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Recurring Decimal
A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be ''terminating'', and is not considered as repeating. It can be shown that a number is rational if and only if its decimal representation is repeating or terminating. 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.... Another example of this is , which becomes periodic after the decimal point, repeating the 13-digit pattern "1886792452830" forever, i.e. 11.18867924528301886792452830.... T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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_\cdots b_0.a_1a_2\cdots Here is the decimal separator, is a nonnegative integer, and b_0, \cdots, b_k, a_1, a_2,\cdots are ''digits'', which are symbols representing integers in the range 0, ..., 9. Commonly, b_k\neq 0 if k \geq 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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India
India, officially the Republic of India, is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area; the List of countries by population (United Nations), most populous country since 2023; and, since its independence in 1947, the world's most populous democracy. 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 near 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 averag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Pakistan
Pakistan, officially the Islamic Republic of Pakistan, is a country in South Asia. It is the List of countries and dependencies by population, fifth-most populous country, with a population of over 241.5 million, having the Islam by country#Countries, second-largest Muslim population as of 2023. Islamabad is the nation's capital, while Karachi is List of cities in Pakistan by population, its largest city and financial centre. Pakistan is the List of countries and dependencies by area, 33rd-largest country by area. Bounded by the Arabian Sea on the south, the Gulf of Oman on the southwest, and the Sir Creek on the southeast, it shares land borders with India to the east; Afghanistan to the west; Iran to the southwest; and China to the northeast. It shares a maritime border with Oman in the Gulf of Oman, and is separated from Tajikistan in the northwest by Afghanistan's narrow Wakhan Corridor. Pakistan is the site of History of Pakistan, several ancient cultures, including the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Morocco
Morocco, officially the Kingdom of Morocco, is a country in the Maghreb region of North Africa. It has coastlines on the Mediterranean Sea to the north and the Atlantic Ocean to the west, and has land borders with Algeria to Algeria–Morocco border, the east, and the disputed territory of Western Sahara to Morocco–Western Sahara border, the south. Morocco also claims the Spain, Spanish Enclave and exclave, exclaves of Ceuta, Melilla and Peñón de Vélez de la Gomera, and several small Plazas de soberanía, Spanish-controlled islands off its coast. It has a population of approximately 37 million. Islam is both the official and predominant religion, while Arabic and Berber are the official languages. Additionally, French and the Moroccan dialect of Arabic are widely spoken. The culture of Morocco is a mix of Arab culture, Arab, Berbers, Berber, Culture of Africa, African and Culture of Europe, European cultures. Its capital is Rabat, while its largest city is Casablanca. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Malaysia
Malaysia is a country in Southeast Asia. Featuring the Tanjung Piai, southernmost point of continental Eurasia, it is a federation, federal constitutional monarchy consisting of States and federal territories of Malaysia, 13 states and three federal territories, separated by the South China Sea into two regions: Peninsular Malaysia on the Mainland Southeast Asia, Indochinese Peninsula and East Malaysia on the island of Borneo. Peninsular Malaysia shares land and maritime Malaysia–Thailand border, borders with Thailand, as well as maritime borders with Singapore, Vietnam, and Indonesia; East Malaysia shares land borders with Brunei and Indonesia, and a maritime border with the Philippines and Vietnam. Kuala Lumpur is the country's national capital, List of cities and towns in Malaysia by population, largest city, and the seat of the Parliament of Malaysia, legislative branch of the Government of Malaysia, federal government, while Putrajaya is the federal administrative capi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Bangladesh
Bangladesh, officially the People's Republic of Bangladesh, is a country in South Asia. It is the List of countries and dependencies by population, eighth-most populous country in the world and among the List of countries and dependencies by population density, most densely populated with a population of over 171 million within an area of . Bangladesh shares land borders with India to the north, west, and east, and Myanmar to the southeast. It has a coastline along the Bay of Bengal to its south and is separated from Bhutan and Nepal by the Siliguri Corridor, and from China by the List of Indian states, Indian state of Sikkim to its north. Dhaka, the capital and list of cities and towns in Bangladesh, largest city, is the nation's political, financial, and cultural centre. Chittagong is the second-largest city and the busiest port of the country. The territory of modern Bangladesh was a stronghold of many List of Buddhist kingdoms and empires, Buddhist and List of Hindu empir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Turkey
Turkey, officially the Republic of Türkiye, is a country mainly located in Anatolia in West Asia, with a relatively small part called East Thrace in Southeast Europe. It borders the Black Sea to the north; Georgia (country), Georgia, Armenia, Azerbaijan, and Iran to the east; Iraq, Syria, and the Mediterranean Sea to the south; and the Aegean Sea, Greece, and Bulgaria to the west. Turkey is home to over 85 million people; most are ethnic Turkish people, Turks, while ethnic Kurds in Turkey, Kurds are the Minorities in Turkey, largest ethnic minority. Officially Secularism in Turkey, a secular state, Turkey has Islam in Turkey, a Muslim-majority population. Ankara is Turkey's capital and second-largest city. Istanbul is its largest city and economic center. Other major cities include İzmir, Bursa, and Antalya. First inhabited by modern humans during the Late Paleolithic, present-day Turkey was home to List of ancient peoples of Anatolia, various ancient peoples. The Hattians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Chile
Chile, officially the Republic of Chile, is a country in western South America. It is the southernmost country in the world and the closest to Antarctica, stretching along a narrow strip of land between the Andes, Andes Mountains and the Pacific Ocean. Chile had a population of 17.5 million as of the latest census in 2017 and has a territorial area of , sharing borders with Peru to the north, Bolivia to the northeast, Argentina to the east, and the Drake Passage to the south. The country also controls several Pacific islands, including Juan Fernández Islands, Juan Fernández, Isla Salas y Gómez, Desventuradas Islands, Desventuradas, and Easter Island, and claims about of Antarctica as the Chilean Antarctic Territory. The capital and largest city of Chile is Santiago, and the national language is Spanish language, Spanish. Conquest of Chile, Spain conquered and colonized the region in the mid-16th century, replacing Incas in Central Chile, Inca rule; however, they Arauco War ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Slovenia
Slovenia, officially the Republic of Slovenia, is a country in Central Europe. It borders Italy to the west, Austria to the north, Hungary to the northeast, Croatia to the south and southeast, and a short (46.6 km) coastline within the Adriatic Sea to the southwest, which is part of the Mediterranean Sea. Slovenia is mostly mountainous and forested, covers , and has a population of approximately 2.1 million people. Slovene language, Slovene is the official language. Slovenia has a predominantly temperate continental climate, with the exception of the Slovene Littoral and the Julian Alps. Ljubljana, the capital and List of cities and towns in Slovenia, largest city of Slovenia, is geographically situated near the centre of the country. Other larger urban centers are Maribor, Ptuj, Kranj, Celje, and Koper. Slovenia's territory has been part of many different states: the Byzantine Empire, the Carolingian Empire, the Holy Roman Empire, the Kingdom of Hungary, the Republic of Venice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Slovakia
Slovakia, officially the Slovak Republic, 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 west, and the Czech Republic to the northwest. Slovakia's mostly mountainous territory spans about , hosting a population exceeding 5.4 million. The capital and largest city is Bratislava, while the second largest city is Košice. The Slavs arrived in the territory of the present-day Slovakia in the 5th and 6th centuries. From the late 6th century, parts of modern Slovakia were incorporated into the Pannonian Avars, Avar Khaghanate. In the 7th century, the Slavs played a significant role in the creation of Samo's Empire. When the Avar Khaghanate dissolved in the 9th century, the Slavs established the Principality of Nitra before it was annexed by the Great Moravia, Principality of Moravia, which later became Great Moravia. When Great Moravia fell in the 10th century, the territory was integrated i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Czech Republic
The Czech Republic, also known as Czechia, and historically known as Bohemia, is a landlocked country in Central Europe. The country 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 Humid continental climate, 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 Estate of the Holy Roman Empire in 1002 and became Kingdom of Bohemia, a kingdom in 1198. Following the Battle of Mohács in 1526, all of the Lands of the Bohemian Crown were gradually integrated into the Habsburg monarchy. Nearly a hundred years later, the Protestantism, Protestant Bohemian Revolt led to the Thirty Years' War. After the Battle of White ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Switzerland
Switzerland, officially the Swiss Confederation, is a landlocked country located in west-central Europe. It is bordered by Italy to the south, France to the west, Germany to the north, and Austria and Liechtenstein to the east. Switzerland is geographically divided among the Swiss Plateau, the Swiss Alps, Alps and the Jura Mountains, Jura; the Alps occupy the greater part of the territory, whereas most of the country's Demographics of Switzerland, 9 million people are concentrated on the plateau, which hosts List of cities in Switzerland, its largest cities and economic centres, including Zurich, Geneva, and Lausanne. Switzerland is a federal republic composed of Cantons of Switzerland, 26 cantons, with federal authorities based in Bern. It has four main linguistic and cultural regions: German, French, Italian and Romansh language, Romansh. Although most Swiss are German-speaking, national identity is fairly cohesive, being rooted in a common historical background, shared ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |