Maasai people Influences from the outside world What kind of influences from the outside world is the article pertaining to? Maintaining a traditional pastoral lifestyle has become increasingly difficult due to outside influences of the modern world.
Has this been a positive thing for them or negative?
Has maintaining a traditional pastoral lifestyle been a positive thing for the Maasai or negative?
Maintaining a traditional pastoral lifestyle has become increasingly difficult due to outside influences of the modern world. Garrett Hardin's article, outlining the "tragedy of the commons", as well as Melville Herskovits' "cattle complex" helped to influence ecologists and policy makers about the harm Maasai pastoralists were causing to savannah rangelands. This concept was later proven false by anthropologists but is still deeply ingrained in the minds of ecologists and Tanzanian officials. This influenced British colonial policy makers in 1951 to remove all Maasai from the Serengeti National Park and relegate them to areas in and around the Ngorongoro Conservation Area (NCA). The plan for the NCA was to put Maasai interests above all else, but this promise was never met.The spread of HIV was rampant. Due to an increase in Maasai population, loss of cattle populations to disease, and lack of available rangelands because of new park boundaries and the incursion of settlements and farms by other tribes (this is also the chief reason for the decline in wildlife-habitat loss, with the second being poaching), the Maasai were forced to develop new ways of sustaining themselves. Many Maasai began to cultivate maize and other crops to get by, a practice that was culturally viewed negatively. Cultivation was first introduced to the Maasai by displaced WaArusha and WaMeru women who were married to Maasai men; subsequent generations practiced a mixed livelihood. To further complicate their situation, in 1975 the Ngorongoro Conservation Area banned cultivation practices. In order to survive they are forced to participate in Tanzania's monetary economy. They have to sell their animals and traditional medicines in order to buy food. The ban on cultivation was lifted in 1992 and cultivation has again become an important part of Maasai livelihood. Park boundaries and land privatisation has continued to limit grazing area for the Maasai and have forced them to change considerably. Over the years, many projects have begun to help Maasai tribal leaders find ways to preserve their traditions while also balancing the education needs of their children for the modern world. The emerging forms of employment among the Maasai people include farming, business (selling of traditional medicine, running of restaurants/shops, buying and selling of minerals, selling milk and milk products by women, embroideries), and wage employment (as security guards/ watchmen, waiters, tourist guides), and others who are engaged in the public and private sectors. Many Maasai have moved away from the nomadic life to positions in commerce and government. Yet despite the sophisticated urban lifestyle they may lead, many will happily head homewards dressed in designer clothes, only to emerge from the traditional family homestead wearing a shuka (colourful piece of cloth), cow hide sandals and carrying a wooden club (o-rinka) - at ease with themselves. CANNOTANSWER
The plan for the NCA was to put Maasai interests above all else, but this promise was never met.The spread of HIV was rampant.
The Maasai () are a Nilotic ethnic group inhabiting northern, central and southern Kenya and northern Tanzania. They are among the best known local populations internationally due to their residence near the many game parks of the African Great Lakes, and their distinctive customs and dress. The Maasai speak the Maa language (ɔl Maa), a member of the Nilotic language family that is related to the Dinka, Kalenjin and Nuer languages. Except for some elders living in rural areas, most Maasai people speak the official languages of Kenya and Tanzania, Swahili and English. The Maasai population has been reported as numbering 1,189,522 in Kenya in the 2019 census, compared to 377,089 in the 1989 census. Many Maasai tribes throughout Tanzania and Kenya welcome visits to their villages to experience their culture, traditions, and lifestyle, in return for a fee.

History
The Maasai inhabit the African Great Lakes region and arrived via the South Sudan. Most Nilotic speakers in the area, including the Maasai, the Turkana and the Kalenjin, are pastoralists, and are famous for their fearsome reputations as warriors and cattle-rustlers. The Maasai and other groups in East Africa have adopted customs and practices from neighboring Cushitic-speaking groups, including the age set system of social organization, circumcision, and vocabulary terms.

Origin, migration and assimilation

Many ethnic groups that had already formed settlements in the region were forcibly displaced by the incoming Maasai, while other, mainly Southern Cushitic groups, were assimilated into Maasai society. The Nilotic ancestors of the Kalenjin likewise absorbed some early Cushitic populations.

Settlement in East Africa
The Maasai territory reached its largest size in the mid-19th century, and covered almost all of the Great Rift Valley and adjacent lands from Mount Marsabit in the north to Dodoma in the south. At this time the Maasai, as well as the larger Nilotic group they were part of, raised cattle as far east as the Tanga coast in Tanganyika (now mainland Tanzania). Raiders used spears and shields, but were most feared for throwing clubs (orinka) which could be accurately thrown from up to 70 paces (approx. 100 metres). In 1852, there was a report of a concentration of 800 Maasai warriors on the move in what is now Kenya. In 1857, after having depopulated the "Wakuafi wilderness" in what is now southeastern Kenya, Maasai warriors threatened Mombasa on the Kenyan coast.

Because of this migration, the Maasai are the southernmost Nilotic speakers. The period of expansion was followed by the Maasai "Emutai" of 1883–1902. This period was marked by epidemics of contagious bovine pleuropneumonia, rinderpest (see 1890s African rinderpest epizootic), and smallpox. The estimate first put forward by a German lieutenant in what was then northwest Tanganyika, was that 90% of cattle and half of wild animals perished from rinderpest. German doctors in the same area claimed that "every second" African had a pock-marked face as the result of smallpox. This period coincided with drought. Rains failed completely in 1897 and 1898.

The Austrian explorer Oscar Baumann travelled in Maasai lands between 1891 and 1893, and described the old Maasai settlement in the Ngorongoro Crater in the 1894 book Durch Massailand zur Nilquelle ("Through the lands of the Maasai to the source of the Nile"). By one estimate two-thirds of the Maasai died during this period.

 Maasai in Tanganyika (now mainland Tanzania) were displaced from the fertile lands between Mount Meru and Mount Kilimanjaro, and most of the fertile highlands near Ngorongoro in the 1940s. More land was taken to create wildlife reserves and national parks: Amboseli National Park, Nairobi National Park, Maasai Mara, Samburu National Reserve, Lake Nakuru National Park and Tsavo in Kenya; and Lake Manyara, Ngorongoro Conservation Area, Tarangire and Serengeti National Park in what is now Tanzania.

Maasai are pastoralist and have resisted the urging of the Tanzanian and Kenyan governments to adopt a more sedentary lifestyle. They have demanded grazing rights to many of the national parks in both countries.

The Maasai people stood against slavery and never condoned traffic of human beings; and outsiders looking for people to enslave avoided the Maasai.

Essentially there are twenty-two geographic sectors or sub tribes of the Maasai community, each one having its own customs, appearance, leadership and dialects. These subdivisions are known as 'nations' or 'iloshon' in the Maa language: the Keekonyokie, Damat, Purko, Wuasinkishu, Siria, Laitayiok, Loitai, Kisonko, Matapato, Dalalekutuk, Loodokolani, Kaputiei, Moitanik, Ilkirasha, Samburu, Lchamus, Laikipia, Loitokitoki, Larusa, Salei, Sirinket and Parakuyo.

Genetics
Recent advances in genetic analyses have helped shed some light on the ethnogenesis of the Maasai people. Genetic genealogy, a tool that uses the genes of modern populations to trace their ethnic and geographic origins, has also helped clarify the possible background of the modern Maasai.

Autosomal DNA
The Maasai's autosomal DNA has been examined in a comprehensive study by Tishkoff et al. (2009) on the genetic affiliations of various populations in Africa. According to the study's authors, the Maasai "have maintained their culture in the face of extensive genetic introgression". Tishkoff et al. also indicate that: "Many Nilo-Saharan-speaking populations in East Africa, such as the Maasai, show multiple cluster assignments from the Nilo-Saharan [...] and Cushitic [...] AACs, in accord with linguistic evidence of repeated Nilotic assimilation of Cushites over the past 3000 years and with the high frequency of a shared East African–specific mutation associated with lactose tolerance."

Y-DNA
A Y chromosome study by Wood et al. (2005) tested various Sub-Saharan populations, including 26 Maasai males from Kenya, for paternal lineages. The authors observed haplogroup E1b1b-M35 (not M78) in 35% of the studied Maasai. E1b1b-M35-M78 in 15%, their ancestor with the more northerly Cushitic males, who possess the haplogroup at high frequencies lived more than 13 000 years ago.  The second most frequent paternal lineage among the Maasai was Haplogroup A3b2, which is commonly found in Nilotic populations, such as the Alur; it was observed in 27% of Maasai males. The third most frequently observed paternal DNA marker in the Maasai was E1b1a1-M2 (E-P1), which is very common in the Sub-Saharan region; it was found in 12% of the Maasai samples. Haplogroup B-M60 was also observed in 8% of the studied Maasai, which is also found in 30% (16/53) of Southern Sudanese Nilotes.

Mitochondrial DNA
According to an mtDNA study by Castri et al. (2008), which tested Maasai individuals in Kenya, the maternal lineages found among the Maasai are quite diverse, but similar in overall frequency to that observed in other Nilo-Hamitic populations from the region, such as the Samburu. Most of the tested Maasai belonged to various macro-haplogroup L sub-clades, including L0, L2, L3, L4 and L5. Some maternal gene flow from North and Northeast Africa was also reported, particularly via the presence of mtDNA haplogroup M lineages in about 12.5% of the Maasai samples.

Culture

The monotheistic Maasai worship a single deity called Enkai or Engai. Engai has a dual nature: Engai Narok (Black God) is benevolent, and Engai Na-nyokie (Red God) is vengeful. There are also two pillars or totems of Maasai society: Oodo Mongi, the Red Cow and Orok Kiteng, the Black Cow with a subdivision of five clans or family trees. The Maasai also have a totemic animal, which is the lion; however, the animal can be killed. The way the Maasai kill the lion differs from trophy hunting as it is used in the rite of passage ceremony. The "Mountain of God", Ol Doinyo Lengai, is located in northernmost Tanzania and can be seen from Lake Natron in southernmost Kenya. The central human figure in the Maasai religious system is the  whose roles include shamanistic healing, divination and prophecy, and ensuring success in war or adequate rainfall. Today, they have a political role as well due to the elevation of leaders. Whatever power an individual laibon had was a function of personality rather than position. Many Maasai have also adopted Christianity and Islam. The Maasai are known for their intricate jewelry and for decades, have sold these items to tourists as a business. 

Educating Maasai women to use clinics and hospitals during pregnancy has enabled more infants to survive. The exception is found in extremely remote areas.  A corpse rejected by scavengers is seen as having something wrong with it, and liable to cause social disgrace; therefore, it is not uncommon for bodies to be covered in fat and blood from a slaughtered ox.<ref name="attitudes">Cultural and Public Attitudes: Improving the Relationship between Humans and Hyaenas from Mills, M.g.L. and Hofer, H. (compilers). (1998) Hyaenas: Status Survey and Conservation Action Plan. IUCN/SSC Hyaena Specialist Group. IUCN, Gland, Switzerland and Cambridge, UK. vi + 154 pp.</ref>

Traditional Maasai lifestyle centres around their cattle, which constitute their primary source of food. A man's wealth is measured in cattle and children. A herd of 50 cattle is respectable, and the more children the better. A man who has plenty of one but not the other is considered to be poor.Africa's Great Rift Valley. Nigel Pavitt. 2001. pages 138. Harry N. Abrams, Incorporated, New York 

All of the Maasai's needs for food are met by their cattle. They eat the meat, drink the milk daily, and drink the blood on occasion. Bulls, goats, and lambs are slaughtered for meat on special occasions and for ceremonies. Though the Maasai's entire way of life has historically depended on their cattle, more recently with their cattle dwindling, the Maasai have grown dependent on food such as sorghum, rice, potatoes and cabbage (known to the Maasai as goat leaves).

One common misconception about the Maasai is that each young man is supposed to kill a lion before he can be circumcised and enter adulthood. Lion hunting was an activity of the past, but it has been banned in East Africa – yet lions are still hunted when they maul Maasai livestock. Nevertheless, killing a lion gives one great value and celebrity status in the community.

Spencer, P. (1988) The Maasai of Matapato: a study of rituals of rebellion Manchester University Press, Manchester. Spencer, P. (2003) Time, Space, and the Unknown: Maasai configurations of power and providence. Routledge, London.

Body modification

The piercing and stretching of earlobes is common among the Maasai as with other tribes, and both men and women wear metal hoops on their stretched earlobes. Various materials have been used to both pierce and stretch the lobes, including thorns for piercing, twigs, bundles of twigs, stones, the cross section of elephant tusks and empty film canisters. Women wear various forms of beaded ornaments in both the ear lobe, and smaller piercings at the top of the ear. Amongst Maasai males, circumcision is practiced as a ritual of transition from boyhood to manhood. Women are also circumcised (as described below in social organization).

This belief and practice is not unique to the Maasai. In rural Kenya a group of 95 children aged between six months and two years were examined in 1991/92. 87% were found to have undergone the removal of one or more deciduous canine tooth buds. In an older age group (3–7 years of age), 72% of the 111 children examined exhibited missing mandibular or maxillary deciduous canines.

Genital Modification

Traditionally, the Maasai conduct elaborate rite of passage rituals which include surgical genital modification in order to initiate children into adulthood. The Maa word for circumcision, "emorata," is applied to this ritual for both males and females. This ritual is typically performed by the elders, who use a sharpened knife and makeshift cattle hide bandages for the procedure.  

The male ceremony refers to the excision of the prepuce (foreskin). In the male ceremony the boy is expected to endure the operation in silence. Expressions of pain bring dishonor upon him; albeit only temporarily. Importantly, any exclamations or unexpected movements on the part of the boy can cause the elder to make a mistake in the delicate and tedious process; which can result in severe lifelong scarring, dysfunction, and pain.  

Young women also undergo excision ("female circumcision", "female genital mutilation," "emorata") as part of an elaborate rite of passage ritual called "Emuatare," the ceremony that initiates young Maasai girls into adulthood through ritual circumcision and then into early arranged marriages. The Maasai believe that female circumcision is necessary and Maasai men may reject any woman who has not undergone it as either not marriageable or worthy of a much-reduced bride price. In Eastern Africa, uncircumcised women, even those highly educated members of parliament like Linah Kilimo, can be accused of not being mature enough to be taken seriously. To others the practice of female circumcision is known as female genital mutilation, and draws a great deal of criticism from both abroad and many women who have undergone it, such as Maasai activist Agnes Pareyio. The female rite of passage ritual has recently seen excision replaced in rare instances with a "cutting with words" ceremony involving singing and dancing in its place. However, despite changes to the law and education drives the practice remains deeply ingrained, highly valued, and near universally practiced by members of the culture. The Last of the Maasai. Mohamed Amin, Duncan Willetts, John Eames. 1987. Pages 168–173. Camerapix Publishers International.  

Graduation from warrior to junior elder takes place at a large gathering known as Eunoto. The long hair of the former warriors is shaved off; elders must wear their hair short. Warriors who do not have sexual relations with women who have not undergone the "Emuatare" ceremony are specially honored at the Eunoto gathering. 

Hair

Broken Spears - a Maasai Journey. Elizabeth L. Gilbert. 2003. Atlantic Monthly Press. page 136. 

The Last of the Maasai. Mohamed Amin, Duncan Willetts, John Eames. 1987. Page 55. Camerapix Publishers International.  This would symbolize the healing of the woman.

Two days before boys are circumcised, their heads are shaved.Maasai. Tepilit Ole Saitoti with photos by Carol Beckwith. Harry N. Abrams, Inc. 1980. pages 126, 129.  When warriors go through the Eunoto'', and become elders, their long plaited hair is shaved off.

Music and dance

Maasai music traditionally consists of rhythms provided by a chorus of vocalists singing harmonies while a song leader, or olaranyani, sings the melody. Unlike most other African tribes, Maasai widely use drone polyphony.

Women chant lullabies, humming songs, and songs praising their sons. Nambas, the call-and-response pattern, repetition of nonsensical phrases, monophonic melodies, repeated phrases following each verse being sung on a descending scale, and singers responding to their own verses are characteristic of singing by females. When many Maasai women gather together, they sing and dance among themselves.

Eunoto, the coming of age ceremony of the warrior, can involve ten or more days of singing, dancing and ritual. The warriors of the Il-Oodokilani perform a kind of march-past as well as the adumu, or aigus, sometimes referred as "the jumping dance" by non-Maasai. (Both adumu and aigus are Maa verbs meaning "to jump" with adumu meaning "To jump up and down in a dance".)

Diet

Traditionally, the Maasai diet consisted of raw meat, raw milk, honey and raw blood from cattle—note that the Maasai cattle are of the Zebu variety.

Most of the milk is consumed as fermented milk or buttermilk (a by-product of butter making). Milk consumption figures are very high by any standards.

The Maasai herd goats and sheep, including the Red Maasai sheep, as well as the more prized cattle.

Although consumed as snacks, fruits constitute a major part of the food ingested by children and women looking after cattle as well as morans in the wilderness.

Shelter

Clothing

Clothing changes by age and location. Young men, for instance, wear black for several months following their circumcision. However, red is a favored colour. Blue, black, striped, and checkered cloth are also worn, as are multicolored African designs. The Maasai began to replace animal skin, calf hides and sheep skin with commercial cotton cloth in the 1960s.

Shúkà is the Maa word for sheets traditionally worn wrapped around the body. These are typically red, though with some other colors (e.g. blue) and patterns (e.g. plaid). Pink, even with flowers, is not shunned by warriors. One piece garments known as kanga, a Swahili term, are common. Maasai near the coast may wear kikoi, a type of sarong that comes in many different colors and textiles. However, the preferred style is stripes.

Influences from the outside world

A traditional pastoral lifestyle has become increasingly difficult due to outside influences of the modern world. Garrett Hardin's article, outlining the "tragedy of the commons", as well as Melville Herskovits' "cattle complex" helped to influence ecologists and policy makers about the harm Maasai pastoralists were causing to savannah rangelands. This concept was later proven false by anthropologists but is still deeply ingrained in the minds of ecologists and Tanzanian officials. This influenced British colonial policy makers in 1951 to remove all Maasai from the Serengeti National Park and relegate them to areas in and around the Ngorongoro Conservation Area (NCA). The plan for the NCA was to put Maasai interests above all else, but this promise was never met. The spread of HIV was rampant.

Due to an increase in Maasai population, loss of cattle populations to disease, and lack of available rangelands because of new park boundaries and the incursion of settlements and farms by other tribes (this is also the chief reason for the decline in wildlife-habitat loss, with the second being poaching), the Maasai were forced to develop new ways of sustaining themselves. Many Maasai began to cultivate maize and other crops to get by, a practice that was culturally viewed negatively. Cultivation was first introduced to the Maasai by displaced WaArusha and WaMeru women who were married to Maasai men; subsequent generations practiced a mixed livelihood. To further complicate their situation, in 1975 the Ngorongoro Conservation Area banned cultivation practices. In order to survive they are forced to participate in Tanzania's monetary economy. They have to sell their animals and traditional medicines in order to buy food. The ban on cultivation was lifted in 1992 and cultivation has again become an important part of Maasai livelihood. Park boundaries and land privatisation has continued to limit grazing area for the Maasai and have forced them to change considerably.

Over the years, many projects have begun to help Maasai tribal leaders find ways to preserve their traditions while also balancing the education needs of their children for the modern world.

The emerging forms of employment among the Maasai people include farming, business (selling of traditional medicine, running of restaurants/shops, buying and selling of minerals, selling milk and milk products by women, embroideries), and wage employment (as security guards/ watchmen, waiters, tourist guides), and others who are engaged in the public and private sectors.

Many Maasai have moved away from the nomadic life to positions in commerce and government.

Notable Maasai

 Joseph Ole Lenku – Cabinet Secretary of Kenya for Interior and Coordination of National Government from 2012 to 2014
 David Rudisha – Middle-distance runner and 800-meter world record holder
 Edward Sokoine – Prime Minister of Tanzania from 1977 to 1980 and again from 1983 to 1984
 Edward Lowassa – Prime Minister of Tanzania from 2005 to 2008. 2nd runner up to president John Pombe Magufuli in the 2015 Tanzania General Elections.
 Olekina Ledama – Founder, Maasai Education Discovery
 James Ole Kiyiapi – associate professor at Moi University and permanent secretary in the Ministries of Education and Local Government
 William Ole Ntimama – Kenyan politician and leader of the Maa community
 Francis Ole Kaparo – Former Speaker of the National Assembly of Kenya
 Joseph Nkaissery – Former Cabinet Secretary of Kenya for Interior and Coordination of National Government from 2014 to his death in 2017
 Nice Nailantei Lengete – First woman to address the Maasai elders council at Mount Kilimanjaro, and persuaded the council to ban female genital mutilation among the Maasai across Kenya and Tanzania
 Katoo Ole Metito – Member of Parliament for Kajiado South sub county

See also
 Maasai mythology
 Maasai language
 Maa Civil Society Forum

References

External links

 African People Ethnography | Maasai
 Maasai online dictionary
 Maasai Aid Association
 Working for a just and self-sustaining community for the Maasai People
 Maasai Trust
 The Maasai People - History and Culture
 Maasai people, Kenya at the Maasai Association
 Indiana University Art Museum Arts of Kenya online collection
 Maasai Mara Tribe Facts

 
 
Ethnic groups in Kenya
Indigenous peoples of East Africa
Nilotic peoples
Indigenous peoples of Arusha Region
Ethnic groups divided by international borders
Members of the Unrepresented Nations and Peoples Organization
Ethnic groups in Tanzania
In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset. If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic. For example, −(−3) = 3 because the opposite of an opposite is the original value.

Negative numbers are usually written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, and is pronounced "minus three" or "negative three". To help tell the difference between a subtraction operation and a negative number, occasionally the negative sign is placed slightly higher than the minus sign (as a superscript). Conversely, a number that is greater than zero is called positive; zero is usually (but not always) thought of as neither positive nor negative. The positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity or positivity of a number is referred to as its sign.

Every real number other than zero is either positive or negative. The non-negative whole numbers are referred to as natural numbers (i.e., 0, 1, 2, 3...), while the positive and negative whole numbers (together with zero) are referred to as integers. (Some definitions of the natural numbers exclude zero.)

In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers.

Negative numbers appeared for the first time in history in the Nine Chapters on the Mathematical Art, which in its present form dates from the period of the Chinese Han Dynasty (202 BC – AD 220), but may well contain much older material. Liu Hui (c. 3rd century) established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers. Islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Prior to the concept of negative numbers, mathematicians such as Diophantus considered negative solutions to problems "false" and equations requiring negative solutions were described as absurd. Western mathematicians like Leibniz (1646–1716) held that negative numbers were invalid, but still used them in calculations.

Introduction

The number line

The relationship between negative numbers, positive numbers, and zero is often expressed in the form of a number line:

Numbers appearing farther to the right on this line are greater, while numbers appearing farther to the left are less. Thus zero appears in the middle, with the positive numbers to the right and the negative numbers to the left.

Note that a negative number with greater magnitude is considered less. For example, even though (positive)  is greater than (positive) , written
 
negative  is considered to be less than negative :
 
(Because, for example, if you have £−8, a debt of £8, you would have less after adding, say £10, to it than if you have £−5.) 
It follows that any negative number is less than any positive number, so
   and

Signed numbers

In the context of negative numbers, a number that is greater than zero is referred to as positive. Thus every real number other than zero is either positive or negative, while zero itself is not considered to have a sign. Positive numbers are sometimes written with a plus sign in front, e.g.  denotes a positive three.

Because zero is neither positive nor negative, the term nonnegative is sometimes used to refer to a number that is either positive or zero, while nonpositive is used to refer to a number that is either negative or zero. Zero is a neutral number.

As the result of subtraction
Negative numbers can be thought of as resulting from the subtraction of a larger number from a smaller. For example, negative three is the result of subtracting three from zero:
 
In general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers. For example,
 
since .

Everyday uses of negative numbers

Sport

 Goal difference in association football and hockey; points difference in rugby football; net run rate in cricket; golf scores relative to par.
 Plus-minus differential in ice hockey: the difference in total goals scored for the team (+) and against the team (−) when a particular player is on the ice is the player's +/− rating. Players can have a negative (+/−) rating.
 Run differential in baseball: the run differential is negative if the team allows more runs than they scored.
 Clubs may be deducted points for breaches of the laws, and thus have a negative points total until they have earned at least that many points that season.
 Lap (or sector) times in Formula 1 may be given as the difference compared to a previous lap (or sector) (such as the previous record, or the lap just completed by a driver in front), and will be positive if slower and negative if faster.
 In some athletics events, such as sprint races, the hurdles, the triple jump and the long jump, the wind assistance is measured and recorded, and is positive for a tailwind and negative for a headwind.

Science
 Temperatures which are colder than 0 °C or 0 °F. 
 Latitudes south of the equator and longitudes west of the prime meridian.
 Topographical features of the earth's surface are given a height above sea level, which can be negative (e.g. the surface elevation of the Dead Sea or Death Valley, or the elevation of the Thames Tideway Tunnel).
 Electrical circuits. When a battery is connected in reverse polarity, the voltage applied is said to be the opposite of its rated voltage. For example, a 6-volt battery connected in reverse applies a voltage of −6 volts.
 Ions have a positive or negative electrical charge.
 Impedance of an AM broadcast tower used in multi-tower directional antenna arrays, which can be positive or negative.

Finance
 Financial statements can include negative balances, indicated either by a minus sign or by enclosing the balance in parentheses. Examples include bank account overdrafts and business losses (negative earnings).
 Refunds to a credit card or debit card are a negative charge to the card.
 The annual percentage growth in a country's GDP might be negative, which is one indicator of being in a recession.
 Occasionally, a rate of inflation may be negative (deflation), indicating a fall in average prices.
 The daily change in a share price or stock market index, such as the FTSE 100 or the Dow Jones.
 A negative number in financing is synonymous with "debt" and "deficit" which are also known as "being in the red".
 Interest rates can be negative, when the lender is charged to deposit their money.

Other

 The numbering of storeys in a building below the ground floor.
 When playing an audio file on a portable media player, such as an iPod, the screen display may show the time remaining as a negative number, which increases up to zero time remaining at the same rate as the time already played increases from zero.
 Television game shows:
 Participants on QI often finish with a negative points score.
 Teams on University Challenge have a negative score if their first answers are incorrect and interrupt the question. 
 Jeopardy! has a negative money score – contestants play for an amount of money and any incorrect answer that costs them more than what they have now can result in a negative score.
 In The Price Is Rights pricing game Buy or Sell, if an amount of money is lost that is more than the amount currently in the bank, it incurs a negative score.
 The change in support for a political party between elections, known as swing.
 A politician's approval rating.
 In video games, a negative number indicates loss of life, damage, a score penalty, or consumption of a resource, depending on the genre of the simulation.
 Employees with flexible working hours may have a negative balance on their timesheet if they have worked fewer total hours than contracted to that point. Employees may be able to take more than their annual holiday allowance in a year, and carry forward a negative balance to the next year.
 Transposing notes on an electronic keyboard are shown on the display with positive numbers for increases and negative numbers for decreases, e.g. "−1" for one semitone down.

Arithmetic involving negative numbers
The minus sign "−" signifies the operator for both the binary (two-operand) operation of subtraction (as in ) and the unary (one-operand) operation of negation (as in , or twice in ). A special case of unary negation occurs when it operates on a positive number, in which case the result is a negative number (as in ).

The ambiguity of the "−" symbol does not generally lead to ambiguity in arithmetical expressions, because the order of operations makes only one interpretation or the other possible for each "−". However, it can lead to confusion and be difficult for a person to understand an expression when operator symbols appear adjacent to one another. A solution can be to parenthesize the unary "−" along with its operand.

For example, the expression  may be clearer if written  (even though they mean exactly the same thing formally). The subtraction expression  is a different expression that doesn't represent the same operations, but it evaluates to the same result.

Sometimes in elementary schools a number may be prefixed by a superscript minus sign or plus sign to explicitly distinguish negative and positive numbers as in
  gives .

Addition

Addition of two negative numbers is very similar to addition of two positive numbers. For example,
.
The idea is that two debts can be combined into a single debt of greater magnitude.

When adding together a mixture of positive and negative numbers, one can think of the negative numbers as positive quantities being subtracted. For example:
  and .
In the first example, a credit of  is combined with a debt of , which yields a total credit of . If the negative number has greater magnitude, then the result is negative:
  and .
Here the credit is less than the debt, so the net result is a debt.

Subtraction
As discussed above, it is possible for the subtraction of two non-negative numbers to yield a negative answer:
 
In general, subtraction of a positive number yields the same result as the addition of a negative number of equal magnitude. Thus
 
and
 

On the other hand, subtracting a negative number yields the same result as the addition a positive number of equal magnitude. (The idea is that losing a debt is the same thing as gaining a credit.) Thus
 
and
 .

Multiplication
When multiplying numbers, the magnitude of the product is always just the product of the two magnitudes. The sign of the product is determined by the following rules:
 The product of one positive number and one negative number is negative.
 The product of two negative numbers is positive.
Thus
 
and
 .
The reason behind the first example is simple: adding three 's together yields :
 .
The reasoning behind the second example is more complicated. The idea again is that losing a debt is the same thing as gaining a credit. In this case, losing two debts of three each is the same as gaining a credit of six:
  debts  each credit.
The convention that a product of two negative numbers is positive is also necessary for multiplication to follow the distributive law.  In this case, we know that
 .
Since , the product  must equal .

These rules lead to another (equivalent) rule—the sign of any product a × b depends on the sign of a as follows:
 if a is positive, then the sign of a × b is the same as the sign of b, and
 if a is negative, then the sign of a × b is the opposite of the sign of b.
The justification for why the product of two negative numbers is a positive number can be observed in the analysis of complex numbers.

Division
The sign rules for division are the same as for multiplication.  For example,
,
,
and
.
If dividend and divisor have the same sign, the result is positive, if they have different signs the result is negative.

Negation

The negative version of a positive number is referred to as its negation. For example,  is the negation of the positive number .  The sum of a number and its negation is equal to zero:
.
That is, the negation of a positive number is the additive inverse of the number.

Using algebra, we may write this principle as an algebraic identity:
.
This identity holds for any positive number . It can be made to hold for all real numbers by extending the definition of negation to include zero and negative numbers.  Specifically:
 The negation of 0 is 0, and
 The negation of a negative number is the corresponding positive number.
For example, the negation of  is .  In general,
.

The absolute value of a number is the non-negative number with the same magnitude. For example, the absolute value of  and the absolute value of  are both equal to , and the absolute value of  is .

Formal construction of negative integers

In a similar manner to rational numbers, we can extend the natural numbers N to the integers Z by defining integers as an ordered pair of natural numbers (a, b). We can extend addition and multiplication to these pairs with the following rules:
(a, b) + (c, d) = (a + c, b + d)
(a, b) × (c, d) = (a × c + b × d, a × d + b × c)

We define an equivalence relation ~ upon these pairs with the following rule:
(a, b) ~ (c, d) if and only if a + d = b + c.
This equivalence relation is compatible with the addition and multiplication defined above, and we may define Z to be the quotient set N²/~, i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense. Note that Z, equipped with these operations of addition and multiplication, is a ring, and is in fact, the prototypical example of a ring.

We can also define a total order on Z''' by writing
(a, b) ≤ (c, d) if and only if a + d ≤ b + c.

This will lead to an additive zero of the form (a, a), an additive inverse of (a, b) of the form (b, a), a multiplicative unit of the form (a + 1, a), and a definition of subtraction
(a, b) − (c, d) = (a + d, b + c).
This construction is a special case of the Grothendieck construction.

Uniqueness
The negative of a number is unique, as is shown by the following proof.

Let x be a number and let y be its negative. Suppose y′ is another negative of x. By an axiom of the real number system

And so, x + y′ = x + y. Using the law of cancellation for addition, it is seen thaty′ = y. Thus y is equal to any other negative of x. That is, y is the unique negative of x.

History

For a long time, understanding of negative numbers was delayed by the impossibility of having a negative-number amount of a physical object, for example "minus-three apples", and negative solutions to problems were considered "false".

In Hellenistic Egypt, the Greek mathematician Diophantus in the 3rd century AD referred to an equation that was equivalent to  (which has a negative solution) in Arithmetica, saying that the equation was absurd. For this reason Greek geometers were able to solve geometrically all forms of the quadratic equation which give positive roots; while they could take no account of others.

Negative numbers appear for the first time in history in the Nine Chapters on the Mathematical Art (九章算術, Jiǔ zhāng suàn-shù), which in its present form dates from the period of the Han Dynasty (202 BC – 220 AD), but may well contain much older material. The mathematician Liu Hui (c. 3rd century) established rules for the addition and subtraction of negative numbers. The historian Jean-Claude Martzloff theorized that the importance of duality in Chinese natural philosophy made it easier for the Chinese to accept the idea of negative numbers. The Chinese were able to solve simultaneous equations involving negative numbers. The Nine Chapters used red counting rods to denote positive coefficients and black rods for negative. This system is the exact opposite of contemporary printing of positive and negative numbers in the fields of banking, accounting, and commerce, wherein red numbers denote negative values and black numbers signify positive values. Liu Hui writes:

The ancient Indian Bakhshali Manuscript carried out calculations with negative numbers, using "+" as a negative sign. The date of the manuscript is uncertain. LV Gurjar dates it no later than the 4th century, Hoernle dates it between the third and fourth centuries, Ayyangar and Pingree dates it to the 8th or 9th centuries, and George Gheverghese Joseph dates it to about AD 400 and no later than the early 7th century,

During the 7th century AD, negative numbers were used in India to represent debts. The Indian mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta (written c. AD 630), discussed the use of negative numbers to produce the general form quadratic formula that remains in use today. He also found negative solutions of quadratic equations and gave rules regarding operations involving negative numbers and zero, such as "A debt cut off from nothingness becomes a credit; a credit cut off from nothingness becomes a debt." He called positive numbers "fortunes", zero "a cipher", and negative numbers "debts".Knowledge Transfer and Perceptions of the Passage of Time, ICEE-2002 Keynote Address by Colin Adamson-Macedo. "Referring again to Brahmagupta's great work, all the necessary rules for algebra, including the 'rule of signs', were stipulated, but in a form which used the language and imagery of commerce and the market place. Thus 'dhana' (=fortunes) is used to represent positive numbers, whereas 'rina' (=debts) were negative".

In the 9th century, Islamic mathematicians were familiar with negative numbers from the works of Indian mathematicians, but the recognition and use of negative numbers during this period remained timid. Al-Khwarizmi in his Al-jabr wa'l-muqabala (from which we get the word "algebra") did not use negative numbers or negative coefficients. But within fifty years, Abu Kamil illustrated the rules of signs for expanding the multiplication , and al-Karaji wrote in his al-Fakhrī that "negative quantities must be counted as terms". In the 10th century, Abū al-Wafā' al-Būzjānī considered debts as negative numbers in A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen.

By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve polynomial divisions. As al-Samaw'al writes:
the product of a negative number—al-nāqiṣ (loss)—by a positive number—al-zāʾid (gain)—is negative, and by a negative number is positive. If we subtract a negative number from a higher negative number, the remainder is their negative difference. The difference remains positive if we subtract a negative number from a lower negative number. If we subtract a negative number from a positive number, the remainder is their positive sum. If we subtract a positive number from an empty power (martaba khāliyya), the remainder is the same negative, and if we subtract a negative number from an empty power, the remainder is the same positive number.

In the 12th century in India, Bhāskara II gave negative roots for quadratic equations but rejected them because they were inappropriate in the context of the problem. He stated that a negative value is "in this case not to be taken, for it is inadequate; people do not approve of negative roots."

Fibonacci allowed negative solutions in financial problems where they could be interpreted as debits (chapter 13 of Liber Abaci, 1202 AD) and later as losses (in Fibonacci's work Flos).

In the 15th century, Nicolas Chuquet, a Frenchman, used negative numbers as exponents but referred to them as "absurd numbers".

Michael Stifel dealt with negative numbers in his 1544 AD Arithmetica Integra, where he also called them numeri absurdi (absurd numbers).

In 1545, Gerolamo Cardano, in his Ars Magna, provided the first satisfactory treatment of negative numbers in Europe. He did not allow negative numbers in his consideration of cubic equations, so he had to treat, for example,  separately from  (with  in both cases). In all, Cardano was driven to the study of thirteen types of cubic equations, each with all negative terms moved to the other side of the = sign to make them positive. (Cardano also dealt with complex numbers, but understandably liked them even less.)

In 1748 Leonhard Euler, by formally manipulating complex power series while using the square root of  obtained Euler's formula of complex analysis:

where 

In 1797 AD, Carl Friedrich Gauss published a proof of the fundamental theorem of algebra but expressed his doubts at the time about "the true metaphysics of the square root of −1".

However, European mathematicians, for the most part, resisted the concept of negative numbers until the middle of the 19th century. In the 18th century it was common practice to ignore any negative results derived from equations, on the assumption that they were meaningless. In 1759 AD, the  English mathematician Francis Maseres wrote that negative numbers "darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple". He came to the conclusion that negative numbers were nonsensical.

 See also 

 Signed zero
 Additive inverse
 History of zero
 Integers
 Positive and negative parts
 Rational numbers
 Real numbers
 Sign function
 Sign (mathematics)
 Signed number representations

 References 

 Citations 

 Bibliography 

 Bourbaki, Nicolas (1998). Elements of the History of Mathematics. Berlin, Heidelberg, and New York: Springer-Verlag. .
 Struik, Dirk J. (1987). A Concise History of Mathematics''. New York: Dover Publications.

External links 

 Maseres' biographical information
 BBC Radio 4 series In Our Time, on "Negative Numbers", 9 March 2006
 Endless Examples & Exercises: Operations with Signed Integers
 Math Forum: Ask Dr. Math FAQ: Negative Times a Negative

Chinese mathematical discoveries
Elementary arithmetic
Numbers In measure theory, given a measurable space  and a signed measure  on it, a set  is called a  for  if every -measurable subset of  has nonnegative measure; that is, for every  that satisfies   holds. 

Similarly, a set  is called a  for  if for every subset  satisfying   holds.

Intuitively, a measurable set  is positive (resp. negative) for  if  is nonnegative (resp. nonpositive) everywhere on  Of course, if  is a nonnegative measure, every element of  is a positive set for 

In the light of Radon–Nikodym theorem, if  is a σ-finite positive measure such that  a set  is a positive set for  if and only if the Radon–Nikodym derivative  is nonnegative -almost everywhere on  Similarly, a negative set is a set where  -almost everywhere.

Properties 

It follows from the definition that every measurable subset of a positive or negative set is also positive or negative. Also, the union of a sequence of positive or negative sets is also positive or negative; more formally, if  is a sequence of positive sets, then

is also a positive set; the same is true if the word "positive" is replaced by "negative".

A set which is both positive and negative is a -null set, for if  is a measurable subset of a positive and negative set  then both  and  must hold, and therefore,

Hahn decomposition 

The Hahn decomposition theorem states that for every measurable space  with a signed measure  there is a partition of  into a positive and a negative set; such a partition  is unique up to -null sets, and is called a Hahn decomposition of the signed measure 

Given a Hahn decomposition  of  it is easy to show that  is a positive set if and only if  differs from a subset of  by a -null set; equivalently, if  is -null. The same is true for negative sets, if  is used instead of

See also

References 

Measure theory