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Repeating Decimals
A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if and only if its decimal representation is repeating or terminating (i.e. all except finitely many digits are zero). For example, the decimal representation of becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333.... A more complicated example is , whose decimal becomes periodic at the ''second'' digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144.... At present, there is no single universally accepted notation or phrasing for repeating decimals. The infinitely repeated digit sequence is called the repetend or reptend. If the repetend is a zero, this decimal representation is called a terminating decimal rather than a repeating decimal, since the zeros ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decimal Representation
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: r = b_k b_\ldots b_0.a_1a_2\ldots Here is the decimal separator, is a nonnegative integer, and b_0, \ldots, b_k, a_1, a_2,\ldots are ''digits'', which are symbols representing integers in the range 0, ..., 9. Commonly, b_k\neq 0 if k > 1. The sequence of the a_i—the digits after the dot—is generally infinite. If it is finite, the lacking digits are assumed to be 0. If all a_i are , the separator is also omitted, resulting in a finite sequence of digits, which represents a natural number. The decimal representation represents the infinite sum: r=\sum_^k b_i 10^i + \sum_^\infty \frac. Every nonnegative real number has at least one such representation; it has two such representations (with b_k\neq 0 if k>0) if and only if one has a trailing infinite sequence of , and the other has a trailing infinite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
India
India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the south, the Arabian Sea on the southwest, and the Bay of Bengal on the southeast, it shares land borders with Pakistan to the west; China, Nepal, and Bhutan to the north; and Bangladesh and Myanmar to the east. In the Indian Ocean, India is in the vicinity of Sri Lanka and the Maldives; its Andaman and Nicobar Islands share a maritime border with Thailand, Myanmar, and Indonesia. Modern humans arrived on the Indian subcontinent from Africa no later than 55,000 years ago., "Y-Chromosome and Mt-DNA data support the colonization of South Asia by modern humans originating in Africa. ... Coalescence dates for most non-European populations average to between 73–55 ka.", "Modern human beings—''Homo sapiens''—originated in Africa. Then, int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
China
China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's most populous country, with a population exceeding 1.4 billion, slightly ahead of India. China spans the equivalent of five time zones and borders fourteen countries by land, the most of any country in the world, tied with Russia. Covering an area of approximately , it is the world's third largest country by total land area. The country consists of 22 provinces, five autonomous regions, four municipalities, and two Special Administrative Regions (Hong Kong and Macau). The national capital is Beijing, and the most populous city and financial center is Shanghai. Modern Chinese trace their origins to a cradle of civilization in the fertile basin of the Yellow River in the North China Plain. The semi-legendary Xia dynasty in the 21st century BCE and the well-attested Shang and Zhou dynasties developed a bureaucratic political system to serve hereditary monarchies, or dyna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
South Korea
South Korea, officially the Republic of Korea (ROK), is a country in East Asia, constituting the southern part of the Korea, Korean Peninsula and sharing a Korean Demilitarized Zone, land border with North Korea. Its western border is formed by the Yellow Sea, while its eastern border is defined by the Sea of Japan. South Korea claims to be the sole legitimate government of the entire peninsula and List of islands of South Korea, adjacent islands. It has a Demographics of South Korea, population of 51.75 million, of which roughly half live in the Seoul Capital Area, the List of metropolitan areas by population, fourth most populous metropolitan area in the world. Other major cities include Incheon, Busan, and Daegu. The Korean Peninsula was inhabited as early as the Lower Paleolithic period. Its Gojoseon, first kingdom was noted in Chinese records in the early 7th century BCE. Following the unification of the Three Kingdoms of Korea into Unified Silla, Silla and Balhae in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Australia
Australia, officially the Commonwealth of Australia, is a Sovereign state, sovereign country comprising the mainland of the Australia (continent), Australian continent, the island of Tasmania, and numerous List of islands of Australia, smaller islands. With an area of , Australia is the largest country by area in Oceania and the world's List of countries and dependencies by area, sixth-largest country. Australia is the oldest, flattest, and driest inhabited continent, with the least fertile soils. It is a Megadiverse countries, megadiverse country, and its size gives it a wide variety of landscapes and climates, with Deserts of Australia, deserts in the centre, tropical Forests of Australia, rainforests in the north-east, and List of mountains in Australia, mountain ranges in the south-east. The ancestors of Aboriginal Australians began arriving from south east Asia approximately Early human migrations#Nearby Oceania, 65,000 years ago, during the Last Glacial Period, last i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
New Zealand
New Zealand ( mi, Aotearoa ) is an island country in the southwestern Pacific Ocean. It consists of two main landmasses—the North Island () and the South Island ()—and over 700 smaller islands. It is the sixth-largest island country by area, covering . New Zealand is about east of Australia across the Tasman Sea and south of the islands of New Caledonia, Fiji, and Tonga. The country's varied topography and sharp mountain peaks, including the Southern Alps, owe much to tectonic uplift and volcanic eruptions. New Zealand's capital city is Wellington, and its most populous city is Auckland. The islands of New Zealand were the last large habitable land to be settled by humans. Between about 1280 and 1350, Polynesians began to settle in the islands and then developed a distinctive Māori culture. In 1642, the Dutch explorer Abel Tasman became the first European to sight and record New Zealand. In 1840, representatives of the United Kingdom and Māori chiefs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
United Kingdom
The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain, is a country in Europe, off the north-western coast of the continental mainland. It comprises England, Scotland, Wales and Northern Ireland. The United Kingdom includes the island of Great Britain, the north-eastern part of the island of Ireland, and many smaller islands within the British Isles. Northern Ireland shares a land border with the Republic of Ireland; otherwise, the United Kingdom is surrounded by the Atlantic Ocean, the North Sea, the English Channel, the Celtic Sea and the Irish Sea. The total area of the United Kingdom is , with an estimated 2020 population of more than 67 million people. The United Kingdom has evolved from a series of annexations, unions and separations of constituent countries over several hundred years. The Treaty of Union between the Kingdom of England (which included Wales, annexed in 1542) and the Kingdom of Scotland in 170 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vinculum (symbol)
\overline = 0. Y = \overline \sqrt /math> a-\overline = a − (b + c) Vinculum usage A vinculum () is a horizontal line used in mathematical notation for various purposes. It may be placed as an overline (or underline) over (or under) a mathematical expression to indicate that the expression is to be considered grouped together. Historically, vincula were extensively used to group items together, especially in written mathematics, but in modern mathematics this function has almost entirely been replaced by the use of parentheses. It was also used to mark Roman numerals whose values are multiplied by 1,000. Today, however, the common usage of a vinculum to indicate the repetend of a repeating decimal is a significant exception and reflects the original usage. History The vinculum, in its general use, was introduced by Frans van Schooten in 1646 as he edited the works of François Viète (who had himself not used this notation). However, earlier versions, such as using a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turkey
Turkey ( tr, Türkiye ), officially the Republic of Türkiye ( tr, Türkiye Cumhuriyeti, links=no ), is a list of transcontinental countries, transcontinental country located mainly on the Anatolia, Anatolian Peninsula in Western Asia, with a East Thrace, small portion on the Balkans, Balkan Peninsula in Southeast Europe. It shares borders with the Black Sea to the north; Georgia (country), Georgia to the northeast; Armenia, Azerbaijan, and Iran to the east; Iraq to the southeast; Syria and the Mediterranean Sea to the south; the Aegean Sea to the west; and Greece and Bulgaria to the northwest. Cyprus is located off the south coast. Turkish people, Turks form the vast majority of the nation's population and Kurds are the largest minority. Ankara is Turkey's capital, while Istanbul is its list of largest cities and towns in Turkey, largest city and financial centre. One of the world's earliest permanently Settler, settled regions, present-day Turkey was home to important Neol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slovakia
Slovakia (; sk, Slovensko ), officially the Slovak Republic ( sk, Slovenská republika, links=no ), is a landlocked country in Central Europe. It is bordered by Poland to the north, Ukraine to the east, Hungary to the south, Austria to the southwest, and the Czech Republic to the northwest. Slovakia's mostly mountainous territory spans about , with a population of over 5.4 million. The capital and largest city is Bratislava, while the second largest city is Košice. The Slavs arrived in the territory of present-day Slovakia in the fifth and sixth centuries. In the seventh century, they played a significant role in the creation of Samo's Empire. In the ninth century, they established the Principality of Nitra, which was later conquered by the Principality of Moravia to establish Great Moravia. In the 10th century, after the dissolution of Great Moravia, the territory was integrated into the Principality of Hungary, which then became the Kingdom of Hungary in 1000. In 1241 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Czech Republic
The Czech Republic, or simply Czechia, is a landlocked country in Central Europe. Historically known as Bohemia, it is bordered by Austria to the south, Germany to the west, Poland to the northeast, and Slovakia to the southeast. The Czech Republic has a hilly landscape that covers an area of with a mostly temperate continental and oceanic climate. The capital and largest city is Prague; other major cities and urban areas include Brno, Ostrava, Plzeň and Liberec. The Duchy of Bohemia was founded in the late 9th century under Great Moravia. It was formally recognized as an Imperial State of the Holy Roman Empire in 1002 and became a kingdom in 1198. Following the Battle of Mohács in 1526, the whole Crown of Bohemia was gradually integrated into the Habsburg monarchy. The Protestant Bohemian Revolt led to the Thirty Years' War. After the Battle of White Mountain, the Habsburgs consolidated their rule. With the dissolution of the Holy Empire in 1806, the Cro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Switzerland
). Swiss law does not designate a ''capital'' as such, but the federal parliament and government are installed in Bern, while other federal institutions, such as the federal courts, are in other cities (Bellinzona, Lausanne, Luzern, Neuchâtel, St. Gallen a.o.). , coordinates = , largest_city = Zürich , official_languages = , englishmotto = "One for all, all for one" , religion_year = 2020 , religion_ref = , religion = , demonym = , german: Schweizer/Schweizerin, french: Suisse/Suissesse, it, svizzero/svizzera or , rm, Svizzer/Svizra , government_type = Federalism, Federal assembly-independent Directorial system, directorial republic with elements of a direct democracy , leader_title1 = Federal Council (Switzerland), Federal Council , leader_name1 = , leader_title2 = , leader_name2 = Walter Thurnherr , legislature = Fe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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