the algebraic symbolism and started to use the word ‘symbolic’ in the relevant, non-ontological sense. This approach has played an important role for many of the great inventions in modern mathematics such as the introduction of the decimal place-value system of numeration, Descartes’ analytic geometry, and Leibniz’s infinitesimal calculus. It was also central for the rigorization movement in mathematics in the late nineteenth century, as well as for the mathematics of modern physics in the 20th century.However, the nature of symbolic mathematics has been concealed and confused due to the strong influence of the heritage from the Euclidean and Aristotelian traditions. This essay sheds some light on what has been concealed by approaching some of the crucial issues from a historical perspective. Furthermore, I argue that the conception of modern mathematics as symbolic mathematics was essential to Wittgenstein’s approach to the foundations and nature of mathematics. This connection between Wittgenstein’s thought and symbolic mathematics provides the resources for countering the still prevalent view that he defended an uttrely idiosyncratic conception, disconnected from the progress of serious science. Instead, his project can be seen as clarifying ideas that have been crucial to the development of mathematics since early modernity.

The Egyptian Number System

2016 ◽  
Author(s):  
Annette Imhausen
Keyword(s):  
Number System ◽  
Decimal Place ◽  
Place Value ◽  
Decimal System ◽  
Absolute Value ◽  
Ancient Egyptian ◽  
System P ◽  
The Absolute ◽  
The Individual

This chapter describes the ancient Egyptian number system. The system can be described in modern terminology as a decimal system without positional (place-value) notation. The basis of the number system was 10 (hence decimal system), but unlike our decimal place-value notation using the ten numerics 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, in which the absolute value is determined by its position within the number (e.g., in the number 125, the absolute value of 1 is 1 × 102, the absolute value of 2 is 2 × 101, and the absolute value of 5 is 5 × 100), the Egyptian system used individual symbols for each power of 10. Although there is no information about the choice of the individual signs for the respective values, some of them seem plausible choices. The most basic, the simple stroke to represent a unit, is used not only in Egypt but also in a variety of other cultures, possibly originating from marks on a tally stick.

scholarly journals Early Texts on Hindu-Arabic Calculation

2001 ◽  
Vol 14 (1-2) ◽  
pp. 13-38 ◽  
Author(s):  
Menso Folkerts
Keyword(s):  
New York ◽  
Fifteenth Century ◽  
Twelfth Century ◽  
Decimal Place ◽  
Latin Translation ◽  
Place Value ◽  
Detailed Knowledge ◽  
The West ◽  
Value System

This article describes how the decimal place value system was transmitted from India via the Arabs to the West up to the end of the fifteenth century. The arithmetical work of al-Khwārizmī's, ca. 825, is the oldest Arabic work on Indian arithmetic of which we have detailed knowledge. There is no known Arabic manuscript of this work; our knowledge of it is based on an early reworking of a Latin translation. Until some years ago, only one fragmentary manuscript of this twelfth-century reworking was known (Cambridge, UL, Ii.6.5). Another manuscript that transmits the complete text (New York, Hispanic Society of America, HC 397/726) has made possible a more exact study of al-Khwārizmī's work. This article gives an outline of this manuscript's contents and discusses some characteristics of its presentation.

Packing Candies

2015 ◽  
Vol 22 (1) ◽  
pp. 13-16
Keyword(s):  
Number System ◽  
Place Value ◽  
Value System ◽  
Mathematical Ideas ◽  
Knowing That ◽  
Experience Difficulty

Do your students understand unitizing and the role it plays in our place-value number system? One of the important mathematical ideas that students need to learn is unitizing, or knowing that individual items can be organized or “packed” into groups and that those groups can then be counted as individual items. Despite having numerous experiences with base-ten blocks and place-value mats, many students may experience difficulty describing the quantity of numbers flexibly within our place-value system. The Packing Candies problem provides students with an authentic purpose to unitize: to organize a collection of loose candies into bags containing 10 candies and boxes containing 10 bags, or 100 candies.

The Influence of Structural Aspects of the English Counting Word System on the Teaching and Learning of Place Value

1998 ◽  
Vol 59 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Ian Thompson
Keyword(s):  
Young Children ◽  
Teaching And Learning ◽  
Place Value ◽  
Single Digit ◽  
Value System ◽  
Number Words ◽  
Large Numbers ◽  
Structural Aspects

The influence of structural aspects of the English counting word system on the teaching and learning of place value In their discussion of the teaching of place value to young children Fuson and Briars (1990) describe the extent to which the English spoken system of number words constitutes a ‘named value’ system for large numbers. They argue that, because two-digit numbers are not ‘named value’, teachers should move from teaching single-digit calculations to teaching calculations with large numbers, only returning to two-digit numbers when children are familiar with the standard written algorithms. This article uses transcriptions of children calculating mentally to suggest that they appear to take advantage of the ‘partitionable’ aspect of the language associated with two-digit numbers - an aspect that Fuson and Briars (1990) appear to have ignored. These examples appear to raise questions about their recommendation that teachers should progress from single-digit to large number calculations.

Multiplication by 10 base-5: Making Sense of Place Value Structure Through an Alternate Base

2015 ◽  
Vol 3 (2) ◽  
pp. 83-98
Author(s):  
Jodi Fasteen ◽  
Kathleen Melhuish ◽  
Eva Thanheiser
Keyword(s):  
Preservice Teachers ◽  
Conceptual Understanding ◽  
Sense Of Place ◽  
Number System ◽  
Place Value ◽  
Mathematical Activity ◽  
Making Sense ◽  
Procedural Fluency ◽  
Whole Number

Prior research has shown that preservice teachers (PSTs) are able to demonstrate procedural fluency with whole number rules and operations, but struggle to explain why these procedures work. Alternate bases provide a context for building conceptual understanding for overly routine rules. In this study, we analyze how PSTs are able to make sense of multiplication by 10five in base five. PSTs' mathematical activity shifted from a procedurally based concatenated digits approach to an explanation based on the structure of the place value number system.

Transformations in ritual practice and social interaction on the Tiwanaku periphery

Antiquity ◽  
2014 ◽  
Vol 88 (341) ◽  
pp. 851-862 ◽  
Author(s):  
Juan Albarracin-Jordan ◽  
José M. Capriles ◽  
Melanie J. Miller
Keyword(s):  
Social Interaction ◽  
Central Andes ◽  
Ritual Practice ◽  
Psychoactive Substances ◽  
Middle Horizon ◽  
Local Elites ◽  
Early States ◽  
Local Elite ◽  
South Central ◽  
Ritual Practices

Ritual practices and their associated material paraphernalia played a key role in extending the reach and ideological impact of early states. The discovery of a leather bag containing snuffing tablets and traces of psychoactive substances at Cueva del Chileno in the southern Andes testifies to the adoption of Tiwanaku practices by emergent local elites. Tiwanaku control spread over the whole of the south-central Andes during the Middle Horizon (AD 500–1100) but by the end of the period it had begun to fragment into a series of smaller polities. The bag had been buried by an emergent local elite who chose at this time to relinquish the former Tiwanaku ritual practices that its contents represent.

Sign in / Sign up

Export Citation Format

Share Document