157 résultats pour "numbers"
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Complex Numbers
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INTRODUCTION
Complex Numbers, in mathematics, the sum of a real number and an imaginary number.
sometimes referred to as an Argand diagram. If a complex number in the plane is thought of as a vector joining the origin to that point, then addition of complexnumbers corresponds to standard vector addition. Figure 1 shows the complex number 3 + 2 i obtained by adding the vectors 1 + 4 i and 2 - 2 i. Figure 1: Complex Plane in Cartesian CoordinatesThis graph illustrates the addition of two complex numbers by using vectors in the complex plane with cartesiancoordinates. The parallelogram shows...
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Algebra
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INTRODUCTION
Algebra, branch of mathematics in which symbols (usually letters) represent unknown numbers in mathematical equations.
B Order of Operations and Grouping Algebra relies on an established sequence for performing arithmetic operations. This ensures that everyone who executes a string of operations arrives at the sameanswer. Multiplication is performed first, then division, followed by addition, then subtraction. For example: 1 + 2 · 3 equals 7 because 2 and 3 are multiplied first and then added to 1. Exponents and roots have even higher priority than multiplication: 3 · 2 2 = 3 · 4 = 12 Grouping symbols override...
- Real Numbers.
- Rational Numbers.
- Irrational Numbers.
- Imaginary Numbers.
- Prime Numbers.
- Numerals I INTRODUCTION Numerals, signs or symbols for graphic representation of numbers.
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- Number Theory I INTRODUCTION Number Theory, branch of mathematics that deals with the properties and relationships of numbers (see Number).
- Number Systems I INTRODUCTION Number Systems, in mathematics, various notational systems that have been or are being used to represent the abstract quantities called numbers.
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Arithmetic
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INTRODUCTION
Arithmetic, branch of mathematics that arises from counting, the most basic mathematical operation.
Subtract the units: 6 - 3 = 3. Then subtract the tens column: 6 – 2 = 4. The results of these two single-digit subtractions, written side by side, provide the answer: Subtraction is a bit more complicated if we need to subtract a larger digit from a smaller one. For example, when subtracting 47 from 92, the units value (7) of 47 isgreater than the units value (2) of 92. We can handle this situation using a procedure called borrowing, which is like carrying in reverse. Ten units can be borrowe...
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Ethnic Groups in Canada - Canadian History.
Ontario and the Atlantic provinces (Nova Scotia, New Brunswick, Newfoundland and Labrador, and Prince Edward Island). Most of the Irish live in rural areas of NovaScotia, New Brunswick, Prince Edward Island, Ontario, and Québec. The Welsh are by far the smallest group among the British Canadians, and they have also settled inthe Atlantic provinces and Ontario. B Culture The language spoken by British Canadians is mostly English, but some Welsh speak their own Celtic language and some Scots, Gae...
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Statistics
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INTRODUCTION
Statistics, branch of mathematics that deals with the collection, organization, and analysis of numerical data and with such problems as experiment design and decision
making.
Professional pollsters typically conduct their surveys among sample populations of 1,000 people. Statistical measurementsshow that reductions in the margin of error flatten out considerably after the sample size reaches 1,000.© Microsoft Corporation. All Rights Reserved. The raw materials of statistics are sets of numbers obtained from enumerations or measurements. In collecting statistical data, adequate precautions must be taken tosecure complete and accurate information. The first problem of...
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Telephone.
digits is dialed. The development of inexpensive and reliable amplification provided by the introduction of the transistor in the 1960s made practical the design of adialing system based on the transmission of relatively low power tones instead of the higher-power dial pulses. Today most telephones have push buttons instead of a rotary dial. Touch-Tone is an optional service, and telephone companies still maintain the ability to receive pulsedialing. Push-button telephones usually have a switch...
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Europe .
D Vegetation Although much of Europe, particularly the west, was originally covered by forest, the vegetation has been transformed by human habitation and the clearing of land.Only in the most northerly mountains and in parts of north central European Russia has the forest cover been relatively unaffected by human activity. On the otherhand, a considerable amount of Europe is covered by woodland that has been planted or has reoccupied cleared lands. The largest vegetation zone in Europe, cuttin...
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Europe - Geography.
D Vegetation Although much of Europe, particularly the west, was originally covered by forest, the vegetation has been transformed by human habitation and the clearing of land.Only in the most northerly mountains and in parts of north central European Russia has the forest cover been relatively unaffected by human activity. On the otherhand, a considerable amount of Europe is covered by woodland that has been planted or has reoccupied cleared lands. The largest vegetation zone in Europe, cuttin...
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Poverty.
economic and demographic trends, and (7) welfare incentives. A Overpopulation Overpopulation, the situation of having large numbers of people with too few resources and too little space, is closely associated with poverty. It can result from highpopulation density (the ratio of people to land area, usually expressed as numbers of persons per square kilometer or square mile) or from low amounts of resources, or from both. Excessively high population densities put stress on available resources....
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Richard Wagner
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INTRODUCTION
Richard Wagner (1813-1883), German composer, conductor, and essayist, one of the most influential cultural figures of the 19th century.
May 1864 he was summoned to Munich by the 18-year-old King Ludwig II of Bavaria, who settled Wagner's debts, paid him a generous allowance, and provided himwith housing. Wagner was soon joined in his new home by Cosima von Bülow, Liszt’s daughter and the wife of German conductor and pianist Hans von Bülow. Wagnerand Cosima began a relationship that produced three children before the dissolution of the Bülows' marriage in 1870. Mounting hostility toward Wagner by members ofLudwig’s court resulted...
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Richard Wagner.
which took place in August 1876. Wagner completed his final opera, Parsifal (which he called a 'festival drama of dedication' for the Festspielhaus), in 1882, and it premiered that July. In September Wagner moved to Venice, where in February 1883, after a heated argument with Cosima, he suffered a fatal heart attack. He was buried in Bayreuth. III MUSIC AND THOUGHT In the early 19th century, an opera was structured as a succession of conventional self-contained forms such as aria (a vocal so...
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Chemical Reaction - chemistry.
Many groups of elements occur so often as ions that they are given names: nitrate, NO 3-; sulfate, SO 42-; and phosphate, PO 43-. The suffix -ate usually indicates the presence of oxygen. The positive ion, NH 4+, is called ammonium, as in NH 4Cl, ammonium chloride, or (NH 4)3PO4, ammonium phosphate. Rules for naming more complicated compounds exist, but many compounds have been given trivial names—for example, Na 2B4O7·10 H 2O, borax—or proprietary names—F(CF 2)nF, Teflon. These nonsystemat...
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Geology.
terminology. Today the geologic time scale is generally agreed upon and used by scientists around the world, dividing time into eons, eras, periods, and epochs. Everyfew years, the numerical time scale is refined based on new evidence, and geologists publish an update. Geologists use several methods to determine geologic time. These methods include physical stratigraphy, or the placement of events in the order of their occurrence,and biostratigraphy, which uses fossils to determine geologic time...
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Chemical Elements - chemistry.
Transition elements (also called transition metals) are a series of 30 chemical elements that share similar chemical properties. They have atomic numbers 21 to 30, 39to 48, and 71 to 80, and include titanium, iron, copper, zinc, gold, and mercury. The rare earth elements series (or rare earth metals) include the elements with atomic numbers 57 through 71, and include lanthanum, cerium, promethium, samarium,europium, gadolinium, ytterbium, and lutetium. (Yttrium [atomic no. 39] and scandium [ato...
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Periodic Law - chemistry.
VI QUANTUM THEORY With the development of the quantum theory and its application to atomic structure by the Danish physicist Niels Bohr and other scientists, most of the detailedfeatures of the periodic table have found a ready explanation. Every electron is characterized by four quantum numbers that designate its orbital motion in space. Bymeans of the selection rules governing these quantum numbers and the exclusion principle of Wolfgang Pauli, which states that two electrons in the same ato...
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North America - Geography.
D Climate Although North America has considerable climatic variety, five principal climatic regions can be identified. The northern two-thirds of Canada and Alaska, as well as all ofGreenland, have subarctic and arctic climates, in which long, dark, bitterly cold winters alternate with brief, mild summers. Most of the region, which receives relativelylittle precipitation, is covered with snow and ice during much of the year. A second climatic region is made up of the eastern two-thirds of the U...
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Los Angeles - geography.
(2006 population, 472,494), located east of the Port of Los Angeles. The city of Compton (95,701) is located north of Long Beach, on the east side of the AlamedaCorridor. On the other side of the corridor are the cities of Torrance (142,350) and Inglewood (114,914). Northwest of Inglewood and west of downtown Los Angeles are the wealthy and fashionable Santa Monica (88,050) and Beverly Hills (34,979). Both cities are enclaves:Santa Monica is surrounded by the City of Los Angeles to the north, ea...
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Los Angeles - geography.
(2006 population, 472,494), located east of the Port of Los Angeles. The city of Compton (95,701) is located north of Long Beach, on the east side of the AlamedaCorridor. On the other side of the corridor are the cities of Torrance (142,350) and Inglewood (114,914). Northwest of Inglewood and west of downtown Los Angeles are the wealthy and fashionable Santa Monica (88,050) and Beverly Hills (34,979). Both cities are enclaves:Santa Monica is surrounded by the City of Los Angeles to the north, ea...
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Statistics
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INTRODUCTION
Statistics, branch of mathematics that deals with the collection, organization, and analysis of numerical data and with such problems as experiment design and decision
making.
frequency, column (d), is the ratio of the frequency of an interval to the total count; the relative frequency is multiplied by 100 to obtain the percent relative frequency.The cumulative frequency, column (e), represents the number of students receiving grades equal to or less than the range in each succeeding interval; thus, thenumber of students with grades of 30 or less is obtained by adding the frequencies in column (c) for the first three intervals, which total 53. The cumulative relativef...
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Radioactivity
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INTRODUCTION
Marie Curie
Working with her husband, Pierre Curie, French physicist Marie Curie discovered the radioactive elements polonium and
radium in 1898.
Rutherford when he allowed an alpha-emitting substance to decay near an evacuated thin-glass vessel. The alpha particles were able to penetrate the glass and werethen trapped in the vessel, and within a few days the presence of elemental helium was demonstrated by use of a spectroscope. Beta particles were subsequentlyshown to be electrons, and gamma rays to consist of electromagnetic radiation of the same nature as X rays but of considerably greater energy. A The Nuclear Hypothesis Rutherford...
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San Francisco - geography.
recognized symbol of the city, opened in 1937. It connects San Francisco to Marin County to the north, one of the wealthiest suburban areas in the nation. With the construction of the Bay and Golden Gate bridges and other links from the city to its suburbs, the San Francisco Bay area has become one large metropolitanregion. San Francisco itself is only 122 sq km (47 sq mi) of land area, but the city’s Primary Metropolitan Statistical Area (defined by the Census Bureau as SanFrancisco, San Mateo,...
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Women's Rights.
In 1840 Lucretia Mott and Elizabeth Cady Stanton traveled to London to attend the World Anti-Slavery Convention. Upon arrival, however, the women were barred fromparticipating in the conference and forced to sit behind a curtain. This experience of discrimination inspired them to organize the first women’s rights convention. Thisconvention met in Seneca Falls, New York, on July 19 and 20, 1848. The Seneca Falls Convention attracted more than 200 women and approximately 40 men. For theconvention,...
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Immigration.
1655, only to lose all of their North American colonies to the British in 1664. These early colonies were often quite cosmopolitan, drawing settlers from many nations.When the English seized New Amsterdam, the city was home to perhaps 1500 residents, including Walloons, Huguenots, Swedes, Dutchmen, and African Americans. C The French and Spanish The French and Spanish also established colonies in North America. The Spanish established the oldest permanent European settlement in Saint Augustine,...
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Immigration - U.
1655, only to lose all of their North American colonies to the British in 1664. These early colonies were often quite cosmopolitan, drawing settlers from many nations.When the English seized New Amsterdam, the city was home to perhaps 1500 residents, including Walloons, Huguenots, Swedes, Dutchmen, and African Americans. C The French and Spanish The French and Spanish also established colonies in North America. The Spanish established the oldest permanent European settlement in Saint Augustine,...
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Extinction (biology) - biology.
III ROLE OF MASS EXTINCTION IN EVOLUTION Historically biologists—most famous among them British naturalist Charles Darwin—assumed that extinction is the natural outcome of competition between newlyevolved, adaptively superior species and their older, more primitive ancestors. These scientists believed that newer, more highly evolved species simply drove less well-adapted species to extinction. That is, historically, extinction was thought to result from evolution. It was also thought that this...
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Slavery in Africa.
The spread of Islam from Arabia into Africa after the religion’s founding in the 7th century AD affected the practice of slavery and slave trading in West, Central, and East Africa. Arabs had practiced slave raiding and trading in Arabia for centuries prior to the founding of Islam, and slavery became a component of Islamic traditions.Both the Qur'an (Koran) (the sacred scripture of Islam) and Islamic religious law served to codify and justify the existence of slavery. As Muslim Arabs conquered...
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Europe - geography.
movement of a segment of the Earth’s crust against the stable shield during the Caledonian orogeny (about 500 to 395 million years ago) raised the mountains of Ireland,Wales, Scotland, and western Norway. Subsequent erosion has rounded and worn down these mountains in the British Isles, but the peaks of Norway still reach 2,472 m(8,110 ft). The second major geological region, a belt of sedimentary materials, sweeps in an arc from southwestern France northward and eastward through the Low Countri...
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Bear - biology.
programs sponsored by zoos or other breeding centers have attempted to breed giant pandas in captivity, although most of these programs have proved unsuccessful.Among the difficulties faced by captive breeders has been the problem of encouraging a female giant panda to mate with a selected male during the two to three daysof the year when she is most fertile, a period known as estrus. B Spectacled Bear Creamy-white rings surrounding the eyes give the spectacled bear its name. Its shaggy coat of...
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Kenya - country.
threatens fish and other water life in the lake by depriving them of oxygen. Kenya is well known for its game parks—including Masai Mara Game Park and Tsavo National Park in the south, and Marsabit National Reserve in the north—whichattract large numbers of tourists and much revenue. Conservation of wildlife within reserves has thus received high priority. About 13 percent (2007) of Kenya’s totalland is protected. There are 229 (2004) threatened species in Kenya. Threatened habitats include the...
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Public Education in the United States.
opportunities for them to break out from whatever social or economic circumstances constrained their development. IV HIGHER EDUCATION During the 20th century participation in higher or postsecondary education in the United States increased as dramatically as it did in American high schools. At thebeginning of the century about 2 percent of Americans from the ages of 18 to 24 were enrolled in a college. There were fewer than 1,000 colleges then, with enrollmenttotaling about 157,000 students. N...
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Islam.
commanded to face the Kaaba, an ancient shrine in the city of Mecca. The Qur’an also refers to the recitation of parts of the Qur’an as a form of prayer. However, evenwith its numerous references, the Qur’an alone does not give exact instructions for this central ritual of prayer. The most detailed descriptions of the rituals for prayer derive from the example set by the prophet Muhammad and are preserved in later Islamic traditions. Somedetails of these rituals vary, however all Muslims agree t...
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Atom
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INTRODUCTION
Water Molecule
A water molecule consists of an oxygen atom and two hydrogen atoms, which are attached at an angle of 105°.
spontaneously break apart and change, or decay, into other atoms. Unlike electrons, which are fundamental particles, protons and neutrons are made up of other, smaller particles called quarks. Physicists know of six different quarks.Neutrons and protons are made up of up quarks and down quarks —two of the six different kinds of quarks. The fanciful names of quarks have nothing to do with their properties; the names are simply labels to distinguish one quark from another. Quarks are unique amo...
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?stanbul - geography.
İstanbul hosts a number of annual cultural events. The İstanbul Cultural Festival, held in the summer, offers music and dance performances in more than 50 venuesthroughout the city, including outdoors at Rumelihisar ı. The city also hosts a summer jazz festival. Turkey is one of the world’s largest producers of motion pictures, anda highly regarded international film festival takes place in İstanbul during the early months of the year. V RECREATION With increasing modernization, recreation is...
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Atom - chemistry.
Atoms have several properties that help distinguish one type of atom from another and determine how atoms change under certain conditions. A Atomic Number Each element has a unique number of protons in its atoms. This number is called the atomic number (abbreviated Z). Because atoms are normally electrically neutral,the atomic number also specifies how many electrons an atom will have. The number of electrons, in turn, determines many of the chemical and physical properties ofthe atom. The ligh...
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Estonia - country.
protest the expansion of open-pit phosphorite mining in northeastern Estonia. Their success in stopping the expansion prompted further demonstrations as part of thecountry’s independence movement. Since independence Estonia has taken measures to protect the environment. The government has ratified international agreementsto reduce emissions of hazardous wastes and greenhouse gases, as well as to protect biodiversity, wetlands, and endangered species. Estonians cherish thecountryside, and 31 perc...
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Korean War.
During the summer of 1949, South Korea had expanded its army to about 90,000 troops, a strength the North matched in early 1950. The North had about 150 SovietT-34 tanks and a small but effective air force of 70 fighters and 62 light bombers—weapons either left behind when Soviet troops evacuated Korea or bought from theUSSR and China in 1949 and 1950. By June 1950 American data showed the two armies at about equal strength, with roughly equal numbers amassed along the 38thparallel. However, thi...
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Korean War - History.
During the summer of 1949, South Korea had expanded its army to about 90,000 troops, a strength the North matched in early 1950. The North had about 150 SovietT-34 tanks and a small but effective air force of 70 fighters and 62 light bombers—weapons either left behind when Soviet troops evacuated Korea or bought from theUSSR and China in 1949 and 1950. By June 1950 American data showed the two armies at about equal strength, with roughly equal numbers amassed along the 38thparallel. However, thi...
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Korean War - U.
During the summer of 1949, South Korea had expanded its army to about 90,000 troops, a strength the North matched in early 1950. The North had about 150 SovietT-34 tanks and a small but effective air force of 70 fighters and 62 light bombers—weapons either left behind when Soviet troops evacuated Korea or bought from theUSSR and China in 1949 and 1950. By June 1950 American data showed the two armies at about equal strength, with roughly equal numbers amassed along the 38thparallel. However, thi...
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Insect - biology.
they almost always have six legs. In some insects, such as beetles, the legs are practically identical, but in other insects each pair is a slightly different shape. Still otherinsects have specialized leg structures. Examples are praying mantises, which have grasping and stabbing forelegs armed with lethal spines, and grasshoppers andfleas, which have large, muscular hind legs that catapult them into the air. Mole crickets’ front legs are modified for digging, and backswimmers have hind legs de...
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Kazakhstan - country.
mismanagement. Between 1949 and 1991 the Soviet government conducted about 70 percent of all of its nuclear testing in Kazakhstan, mostly in the northeastern area near the city ofSemipalatinsk (now Semey). Nearly 500 nuclear explosions occurred both above and below ground near Semipalatinsk, while more than 40 nuclear detonationsoccurred at other testing grounds in western Kazakhstan and in the Qyzylqum desert. More than 1 million of Kazakhstan’s inhabitants were exposed to dangerous levelsof ra...
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African American History - U.
In their day-to-day lives, slaves and servants shared similar grievances and frequently formed alliances. Advertisements seeking the return of slaves and servants whohad run away together filled colonial newspapers. When a slave named Charles escaped in 1740, the Pennsylvania Gazette reported that two white servants, a 'Scotch man' and an Englishman, escaped with him. Sometimes interracial alliances involved violence. During Bacon's Rebellion in 1676, slaves and servants took up armsagainst Na...
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Federal Republic of Germany - country.
B Rivers and Lakes Rivers have played a major role in Germany’s economic development. The Rhine River flows in a northwesterly direction from Switzerland through much of westernGermany and The Netherlands into the North Sea. It is a major European waterway and a pillar of commerce and trade. Its primary German tributaries include theMain, Mosel, Neckar, and Ruhr rivers. The Oder (Odra) River, along the border between Poland and Germany, runs northward and empties into the Baltic; it provides an...