3x+4yplease solve it?
Each province, district, and municipality is named after its seat, which is usually the largest city. The administrative divisions have changed several times since independence. When introducing new provinces, the numbers of old provinces are kept, hence the non-alphabetical order. With their official numbers, currently (since 1983) they are Algeria is classified as an upper middle income country by the World Bank. Algeria's currency is the dinar (DZD). The economy remains dominated by the state, a legacy of the country's socialist post-independence development model. In recent years, the Algerian government has halted the privatization of state-owned industries and imposed restrictions on imports and foreign involvement in its economy. These restrictions are just starting to be lifted off recently although questions about Algeria's slowly-diversifying economy remain. Algeria has struggled to develop industries outside hydrocarbons in part because of high costs and an inert state bureaucracy. The government's efforts to diversify the economy by attracting foreign and domestic investment outside the energy sector have done little to reduce high youth unemployment rates or to address housing shortages. The country is facing a number of short-term and medium-term problems, including the need to diversify the economy, strengthen political, economic and financial reforms, improve the business climate and reduce inequalities amongst regions.A wave of economic protests in February and March 2011 prompted the Algerian government to offer more than $23 billion in public grants and retroactive salary and benefit increases. Public spending has increased by 27% annually during the past 5 years. The 2010–14 public-investment programme will cost US$286 billion, 40% of which will go to human development. The Algerian economy grew by 2.6% in 2011, driven by public spending, in particular in the construction and public-works sector, and by growing internal demand. If hydrocarbons are excluded, growth has been estimated at 4.8%. Growth of 3% is expected in 2012, rising to 4.2% in 2013. The rate of inflation was 4% and the budget deficit 3% of GDP. The current-account surplus is estimated at 9.3% of GDP and at the end of December 2011, official reserves were put at US$182 billion. Inflation, the lowest in the region, has remained stable at 4% on average between 2003 and 2007. In 2011 Algeria announced a budgetary surplus of $26.9 billion, 62% increase in comparison to 2010 surplus. In general, the country exported $73 billion worth of commodities while it imported $46 billion.Thanks to strong hydrocarbon revenues, Algeria has a cushion of $173 billion in foreign currency reserves and a large hydrocarbon stabilization fund. In addition, Algeria's external debt is extremely low at about 2% of GDP. The economy remains very dependent on hydrocarbon wealth, and, despite high foreign exchange reserves (US$178 billion, equivalent to three years of imports), current expenditure growth makes Algeria's budget more vulnerable to the risk of prolonged lower hydrocarbon revenues.In 2011, the agricultural sector and services recorded growth of 10% and 5.3%, respectively. About 14% of the labor force are employed in the agricultural sector. Fiscal policy in 2011 remained expansionist and made it possible to maintain the pace of public investment and to contain the strong demand for jobs and housing.Algeria has not joined the WTO, despite several years of negotiations.In March 2006, Russia agreed to erase $4.74 billion of Algeria's Soviet-era debt during a visit by Russian President Vladimir Putin to the country, the first by a Russian leader in half a century. In return, Algerian President Abdelaziz Bouteflika agreed to buy $7.5 billion worth of combat planes, air-defence systems and other arms from Russia, according to the head of Russia's state arms exporter Rosoboronexport.Dubai-based conglomerate Emarat Dzayer Group said it had signed a joint venture agreement to develop a $1.6 billion steel factory in Algeria. Algeria, whose economy is reliant on petroleum, has been an OPEC member since 1969. Its crude oil production stands at around 1.1 million barrels/day, but it is also a major gas producer and exporter, with important links to Europe. Hydrocarbons have long been the backbone of the economy, accounting for roughly 60% of budget revenues, 30% of GDP, and over 95% of export earnings. Algeria has the 10th-largest reserves of natural gas in the world and is the sixth-largest gas exporter. The U.S. Energy Information Administration reported that in 2005, Algeria had 4.5 trillion cubic metres (160×10^12 cu ft) of proven natural-gas reserves. It also ranks 16th in oil reserves.Non-hydrocarbon growth for 2011 was projected at 5%. To cope with social demands, the authorities raised expenditure, especially on basic food support, employment creation, support for SMEs, and higher salaries. High hydrocarbon prices have improved the current account and the already large international reserves position.Income from oil and gas rose in 2011 as a result of continuing high oil prices, though the trend in production volume is downwards. Production from the oil and gas sector in terms of volume, continues to decline, dropping from 43.2 million tonnes to 32 million tonnes between 2007 and 2011. Nevertheless, the sector accounted for 98% of the total volume of exports in 2011, against 48% in 1962, and 70% of budgetary receipts, or US$71.4 billion.The Algerian national oil company is Sonatrach, which plays a key role in all aspects of the oil and natural gas sectors in Algeria. All foreign operators must work in partnership with Sonatrach, which usually has majority ownership in production-sharing agreements.Access to biocapacity in Algeria is lower than world average. In 2016, Algeria had 0.53 global hectares of biocapacity per person within its territory, much less than the world average of 1.6 global hectares per person. In 2016 Algeria used 2.4 global hectares of biocapacity per person - their ecological footprint of consumption. This means they use just under 4.5 times as much biocapacity as Algeria contains. As a result, Algeria is running a biocapacity deficit. Algeria has invested an estimated 100 billion dinars towards developing research facilities and paying researchers. This development program is meant to advance alternative energy production, especially solar and wind power. Algeria is estimated to have the largest solar energy potential in the Mediterranean, so the government has funded the creation of a solar science park in Hassi R'Mel. Currently, Algeria has 20,000 research professors at various universities and over 780 research labs, with state-set goals to expand to 1,000. Besides solar energy, areas of research in Algeria include space and satellite telecommunications, nuclear power and medical research. Despite a decline in total unemployment, youth and women unemployment is high. Unemployment particularly affects the young, with a jobless rate of 21.5% among the 15–24 age group.The overall rate of unemployment was 10% in 2011, but remained higher among young people, with a rate of 21.5% for those aged between 15 and 24. The government strengthened in 2011 the job programmes introduced in 1988, in particular in the framework of the programme to aid those seeking work (Dispositif d'Aide à l'Insertion Professionnelle). The development of the tourism sector in Algeria had previously been hampered by a lack of facilities, but since 2004 a broad tourism development strategy has been implemented resulting in many hotels of a high modern standard being built. There are several UNESCO World Heritage Sites in Algeria including Al Qal'a of Beni Hammad, the first capital of the Hammadid empire; Tipasa, a Phoenician and later Roman town; and Djémila and Timgad, both Roman ruins; M'Zab Valley, a limestone valley containing a large urbanized oasis; and the Casbah of Algiers, an important citadel. The only natural World Heritage Site is the Tassili n'Ajjer, a mountain range. The Algerian road network is the densest in Africa; its length is estimated at 180,000 km (110,000 mi) of highways, with more than 3,756 structures and a paving rate of 85%. This network will be complemented by the East-West Highway, a major infrastructure project currently under construction. It is a 3-way, 1,216-kilometre-long (756 mi) highway, linking Annaba in the extreme east to the Tlemcen in the far west. Algeria is also crossed by the Trans-Sahara Highway, which is now completely paved. This road is supported by the Algerian government to increase trade between the six countries crossed: Algeria, Mali, Niger, Nigeria, Chad, and Tunisia. In January 2016 Algeria's population was an estimated 40.4 million, who are mainly Arab-Berber ethnically. At the outset of the 20th century, its population was approximately four million. About 90% of Algerians live in the northern, coastal area; the inhabitants of the Sahara desert are mainly concentrated in oases, although some 1.5 million remain nomadic or partly nomadic. 28.1% of Algerians are under the age of 15.Women make up 70% of the country's lawyers and 60% of its judges and also dominate the field of medicine. Increasingly, women are contributing more to household income than men. 60% of university students are women, according to university researchers.Between 90,000 and 165,000 Sahrawis from Western Sahara live in the Sahrawi refugee camps, in the western Algerian Sahara desert. There are also more than 4,000 Palestinian refugees, who are well integrated and have not asked for assistance from the United Nations High Commissioner for Refugees (UNHCR). In 2009, 35,000 Chinese migrant workers lived in Algeria.The largest concentration of Algerian migrants outside Algeria is in France, which has reportedly over 1.7 million Algerians of up to the second generation. Indigenous Berbers as well as Phoenicians, Romans, Byzantine Greeks, Arabs, Turks, various Sub-Saharan Africans, and French have contributed to the history of Algeria. Descendants of Andalusian refugees are also present in the population of Algiers and other cities. Moreover, Spanish was spoken by these Aragonese and Castillian Morisco descendants deep into the 18th century, and even Catalan was spoken at the same time by Catalan Morisco descendants in the small town of Grish El-Oued. Despite the dominance of the Berber ethnicity in Algeria, the majority of Algerians identify with an Arabic-based identity, especially after the Arab nationalism rising in the 20th century. Berbers and Berber-speaking Algerians are divided into many groups with varying languages. The largest of these are the Kabyles, who live in the Kabylie region east of Algiers, the Chaoui of Northeast Algeria, the Tuaregs in the southern desert and the Shenwa people of North Algeria.During the colonial period, there was a large (10% in 1960) European population who became known as Pied-Noirs. They were primarily of French, Spanish and Italian origin. Almost all of this population left during the war of independence or immediately after its end. Modern Standard Arabic and Berber are the official languages. Algerian Arabic (Darja) is the language used by the majority of the population. Colloquial Algerian Arabic is heavily infused with borrowings from French and Berber. Berber has been recognised as a "national language" by the constitutional amendment of 8 May 2002. Kabyle, the predominant Berber language, is taught and is partially co-official (with a few restrictions) in parts of Kabylie. In February 2016, the Algerian constitution passed a resolution that would make Berber an official language alongside Arabic. Although French has no official status, Algeria is the second-largest Francophone country in the world in terms of speakers, and French is widely used in government, media (newspapers, radio, local television), and both the education system (from primary school onwards) and academia due to Algeria's colonial history. It can be regarded as a lingua franca of Algeria. In 2008, 11.2 million Algerians could read and write in French. An Abassa Institute study in April 2000 found that 60% of households could speak and understand French or 18 million in a population of 30 million then. After an earlier period during which the Algerian government tried to phase out French, in recent decades the government has backtracked and reinforced the study of French, and some television programs are broadcast in the language. Algeria emerged as a bilingual state after 1962. Colloquial Algerian Arabic is spoken by about 72% of the population and Berber by 27–30%. Islam is the predominant religion in Algeria, with its adherents, mostly Sunnis, accounting for 99% of the population according to a 2012 CIA World Factbook estimate, and 97.9% according to Pew Research in 2010. There are about 150,000 Ibadis in the M'zab Valley in the region of Ghardaia. Estimates of the Christian population range from 60,000 to 200,000. Algerian citizens who are Christians predominantly belong to Protestant groups, which have seen increased pressure from the government in recent years including many forced closures.Algeria has given the Muslim world a number of prominent thinkers, including Emir Abdelkader, Abdelhamid Ben Badis, Mouloud Kacem Naît Belkacem, Malek Bennabi and Mohamed Arkoun. In 2002, Algeria had inadequate numbers of physicians (1.13 per 1,000 people), nurses (2.23 per 1,000 people), and dentists (0.31 per 1,000 people). Access to "improved water sources" was limited to 92% of the population in urban areas and 80% of the population in the rural areas. Some 99% of Algerians living in urban areas, but only 82% of those living in rural areas, had access to "improved sanitation". According to the World Bank, Algeria is making progress toward its goal of "reducing by half the number of people without sustainable access to improved drinking water and basic sanitation by 2015". Given Algeria's young population, policy favors preventive health care and clinics over hospitals. In keeping with this policy, the government maintains an immunization program. However, poor sanitation and unclean water still cause tuberculosis, hepatitis, measles, typhoid fever, cholera and dysentery. The poor generally receive health care free of charge.Health records have been maintained in Algeria since 1882 and began adding Muslims living in the south to their vital record database in 1905 during French rule. Since the 1970s, in a centralised system that was designed to significantly reduce the rate of illiteracy, the Algerian government introduced a decree by which school attendance became compulsory for all children aged between 6 and 15 years who have the ability to track their learning through the 20 facilities built since independence, now the literacy rate is around 78.7%. Since 1972, Arabic is used as the language of instruction during the first nine years of schooling. From the third year, French is taught and it is also the language of instruction for science classes. The students can also learn English, Italian, Spanish and German. In 2008, new programs at the elementary appeared, therefore the compulsory schooling does not start at the age of six anymore, but at the age of five. Apart from the 122 private schools, the Universities of the State are free of charge. After nine years of primary school, students can go to the high school or to an educational institution. The school offers two programs: general or technical. At the end of the third year of secondary school, students pass the exam of the baccalaureate, which allows once it is successful to pursue graduate studies in universities and institutes.Education is officially compulsory for children between the ages of six and 15. In 2008, the illiteracy rate for people over 10 was 22.3%, 15.6% for men and 29.0% for women. The province with the lowest rate of illiteracy was Algiers Province at 11.6%, while the province with the highest rate was Djelfa Province at 35.5%.Algeria has 26 universities and 67 institutions of higher education, which must accommodate a million Algerians and 80,000 foreign students in 2008. The University of Algiers, founded in 1879, is the oldest, it offers education in various disciplines (law, medicine, science and letters). 25 of these universities and almost all of the institutions of higher education were founded after the independence of the country. Even if some of them offer instruction in Arabic like areas of law and the economy, most of the other sectors as science and medicine continue to be provided in French and English. Among the most important universities, there are the University of Sciences and Technology Houari Boumediene, the University of Mentouri Constantine, and University of Oran Es-Senia. The University of Abou Bekr Belkaïd in Tlemcen and University of Batna Hadj Lakhdar occupy the 26th and 45th row in Africa. Below is a list of the most important Algerian cities: Modern Algerian literature, split between Arabic, Tamazight and French, has been strongly influenced by the country's recent history. Famous novelists of the 20th century include Mohammed Dib, Albert Camus, Kateb Yacine and Ahlam Mosteghanemi while Assia Djebar is widely translated. Among the important novelists of the 1980s were Rachid Mimouni, later vice-president of Amnesty International, and Tahar Djaout, murdered by an Islamist group in 1993 for his secularist views.Malek Bennabi and Frantz Fanon are noted for their thoughts on decolonization; Augustine of Hippo was born in Tagaste (modern-day Souk Ahras); and Ibn Khaldun, though born in Tunis, wrote the Muqaddima while staying in Algeria. The works of the Sanusi family in pre-colonial times, and of Emir Abdelkader and Sheikh Ben Badis in colonial times, are widely noted. The Latin author Apuleius was born in Madaurus (Mdaourouch), in what later became Algeria. Contemporary Algerian cinema is various in terms of genre, exploring a wider range of themes and issues. There has been a transition from cinema which focused on the war of independence to films more concerned with the everyday lives of Algerians. Algerian painters, like Mohamed Racim or Baya, attempted to revive the prestigious Algerian past prior to French colonization, at the same time that they have contributed to the preservation of the authentic values of Algeria. In this line, Mohamed Temam, Abdelkhader Houamel have also returned through this art, scenes from the history of the country, the habits and customs of the past and the country life. Other new artistic currents including the one of M'hamed Issiakhem, Mohammed Khadda and Bachir Yelles, appeared on the scene of Algerian painting, abandoning figurative classical painting to find new pictorial ways, in order to adapt Algerian paintings to the new realities of the country through its struggle and its aspirations. Mohammed Khadda and M'hamed Issiakhem have been notable in recent years. The historic roots of Algerian literature go back to the Numidian and Roman African era, when Apuleius wrote The Golden Ass, the only Latin novel to survive in its entirety. This period had also known Augustine of Hippo, Nonius Marcellus and Martianus Capella, among many others. The Middle Ages have known many Arabic writers who revolutionized the Arab world literature, with authors like Ahmad al-Buni, Ibn Manzur and Ibn Khaldoun, who wrote the Muqaddimah while staying in Algeria, and many others. Albert Camus was an Algerian-born French Pied-Noir author. In 1957 he was awarded the Nobel Prize in literature. Today Algeria contains, in its literary landscape, big names having not only marked the Algerian literature, but also the universal literary heritage in Arabic and French. As a first step, Algerian literature was marked by works whose main concern was the assertion of the Algerian national entity, there is the publication of novels as the Algerian trilogy of Mohammed Dib, or even Nedjma of Kateb Yacine novel which is often regarded as a monumental and major work. Other known writers will contribute to the emergence of Algerian literature whom include Mouloud Feraoun, Malek Bennabi, Malek Haddad, Moufdi Zakaria, Abdelhamid Ben Badis, Mohamed Laïd Al-Khalifa, Mouloud Mammeri, Frantz Fanon, and Assia Djebar. In the aftermath of the independence, several new authors emerged on the Algerian literary scene, they will attempt through their works to expose a number of social problems, among them there are Rachid Boudjedra, Rachid Mimouni, Leila Sebbar, Tahar Djaout and Tahir Wattar. Currently, a part of Algerian writers tends to be defined in a literature of shocking expression, due to the terrorism that occurred during the 1990s, the other party is defined in a different style of literature who staged an individualistic conception of the human adventure. Among the most noted recent works, there is the writer, the swallows of Kabul and the attack of Yasmina Khadra, the oath of barbarians of Boualem Sansal, memory of the flesh of Ahlam Mosteghanemi and the last novel by Assia Djebar nowhere in my father's House. Chaâbi music is a typically Algerian musical genre characterized by specific rhythms and of Qacidate (popular poems) in Arabic dialect. The undisputed master of this music is El Hadj M'Hamed El Anka. The Constantinois Malouf style is saved by musician from whom Mohamed Tahar Fergani is a performer. Folk music styles include Bedouin music, characterized by the poetic songs based on long kacida (poems); Kabyle music, based on a rich repertoire that is poetry and old tales passed through generations; Shawiya music, a folklore from diverse areas of the Aurès Mountains. Rahaba music style is unique to the Aures. Souad Massi is a rising Algerian folk singer. Other Algerian singers of the diaspora include Manel Filali in Germany and Kenza Farah in France. Tergui music is sung in Tuareg languages generally, Tinariwen had a worldwide success. Finally, the staïfi music is born in Sétif and remains a unique style of its kind. Modern music is available in several facets, Raï music is a style typical of western Algeria. Rap, a relatively recent style in Algeria, is experiencing significant growth. The Algerian state's interest in film-industry activities can be seen in the annual budget of DZD 200 million (EUR 1.3 million) allocated to production, specific measures and an ambitious programme plan implemented by the Ministry of Culture in order to promote national production, renovate the cinema stock and remedy the weak links in distribution and exploitation. The financial support provided by the state, through the Fund for the Development of the Arts, Techniques and the Film Industry (FDATIC) and the Algerian Agency for Cultural Influence (AARC), plays a key role in the promotion of national production. Between 2007 and 2013, FDATIC subsidised 98 films (feature films, documentaries and short films). In mid-2013, AARC had already supported a total of 78 films, including 42 feature films, 6 short films and 30 documentaries. According to the European Audiovisual Observatory's LUMIERE database, 41 Algerian films were distributed in Europe between 1996 and 2013; 21 films in this repertoire were Algerian-French co-productions. Days of Glory (2006) and Outside the Law (2010) recorded the highest number of admissions in the European Union, 3,172,612 and 474,722, respectively.Algeria won the Palme d'Or for Chronicle of the Years of Fire (1975), two Oscars for Z (1969), and other awards for the Italian-Algerian movie The Battle of Algiers. Algerian cuisine is rich and diverse. The country was considered as the "granary of Rome". It offers a component of dishes and varied dishes, depending on the region and according to the seasons. The cuisine uses cereals as the main products, since they are always produced with abundance in the country. There is not a dish where cereals are not present. Algerian cuisine varies from one region to another, according to seasonal vegetables. It can be prepared using meat, fish and vegetables. Among the dishes known, couscous, chorba, rechta, chakhchoukha, berkoukes, shakshouka, mthewem, chtitha, mderbel, dolma, brik or bourek, garantita, lham'hlou, etc. Merguez sausage is widely used in Algeria, but it differs, depending on the region and on the added spices. Cakes are marketed and can be found in cities either in Algeria, in Europe or North America. However, traditional cakes are also made at home, following the habits and customs of each family. Among these cakes, there are Tamina, Baklawa, Chrik, Garn logzelles, Griouech, Kalb el-louz, Makroud, Mbardja, Mchewek, Samsa, Tcharak, Baghrir, Khfaf, Zlabia, Aarayech, Ghroubiya and Mghergchette. Algerian pastry also contains Tunisian or French cakes. Marketed and home-made bread products include varieties such as Kessra or Khmira or Harchaya, chopsticks and so-called washers Khoubz dar or Matloue. Other traditional meals sold often as street food include mhadjeb or mahjouba, karantika, doubara, chakhchoukha, hassouna, and t'chicha. Various games have existed in Algeria since antiquity. In the Aures, people played several games such as El Kherba or El khergueba (chess variant). Playing cards, checkers and chess games are part of Algerian culture. Racing (fantasia) and rifle shooting are part of cultural recreation of the Algerians.The first Algerian and African gold medalist is Boughera El Ouafi in 1928 Olympics of Amsterdam in the Marathon. The second Algerian Medalist was Alain Mimoun in 1956 Summer Olympics in Melbourne. Several men and women were champions in athletics in the 1990s including Noureddine Morceli, Hassiba Boulmerka, Nouria Merah-Benida, and Taoufik Makhloufi, all specialized in middle-distance running.Football is the most popular sport in Algeria. Several names are engraved in the history of the sport, including Lakhdar Belloumi, Rachid Mekhloufi, Hassen Lalmas, Rabah Madjer, Salah Assad and Djamel Zidane. The Algeria national football team qualified for the 1982 FIFA World Cup, 1986 FIFA World Cup, 2010 FIFA World Cup and 2014 FIFA World Cup. In addition, several football clubs have won continental and international trophies as the club ES Sétif or JS Kabylia. The Algerian Football Federation is an association of Algeria football clubs organizing national competitions and international matches of the selection of Algeria national football team. In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system. This contrasts with the faster fixed-precision arithmetic found in most arithmetic logic unit (ALU) hardware, which typically offers between 8 and 64 bits of precision. Several modern programming languages have built-in support for bignums, and others have libraries available for arbitrary-precision integer and floating-point math. Rather than store values as a fixed number of bits related to the size of the processor register, these implementations typically use variable-length arrays of digits. Arbitrary precision is used in applications where the speed of arithmetic is not a limiting factor, or where precise results with very large numbers are required. It should not be confused with the symbolic computation provided by many computer algebra systems, which represent numbers by expressions such as π·sin(2), and can thus represent any computable number with infinite precision. A common application is public-key cryptography, whose algorithms commonly employ arithmetic with integers having hundreds of digits. Another is in situations where artificial limits and overflows would be inappropriate. It is also useful for checking the results of fixed-precision calculations, and for determining optimal or near-optimal values for coefficients needed in formulae, for example the √⅓ that appears in Gaussian integration.Arbitrary precision arithmetic is also used to compute fundamental mathematical constants such as π to millions or more digits and to analyze the properties of the digit strings or more generally to investigate the precise behaviour of functions such as the Riemann zeta function where certain questions are difficult to explore via analytical methods. Another example is in rendering fractal images with an extremely high magnification, such as those found in the Mandelbrot set. Arbitrary-precision arithmetic can also be used to avoid overflow, which is an inherent limitation of fixed-precision arithmetic. Similar to a 5-digit odometer's display which changes from 99999 to 00000, a fixed-precision integer may exhibit wraparound if numbers grow too large to represent at the fixed level of precision. Some processors can instead deal with overflow by saturation, which means that if a result would be unrepresentable, it is replaced with the nearest representable value. (With 16-bit unsigned saturation, adding any positive amount to 65535 would yield 65535.) Some processors can generate an exception if an arithmetic result exceeds the available precision. Where necessary, the exception can be caught and recovered from—for instance, the operation could be restarted in software using arbitrary-precision arithmetic. In many cases, the task or the programmer can guarantee that the integer values in a specific application will not grow large enough to cause an overflow. Such guarantees may be based on pragmatic limits: a school attendance program may have a task limit of 4,000 students. A programmer may design the computation so that intermediate results stay within specified precision boundaries. Some programming languages such as Lisp, Python, Perl, Haskell and Ruby use, or have an option to use, arbitrary-precision numbers for all integer arithmetic. Although this reduces performance, it eliminates the possibility of incorrect results (or exceptions) due to simple overflow. It also makes it possible to guarantee that arithmetic results will be the same on all machines, regardless of any particular machine's word size. The exclusive use of arbitrary-precision numbers in a programming language also simplifies the language, because a number is a number and there is no need for multiple types to represent different levels of precision. Arbitrary-precision arithmetic is considerably slower than arithmetic using numbers that fit entirely within processor registers, since the latter are usually implemented in hardware arithmetic whereas the former must be implemented in software. Even if the computer lacks hardware for certain operations (such as integer division, or all floating-point operations) and software is provided instead, it will use number sizes closely related to the available hardware registers: one or two words only and definitely not N words. There are exceptions, as certain variable word length machines of the 1950s and 1960s, notably the IBM 1620, IBM 1401 and the Honeywell Liberator series, could manipulate numbers bound only by available storage, with an extra bit that delimited the value. Numbers can be stored in a fixed-point format, or in a floating-point format as a significand multiplied by an arbitrary exponent. However, since division almost immediately introduces infinitely repeating sequences of digits (such as 4/7 in decimal, or 1/10 in binary), should this possibility arise then either the representation would be truncated at some satisfactory size or else rational numbers would be used: a large integer for the numerator and for the denominator. But even with the greatest common divisor divided out, arithmetic with rational numbers can become unwieldy very quickly: 1/99 − 1/100 = 1/9900, and if 1/101 is then added, the result is 10001/999900. The size of arbitrary-precision numbers is limited in practice by the total storage available
