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Indian mathematics
The chronology of Indian mathematics spans from the Indus valley civilization and the Vedas to modern times.
Indian mathematics has made outstanding contributions to the development of mathematics as we know it today. The Indian decimal notation of numbers, negative numbers and concept of zero have probably provided some of the biggest impetus' to advances in the field. Concepts from ancient India were carried to the Middle East, where they were studied extensively. From there they made their way to Europe.
Indian contributions to mathematics
- Decimal system — goes back as far as the Indus Valley civilization
- Hindu-Arabic numeral system, the modern positional notation numeral system used universally
- Zero — see Hindu-Arabic numerals
- Negative numbers — see Brahmagupta
- Infinity — see Yajur Veda
- Transfinite numbers — see [http://www.infinityfoundation.com/mandala/t_es/t_es_agraw_jaina_frameset.htm Ancient Jaina Mathematics: an Introduction], [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html Jaina mathematics]
- Irrational numbers — see Sulba Sutras
- Geometry — see Sulba Sutras
- Square roots and cube roots — see Baudhayana
- Metarules, transformations, recursions — see Panini
- Binary numbers — see Pingala
- Pascal's triangle — see Pingala
- The series popularly known as Fibonacci series
- The theorem popularly known as Pythagorean theorem
- Pythagorean triples — see Sulba Sutras
- "James Gregory" series expansion — see Madhava
- Logarithms — see [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html Jaina mathematics]
- Quadratic equations and cubic equations — see Sulba_Sutras
- The Panini Backus Normal Form
- Trigonometry and trigonometric tables — see Aryabhata
- Algebra — see for example Aryabhata
- Algorithms — see [http://www.crystalinks.com/indiamathematics.html Ancient India - Mathematics]
- Mathematical analysis — see Madhava
- Calculus — see Bhaskara, Madhava, Jyeshtadeva
Charges of Eurocentrism
Unfortunately, Indian contributions have not been given due acknowledgement in modern history, with many discoveries/inventions by Indian mathematicians now attributed to their western counterparts. More recently, evidence has been unearthed that reveals that the foundations of calculus were laid in India, at the Kerala school. Some allege that calculus and other mathematics of India were transmitted to Europe through the trade route from Kerala by Jesuit missionaries and traders. They point to evidence of methodological similarities, communication routes and a suitable chronology for transmission.
See also
- Indian mathematicians
- History of mathematics
External links
- [http://members.tripod.com/~INDIA_RESOURCE/mathematics.htm History of Indian Mathematics]
- [http://www-history.mcs.st-and.ac.uk/history/Projects/Pearce/index.html Indian Mathematics: Redressing the balance]
- [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Bakhshali_manuscript.html Bakhshali Manuscript]
- [http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch10.html Eurocentrism in Mathematics; Transmission of Calculus to Europe]
- [http://f2.grp.yahoofs.com/v1/ULPXQm9le1xk-VjK67S6K30TGTC_jPFaSgMQDvZP4dALlvjZbhnS4Tpcq5YV1odwyijUBl9wYLu3FjuOfVoa2w/profvk/History%20of%20Mathematics%20-%20Indian%20Contribution.doc History of Mathematics -Indian Contribution]
Category:History of mathematics
Indus Valley Civilization-Ghaggar river and their tributaries. The Mohenjo-daro ruins pictured above were once the center of this ancient society.]]
The Indus Valley Civilization, 3300 BCE–1800 BCE, was an ancient civilization thriving along the lower Indus River and the Ghaggar-Hakra river in what is now Pakistan and western India. Among other names for this civilization is the Harappan Civilization of the Indus Valley, in reference to its first excavated city of Harappa.
Overview
The Indus Civilization is among the world's earliest civilizations, contemporary to the Bronze Age civilizations of Mesopotamia and Ancient Egypt. It peaked around 2500 BCE in the western part of South Asia, declined during the mid-2nd millennium BCE and was forgotten until its rediscovery in the 1920s by RD Banerjee.
Geographically, it was spread over an area of some 1,260,000 km², comprising the whole of modern day Pakistan and parts of modern-day India and Afghanistan. There were Indus civilization settlements spread as far south as Mumbai, as far east as Delhi, as far west as the Iranian border, and as far north as the Himalayas. Thus there is an Indus Valley site on the Oxus river at Shortughai in northern Afghanistan (Kenoyer 1998:96) and the Indus Valley site Alamgirpur at the Hindon river is located only 28 km from Delhi (S.P. Gupta 1995:183). At its peak, the Indus Civilization may have had a population of well over five million.
The Indus civilization is still poorly understood. Its very existence was forgotten until the 20th century. Its writing system remains undeciphered. Among the Indus civilization's mysteries are fundamental questions, including its means of subsistence and the causes for its sudden disappearance beginning around 1900 BCE. We do not know what language the people spoke. We do not know what they called themselves. All of these facts stand in stark contrast to what is known about its contemporaries, Mesopotamia and ancient Egypt.
The native name of the Indus civilization may be preserved in the Sumerian Me-lah-ha, which Asko Parpola, editor of the Indus script corpus, identifies with the Dravidian Met-akam "high abode/country" (see also Proto-Dravidian). He further suggets that the Sanskrit word mleccha for "foreigner, barbarian, non-Aryan" may be derived from that name.
Settlements
Aryan
To date, over 1,052 cities and settlements have been found, mainly in the general region of the Hakra-Ghaggar river and its tributaries. Among the settlements were the major urban centers of Harappa and Mohenjo-daro, as well as Dholavira, Ganweriwala, Kalibanga, Lothal, and Rakhigarhi.
Additionally, there is some disputed evidence indicative of another large river, now long dried up, running parallel and to the east of the Indus. The dried-up river beds overlap with the Hakra channel in Pakistan, and the seasonal Ghaggar river in India. Over 500 ancient sites belonging to the Indus Valley Civilization have been discovered along the Hakra-Ghaggar river and its tributaries (S.P. Gupta 1995: 183). In contrast to this, only 90 to 96 of the over 800 known Indus Valley sites have been discovered on the Indus and its tributaries. A section of scholars claim that this was a major river during the third and fourth millennia BCE, and propose that it may have been the Sarasvati River of the Rig Veda. Some of those who accept this hypothesis advocate designating the Indus Valley culture the "Sarasvati-Sindhu Civilization", Sindhu being the ancient name of the Indus River. Most archeologists dispute this view, arguing that the old and dry river died out during the mesolithic age at the latest, and was reduced to a seasonal stream thousands of years before the Vedic period.
Predecessors
The Indus civilization was predated by the first farming cultures in south Asia, which emerged in the hills of what is now called Balochistan, to the west of the Indus Valley. The best-known site of this culture is Mehrgarh, established around 6500 BCE. These early farmers domesticated wheat and a variety of animals, including cattle. Pottery was in use by around 5500 BCE. The Indus civilization grew out of this culture's technological base, as well as its geographic expansion into the alluvial plains of what are now the provinces of Sindh and Punjab in contemporary Pakistan and Northern India.
By 4000 BCE, a distinctive, regional culture, called pre-Harappan, had emerged in this area. (It is called pre-Harappan because remains of this widespread culture are found in the early strata of Indus civilization cities.) Trade networks linked this culture with related regional cultures and distant sources of raw materials, including lapis lazuli and other materials for bead-making. Villagers had, by this time, domesticated numerous crops, including peas, sesame seeds, dates, and cotton, as well as a wide range of domestic animals, including the water buffalo, an animal that remains essential to intensive agricultural production throughout Asia today.
Emergence of Civilization
The first appearance of the Indus civilization was the early Harappan/Ravi Phase. This Ravi Phase, named after the nearby Ravi River, lasted from approximately 3300 BC, or even 3500 BC, to 2800 BC. This phase is related to the Hakra Phase, identified in the Ghaggar-Hakra River valley to the west, and predates the Kot Diji Phase (2800-2600 BC), named after a site in northern Sindh near Mohenjo-daro. Increasing knowledge of the Ravi and Kot Diji Phase occupations at Harappa, and of contemporary settlements throughout northwestern South Asia, permits glimpses of later Indus Civilization. Some of the most exciting discoveries in Ravi Phase levels have been of early writing. The origins of the Indus script-like signs dates from 3300-2800 BC. This would make the origins of writing in South Asia approximately the same time as in ancient Egypt and Mesopotamia. [http://www.essays.cc/free_essays/a3/myv94.shtml] The civilization's mature Harappan period began from 2600 BC.
Cities
A sophisticated and technologically advanced urban culture is evident in the Indus Valley civilization. The quality of municipal town planning suggests knowledge of urban planning and efficient municipal governments which placed a high priority on hygiene. The streets of major cities such as Mohenjo-daro or Harappa were laid out in a perfect grid pattern, comparable to that of present day New York. The houses were protected from noise, odors, and thieves.
As seen in Harappa, Mohenjo-daro, and the recently discovered Rakhigarhi, this urban plan included the world's first urban sanitation systems. Within the city, individual homes or groups of homes obtained water from wells. From a room that appears to have been set aside for bathing, waste water was directed to covered drains, which lined the major streets. Houses opened only to inner courtyards and smaller lanes.
The ancient Indus systems of sewage and drainage that were developed and used in cities throughout the Indus Empire were far more advanced than any found in contemporary urban sites in the Middle East and even more efficient than those in some areas of modern Pakistan and India today. The advanced architecture of the Harappans is shown by their impressive dockyards, granaries, warehouses, brick platforms, and protective walls. The massive citadels of Indus cities that protected the Harappans from floods and attackers were larger than most Mesopotamian ziggurats.
The purpose of the "Citadel" remains a matter of debate. In sharp contrast to this civilization's contemporaries, Mesopotamia and ancient Egypt, no large monumental structures were built. There is no conclusive evidence of palaces or temples—or, indeed, of kings, armies, or priests. Some structures are thought to have been granaries. Found at one city is an enormous well-built bath, which may have been a public bath. Although the "Citadels" are walled, it is far from clear that these structures were defensive. They may have been built to divert flood waters.
Most city dwellers appear to have been traders or artisans, who lived with others pursuing the same occupation in well-defined neighborhoods. Materials from distant regions were used in the cities for constructing seals, beads, and other objects. Among the artifacts made were beautiful beads made of glazed stone called faïence. The seals have images of animals, gods, etc., and inscriptions. Some of the seals were used to stamp clay on trade goods, but they probably had other uses. Although some houses were larger than others, Indus civilization cities were remarkable for their apparent egalitarianism. For example, all houses had access to water and drainage facilities. One gets the impression of a vast middle-class society.
Science
The people of the Indus Civilization achieved great accuracy in measuring length, mass, and time. They were among the first to develop a system of uniform weights and measures. Their measurements were extremely precise. Their smallest division, which is marked on an ivory scale found in Lothal, was approximately 1.704mm, the smallest division ever recorded on a scale of the Bronze Age. Harappan engineers followed the decimal division of measurement for all practical purposes, including the measurement of mass as revealed by their hexahedron weights.
Brick sizes were in a perfect ratio of 4:2:1, and the decimal system was used. Weights were based on units of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with each unit weighing approximately 28 grams, similar to the English ounce or Greek uncia, and smaller objects were weighed in similar ratios with the units of 0.871.
Unique Harappan inventions include an instrument which was used to measure whole sections of the horizon and the tidal dock. In addition, they evolved new techniques in metallurgy, and produced copper, bronze, lead, and tin. The engineering skill of the Harappans was remarkable, especially in building docks after a careful study of tides, waves, and currents.
In 2001, archaeologists studying the remains of two men from Mehrgarh, Pakistan made the startling discovery that the people of Indus Civilization, even from the early Harappan periods, had knowledge of medicine and dentistry. The physical anthropologist that carried out the examinations, Professor Andrea Cucina from the University of Missouri-Columbia, made the discovery when he was cleaning the teeth from one of the men.
Arts and Culture
Various sculptures, seals, pottery, gold jewelry, and anatomically detailed figurines in terracotta, bronze, steatite have been found at the excavation sites.
A number of bronze, terracotta, and stone figurines of girls in dancing poses reveal the presence of some dance form. Sir John Marshall is known to have reacted with surprise when he saw the famous Indus bronze statuette of the slender-limbed "dancing girl" in Mohenjo-daro:
:"… When I first saw them I found it difficult to believe that they were prehistoric; they seemed to completely upset all established ideas about early art. Modeling such as this was unknown in the ancient world up to the Hellenistic age of Greece, and I thought, therefore, that some mistake must surely have been made; that these figures had found their way into levels some 3000 years older than those to which they properly belonged. … Now, in these statuettes, it is just this anatomical truth which is so startling; that makes us wonder whether, in this all-important matter, Greek artistry could possibly have been anticipated by the sculptors of a far-off age on the banks of the Indus."
A harp-like instrument depicted on an Indus seal and two shell objects found at Lothal indicate the use of stringed musical instruments.
Seals have been found at Mohenjo-daro depicting a figure standing on its head, and one sitting cross-legged; perhaps the earliest indication, at least illustration, of the practice of yoga. A horned figure in a meditation pose (see image, Pashupati, below right) has been interpreted as one of the earliest depictions of the god Shiva.
Trade
The Indus civilization's economy appears to have depended significantly on trade, which was facilitated by major advances in transport technology. These advances included bullock-driven carts that are identical to those seen throughout South Asia today, as well as boats. Most of these boats were probably small, flat-bottomed craft, perhaps driven by sail, similar to those one can see on the Indus River today; however, there is secondary evidence of sea-going craft. Archaeologists have discovered a massive, dredged canal and docking facility at the coastal city of Lothal.
Judging from the dispersal of Indus civilization artifacts, the trade networks, economically, integrated a huge area, including portions of Afghanistan, the coastal regions of Persia, northern and central India, and Mesopotamia (see Meluhha).
Agriculture
The nature of the Indus civilization's agricultural system is still largely a matter of conjecture due to the limited amount of information surviving through the ages. Some speculation is possible, however.
Indus civilization agriculture must have been highly productive; after all, it was capable of generating surpluses sufficient to support tens of thousands of urban residents who were not primarily engaged in agriculture. It relied on the considerable technological achievements of the pre-Harappan culture, including the plough. Still, very little is known about the farmers who supported the cities or their agricultural methods. Some of them undoubtedly made use of the fertile alluvial soil left by rivers after the flood season, but this simple method of agriculture is not thought to be productive enough to support cities. There is no evidence of irrigation, but such evidence could have been obliterated by repeated, catastrophic floods.
The Indus civilization appears to contradict the hydraulic despotism hypothesis of the origin of urban civilization and the state. According to this hypothesis, cities could not have arisen without irrigation systems capable of generating massive agricultural surpluses. To build these systems, a despotic, centralized state emerged that was capable of suppressing the social status of thousands of people and harnessing their labor as slaves. It is very difficult to square this hypothesis with what is known about the Indus civilization. There is no evidence of kings, slaves, or forced mobilization of labor.
It is often assumed that intensive agricultural production requires dams and canals. This assumption is easily refuted. Throughout Asia, rice farmers produce significant agricultural surpluses from terraced, hillside rice paddies, which result not from slavery but rather the accumulated labor of many generations of people. Instead of building canals, Indus civilization people may have built water diversion schemes, which—like terrace agriculture—can be elaborated by generations of small-scale labor investments. In addition, it is known that Indus civilization people practiced rainfall harvesting, a powerful technology that was brought to fruition by classical Indian civilization but nearly forgotten in the 20th century. It should be remembered that Indus civilization people, like all peoples in South Asia, built their lives around the monsoon, a weather pattern in which the bulk of a year's rainfall occurs in a four-month period. At a recently discovered Indus civilization city in western India, archaeologists discovered a series of massive reservoirs, hewn from solid rock and designed to collect rainfall, that would have been capable of meeting the city's needs during the dry season.
Writing or Symbol System
monsoon
Main article: Indus script.
It has long been claimed that the Indus Valley was the home of a literate civilization, but this has recently been challenged on linguistic and archaeological grounds. Well over 400 Indus symbols have been found on seals or ceramic pots and over a dozen other materials, including a 'signboard' that apparently once hung over the gate of the inner citadel of the Indus city of Dholavira. Typical Indus inscriptions are no more than four or five characters in length, most of which (aside from the Dholavira 'signboard') are exquisitely tiny; the longest on a single surface, which is less than 1 inch (2.54 cm) square, is 17 signs long; the longest on any object (found on three different faces of a mass-produced object) carries only 26 symbols. It has been recently pointed out that the brevity of the inscriptions is unparalleled in any known premodern literate society, including those that wrote extensively on leaves, bark, wood, cloth, wax, animal skins, and other perishable materials.
Based partly on this evidence, a controversial recent paper by Farmer, Sproat, and Witzel (2004)[http://www.safarmer.com/fsw2.pdf], which has been widely discussed in the world press (see external links), argues that the Indus system did not encode language, but was related instead to a variety of non-linguistic sign systems used extensively in the Near East. It has also been claimed on occasion that the symbols were exclusively used for economic transactions, but this claim leaves unexplained the appearance of Indus symbols on many ritual objects, many of which were mass produced in molds. No parallels to these mass-produced inscriptions are known in any other early ancient civilizations.
Photos of many of the thousands of extant inscriptions are published in the Corpus of Indus Seals and Inscriptions (1987, 1991), edited by A. Parpola and his colleagues. Publication of a final third volume, which will reportedly republish photos taken in the 20s and 30s of hundreds of lost or stolen inscriptions, along with many discovered in the last few decades, has been announced for several years, but has not yet found its way into print. For now, researchers must supplement the materials in the Corpus by study of the tiny photos in the excavation reports of Marshall (1931), Mackay (1938, 1943), Wheeler (1947), or reproductions in more recent scattered sources.
Geography
The Indus valley was by main rivers, the Indus River. The Indus River was very important to Indus life. The river provided irrigation, and also created fertile land for farming.
In the middle of India is the Deccan Plateau, which might have helped protect the Indus people from foriegn invaders. The Himalayas are also located near the Indus Valley, as is the Hindu Kush mountain range.
Decline, collapse and legacy
Around 1900 BC, signs of a gradual decline begin to emerge. People started to leave the cities. Those who remained were poorly nourished. By around 1800 BC, most of the cities were abandoned.
In the aftermath of the Indus civilization's collapse, regional cultures emerged, to varying degrees showing the influence of the Indus civilization. In the formerly great city of Harappa, burials have been found that correspond to a regional culture called the Cemetery H culture. At the same time, the Ochre Coloured Pottery culture expands from Rajasthan into the Gangetic Plain.
It is in this context of the aftermath of a civilization's collapse that the Indo-Aryan migration into northern India is discussed. In the early twentieth century, this migration was forwarded in the guise of an "Aryan invasion", and when the civilization was discovered in the 1920s, its collapse at precisely the time of the conjectured invasion was seen as an independent confirmation. In the words of the archaeologist Mortimer Wheeler, the Indo-Aryan war god Indra "stands accused" of the destruction. It is however far from certain whether the collapse of the IVC is a result of an Indo-Aryan migration. It seems rather likely that, to the contrary, the Indo-Aryan migration was as a result of the collapse, comparable with the decline of the Roman Empire and the incursions of relatively primitive peoples during the Migrations Period.
A possible natural reason of the IVC's decline is connected with climate change. In 2600 BC, the Indus Valley was verdant, forested, and teeming with wildlife. It was wetter, too; floods were a problem and appear, on more than one occasion, to have overwhelmed certain settlements. As a result, Indus civilization people supplemented their diet with hunting. By 1800 BC, the climate is known to have changed. It became significantly cooler and drier.
The crucial factor may have been the disappearance of substantial portions of the Ghaggar-Hakra river system. A tectonic event may have diverted the system's sources toward the Ganges Plain, though there is some uncertainty about the date of this event. Such a statement may seem dubious if one does not realize that the transition between the Indus and Gangetic plains amounts to a matter of inches. The region in which the river's waters formerly arose is known to be geologically active, and there is evidence of major tectonic events at the time the Indus civilization collapsed. Although this particular factor is speculative, and not generally accepted, the decline of the IVC, as with any other civilization, will have been due to a combination of a variety of reasons.
In the course of the 2nd millennium BC, remnants of the IVC's culture will have amalgamated with that of other peoples, likely contributing to what eventually resulted in the rise of historical Hinduism. Judging from the abundant figurines depicting female fertility that they left behind, indicate worship of a Mother goddess (compare Shakti and Kali). IVC seals depict animals, perhaps as the object of veneration, comparable to the zoomorphic aspects of some Hindu gods. Seals resembling Pashupati in a yogic posture have also been discovered. Like Hindus today, Indus civilization people seemed to have placed a high value on bathing and personal cleanliness. The houses of Mohenjo-Daro usually had a private well and bathing platforms were often near the well (Kenoyer 1998: 58-60).
Unlike other ancient civilizations, the archaeological record of the Indus civilization provides practically no evidence of armies, kings, slaves, social conflict, prisons, and other oft-negative traits that we traditionally associate with early civilization, although this could simply be due to the sheer completeness of its collapse and subsequent disappearance.
See also
- Gandhara culture, a later Buddhist culture also situated on the Indus
- Meluhha
- Dilmun, contemporary civilization based in present day Bahrain with extensive trade links with the Indus Valley Civilization
Bibliography
- Gupta, S.P. (ed.). 1995. The lost Sarasvati and the Indus Civilization. Kusumanjali Prakashan, Jodhpur.
- Kenoyer, J. Mark. 1998. Ancient cities of the Indus Valley Civilization. Oxford University Press. ISBN 0195779401
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- Jim G. Shaffer. 1992. "The Indus Valley, Baluchistan and Helmand Traditions: Neolithic Through Bronze Age." In Chronologies in Old World Archaeology. Second Edition. R.W. Ehrich, (Ed.). Chicago: University of Chicago Press. I:441-464, II:425-446.
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External links and references
- [http://www.safarmer.com/fsw2.pdf Paper published in 2004 by Farmer, Sproat, and Witzel questioning the traditional Indus-script thesis ]
- [http://pubweb.cc.u-tokai.ac.jp/indus/english/index.html An invitation to the Indus Civilization (Tokyo Metropolitan Museum)]
- [http://www.harappa.com/ Photos and descriptions of archaeological excavations — 90 page intro to Indus] (harappa.com)
- [http://www.archaeologyonline.net/artifacts/harappa-mohenjodaro.html The Harappan Civilization]
- [http://news.bbc.co.uk/1/hi/health/1272010.stm Prehistoric dentistry evidence found in Indus - BBC News]
- [http://www.hindunet.org/hindu_history/sarasvati/html/artefacts.htm Indus artefacts]
- [http://www.upenn.edu/researchatpenn/article.php?674&soc Cache of Seal Impressions Discovered in Western India ]
- [http://www.ucl.ac.uk/~ucgadkw/members/indus.html Sarasvati-Sindhu civilization]
- [http://lah.ru/fotoarh/megalit/asia/india.htm image collection]
Category:Ancient peoples
Category:Bronze Age
Category:History of India
Category:History of Pakistan
Category:Civilizations
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Vedas
The Vedas (Sanskrit:- वेद), collectively refers to a corpus of ancient Indo Aryan religious literature that are considered by adherents of Hinduism to be revealed knowledge. The word Veda means Knowledge and is cognate with the word "wit" in English (as well as "vision" through Latin). Many Hindus believe the Vedas existed since the beginning of creation. The texts of the Vedas have several references to specific patterns in the ancient flows of the Ganges River, which coincide with the sites of its ancient (but now dried) tributaries.
The newest parts of the Vedas are estimated to date back to around 500 BCE. The oldest text (RigVeda) found is now dated to around 1,500 BCE, but most Indologists agree that a long oral tradition possibly existed before it was written down. They represent the oldest stratum of Indian Literature and according to modern scholars are written in forms of a language which evolved into Sanskrit. They consider the use of Vedic Sanskrit for the language of the texts an anachronism, although it is generally accepted.
Contents
The Vedas consist of several kinds of texts, all of which date back to early times. The core is formed by the Mantras which represent hymns, prayers, incantations, magic and ritual formulas, charms etc. The hymns and prayers are addressed to a pantheon of gods (and a few goddesses), important members of which are Shiva, Varuna, Indra, Agni, etc. The mantras are supplemented by texts regarding the sacrificial rituals in which these mantras are used as well as texts exploring the philosophical aspects of the ritual tradition, narratives etc.
Organization
The Mantras are collected into anthologies called Samhitas. There are four Samhitas, the Rk (= Poetry), Sāman (=Song), Yajus(=Prayer) and Atharvan (=A kind of priest) commonly referred to as the Rig Veda, Yajur Veda, Sama Veda and Atharva Veda. Each Samhita is preserved in a number of versions, the differences among them being minor, except in the case of the Yajur Veda, where there are the "Black" (krishna) and "White" (shukla) versions, with the Black also containing explanatory material apart from the Mantras. The Rig Veda contains the oldest part of the corpus, and consists of 1028 hymns. The Sama Veda is mostly a rearrangement of the Rig Veda for musical rendering. The Yajur Veda gives sacrificial prayers and the Atharva Veda gives charms, incantations, magic formulas etc. Apart from these there are some stray secular material, legends, etc.
The next category of texts are the Brahmanas. These are ritual texts that describe in detail the sacrifices in which the Mantras were to be used, as well as commenting on the meaning of the sacrificial ritual. The Brahmanas are associated with one of the Samhitas. The Brahmanas may either form separate texts, or in the case of the Black Yajur Veda, can be partly integrated into the text of the Samhita. The most important of the Brahmanas is the Shatapatha Brahmana of the White Yajur Veda.
The Aranyakas and Upanishads are theological and philosophical works. They often form part of the Brahmanas (e.g. the Brhadaranyaka Upanishad). They are the basis of the Vedanta school of Darsana.
Position and compilation
Hindu tradition regards the Vedas as uncreated, eternal and being revealed to sages (Rishis). The hymns of the Rig-Veda Samhita are believed to have been collected and arranged by Paila under the supervision of Vyasa. Others were chanted during religious and social ceremonies and were compiled by Vaishampayana under the title Yajus mantra Samhita (see Yajur-Veda). Jaimini is said to have collected hymns that were set to music and melody — 'Saman' (see Sama-Veda). The fourth collection of hymns and chants known as Atharva Samhita.
Philosophies and sects that developed in the Indian subcontinent have taken differing positions on the Vedas. In Buddhism and Jainism, the authority of the Veda is repudiated, and both evolved into separate religions. The sects which did not explicitly reject the Vedas remained followers of the Sanatana Dharma, which is known in modern times as Hinduism.
In later Hinduism, the Vedas hold an exalted position. They are regarded as Shruti, i.e. Revelation, and the Brahminical caste based on the Vedas forms an important part of Hindu religious life to this day. Vedanta, Yoga, Tantra and even Bhakti acknowledge the Vedas as revelation.
Study
In the dharmashastras the study of the Vedas was regarded as a religious duty of the three upper varnas (Brahmins, Kshatriyas and Vaishyas). Women and Shudras were neither required nor allowed to study the Veda (this came to happen only in the very Later Vedic or the Sutra Age, because numerous evidences suggest that all humans were equally allowed to study the Vedas, and many Vedic "authors" were women). Elaborate methods for preserving the text (by learning them by heart and not by writing), subsidiary disciplines (Vedanga), exegetical literature, etc., were developed in the Vedic schools. In the fourteenth century Sayana wrote famous commentaries on the Vedic texts.
In modern times, Vedic studies are crucial in the understanding of Indo-European linguistics, as well as ancient Indian history.
It may be interesting to note that Hinduism encourages the Vedic mantras to be interpreted as liberally and as philosophically as possible unlike the Abrahamic religions (concerning the Tanakh, the Bible and the Koran). Infact, too literal interpretation of the mantras is actually discouraged, and even the three layers of commentaries (Brahmanas, Aranyakas and Upanishads), which form an intergral part of the shruti literature, actually interpret the seemingly polytheistic, ritualistic and highly complex Samhitas in a philosophical and metaphorical way to explain the "hidden" concepts of God (Ishwara), the Supreme Being (Brahman) and the soul or the self (Atman). Also, many Hindus believe that the very sound of the Vedic mantras is purifying for the environment and human mind.
The religion of the Vedic period, particularly at its earliest, was distinct in a number of respects, including reference to females in positions of religious authority (female rishis, or sages), an apparent lack of belief in reincarnation, and a markedly different pantheon, with Indra generally the chief god, and little mention of the later primary gods Vishnu and Shiva, although Brahma does appear quite frequently.
While Hinduism is generally monistic or monotheistic admitting emanating deities, the early Rig Veda (undeveloped early Hinduism) was what Max Müller based his views of henotheism on. In the four Vedas, Müller believed, a striving towards One was being aimed at by the worship of different cosmic principles, such as Agni (fire), Vayu (wind), Indra (rain, thunder, the sky; also the King of gods), etc. each of which was variously, by clearly different writers, hailed as supreme in different sections of the books. Indeed, however, what was confusing was an early idea of Rita, or supreme order, that bound all the gods. Other phrases such as ekam sat, vipraha bahudha vadanti (Truth is One though the sages know it as many) lead to understandings that the Vedic people admitted of fundamental oneness. There were attempts at monism by subordinating other gods to singular entities or gods of supreme power, three most notably being Vishvakarma, Indra and Varuna, though Indra was the most eulogized as supreme in his 200 Rig Vedic verses. From this mix of monism, monotheism and naturalist polytheism Max Müller decided to name the early Vedic religion henotheistic. He decided that while polytheism did not fit with views so clearly admitting of fundamental unity, monism in his opinion was not yet fully developed.
This, however, is clearly a one-man view. Extremely advanced, indeed unprecedented and thitherto unduplicated ideas of pure monism are to be found in the early Vedas, notwithstanding clearly monist and monotheist movements of Hinduism that developed with the advent of the Upanishads. One such example of early Vedic monotheism is the Nasadiya hymn of the Rig Veda: "That One breathed by itself without breath, other than it there has been nothing." To collectively term the Vedas henotheistic, and thus further leaning towards polytheism, rather than monotheism, is to ignore the bent of the Vedas that laid the foundation for the Upanishads as early as 1000 BCE.
Cosmogony
The Vedic view of the world and cosmogony sees one true divine principle self-projecting as the divine word, Vaak, 'birthing' the cosmos that we know from 'Hiranyagarbha' or Golden Womb, a primordial sun figure that is equivalent to Surya. The varied gods like Vayu, Indra, Rudra (the Destroyer), Agni (Fire, the sacrifical medium) and the goddess Saraswati (the Divine Word, aka Vaak) are just some examples of the myriad aspects of the one underlying nature of the universe.
See also
- Pandit
- Vedic chant
External links
- [http://www.vedmandir.com A great source of information]
- [http://www.stephen-knapp.com/complete_review_of_vedic_literature.htm A Complete Review of Vedic Literature]
- [http://www.comparative-religion.com/hinduism/vedas/ Vedas: Rig, Sama, Yajur, and Artharva]
- [http://www.vedah.com/org/index.asp Excellent site about Vedas (Aurobindo)]
- [http://san.beck.org/EC7-Vedas.html Veda and Upanishads]
- [http://veda.harekrsna.cz/ VEDA - Vedas and Vedic Knowledge Online (Vaishnava and general)]
- [http://sarasvati.tripod.com/veda.htm Vedic Chanting .mp3 audio files]
- [http://www.india-picture.net/vedic_ritual Photos of the performance of Vedic rituals in India]
References
- Winternitz, Moritz : History of Indian Literature, Vol.1 (Calcutta 1926)l.
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Mathematics
Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions.
Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries.
The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.
History
:Main article: History of mathematics
The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought.
In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry.
Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse.
Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change.
Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both.
Mathematical discoveries have been made throughout history and continue to be made today.
Inspiration, pure and applied mathematics, and aesthetics
Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts.
As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science.
Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.
Notation, language, and rigor
Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales.
The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary.
Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis).
Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.
Is mathematics a science?
Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)."
If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm]
In any case, mathematics shares much in common with many fields in the physical sciences, notably
the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.
Overview of fields of mathematics
As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics).
The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory.
The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space.
The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics.
Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.
Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science.
An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.
Major themes in mathematics
An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.
Quantity
This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements.
:
:Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base
Structure
:Pinning down ideas of size, symmetry, and mathematical structure.
:
:Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory
Space
:A more visual approach to mathematics.
:
:Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry
Change
:Ways to express and handle change in mathematical functions, and changes between numbers.
:
:Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions
Foundations and methods
:Approaches to understanding the nature of mathematics.
:philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols
Discrete mathematics
:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values.
:
:Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory
Applied mathematics
:Applied mathematics uses the full knowledge of mathematics to solve real-world problems.
:Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory
Important theorems
:These theorems have interested mathematicians and non-mathematicians alike.
:See list of theorems for more
:Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.
Important conjectures
See list of conjectures for more
:These are some of the major unsolved problems in mathematics.
:Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.
History and the world of mathematicians
See also list of mathematics history topics
:History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues
Mathematics and other fields
:Mathematics and architecture – Mathematics and education – Mathematics of musical scales
Common misconceptions
Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems.
Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature.
The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne.
Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter.
Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions.
Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.
See also
- Mathematical game
- Mathematical problem
- Mathematical puzzle
- Puzzle
Bibliography
- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).
External links
- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
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Negative numbersA negative number is a number that is less than zero, such as −3. A positive number is a number that is greater than zero, such as 3. Zero itself is neither negative nor positive, though in computing zero is sometimes treated as though it were a positive number. The non-negative numbers are the real numbers that are not negative (positive or zero). The non-positive numbers are the real numbers that are not positive (negative or zero).
In the context of complex numbers positive implies real, but for clarity one may say "positive real number".
Negative numbers
Negative integers can be regarded as an extension of the natural numbers, such that the equation x − y = z has a meaningful solution for all values of x and y. The other sets of numbers are then derived as progressively more elaborate extensions and generalizations from the integers.
Negative numbers are useful to describe values on a scale that goes below zero, such as temperature, and also in bookkeeping where they can be used to represent debts. In bookkeeping, debts are often represented by red numbers, or a number in parentheses.
Non-negative numbers
A number is nonnegative if and only if it is greater than or equal to zero, i.e. positive or zero. Thus the nonnegative integers are all the integers from zero on upwards, and the nonnegative reals are all the real numbers from zero on upwards.
A real matrix A is called nonnegative if every entry of A is nonnegative.
A real matrix A is called totally nonnegative by matrix theorists or totally positive by computer scientists if the determinant of every square submatrix of A is nonnegative.
Sign function
It is possible to define a function sgn(x) on the real numbers which is 1 for positive numbers, −1 for negative numbers and 0 for zero (sometimes called the signum function):
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