112-414: An ahnentafel ( German for "ancestor table"; German: [ˈʔaːnənˌtaːfəl] ) or ahnenreihe ("ancestor series"; German: [ˈʔaːnənˌʁaɪə] ) is a genealogical numbering system for listing a person's direct ancestors in a fixed sequence of ascent. The subject (or proband ) of the ahnentafel is listed as No. 1, the subject's father as No. 2 and the mother as No. 3,
224-465: A de facto official language of Namibia after the end of German colonial rule alongside English and Afrikaans , and had de jure co-official status from 1984 until its independence from South Africa in 1990. However, the Namibian government perceived Afrikaans and German as symbols of apartheid and colonialism, and decided English would be the sole official language upon independence, stating that it
336-461: A rational number that has a finite representation in the binary numeral system, that is, the quotient of an integer by a power of two. The base-2 numeral system is a positional notation with a radix of 2 . Each digit is referred to as bit , or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates , the binary system is used by almost all modern computers and computer-based devices , as
448-402: A second language , and 75–100 million as a foreign language . This would imply the existence of approximately 175–220 million German speakers worldwide. German sociolinguist Ulrich Ammon estimated a number of 289 million German foreign language speakers without clarifying the criteria by which he classified a speaker. As of 2012 , about 90 million people, or 16% of
560-612: A "commonly used" language and the Pan South African Language Board is obligated to promote and ensure respect for it. Cameroon was also a colony of the German Empire from the same period (1884 to 1916). However, German was replaced by French and English, the languages of the two successor colonial powers, after its loss in World War I . Nevertheless, since the 21st century, German has become
672-610: A binary system for describing prosody . He described meters in the form of short and long syllables (the latter equal in length to two short syllables). They were known as laghu (light) and guru (heavy) syllables. Pingala's Hindu classic titled Chandaḥśāstra (8.23) describes the formation of a matrix in order to give a unique value to each meter. "Chandaḥśāstra" literally translates to science of meters in Sanskrit. The binary representations in Pingala's system increases towards
784-531: A great interval of time, will seem all the more curious." The relation was a central idea to his universal concept of a language or characteristica universalis , a popular idea that would be followed closely by his successors such as Gottlob Frege and George Boole in forming modern symbolic logic . Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet , who visited China in 1685 as
896-559: A greater need for regularity in written conventions. While the major changes of the MHG period were socio-cultural, High German was still undergoing significant linguistic changes in syntax, phonetics, and morphology as well (e.g. diphthongization of certain vowel sounds: hus (OHG & MHG "house") → haus (regionally in later MHG)→ Haus (NHG), and weakening of unstressed short vowels to schwa [ə]: taga (OHG "days")→ tage (MHG)). A great wealth of texts survives from
1008-512: A missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own religious beliefs as a Christian. Binary numerals were central to Leibniz's theology. He believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. [A concept that] is not easy to impart to the pagans, is the creation ex nihilo through God's almighty power. Now one can say that nothing in
1120-458: A native tongue today, mostly descendants of German colonial settlers . The period of German colonialism in Namibia also led to the evolution of a Standard German-based pidgin language called " Namibian Black German ", which became a second language for parts of the indigenous population. Although it is nearly extinct today, some older Namibians still have some knowledge of it. German remained
1232-524: A number of simple basic principles or categories, for which he has been considered a predecessor of computing science and artificial intelligence. In 1605, Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could then be encoded as scarcely visible variations in the font in any random text. Importantly for the general theory of binary encoding, he added that this method could be used with any objects at all: "provided those objects be capable of
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#17327867081181344-696: A popular foreign language among pupils and students, with 300,000 people learning or speaking German in Cameroon in 2010 and over 230,000 in 2020. Today Cameroon is one of the African countries outside Namibia with the highest number of people learning German. In the United States, German is the fifth most spoken language in terms of native and second language speakers after English, Spanish , French , and Chinese (with figures for Cantonese and Mandarin combined), with over 1 million total speakers. In
1456-601: A preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation. The modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot , Gottfried Leibniz . However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, and India. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions (not related to
1568-483: A result, the surviving texts are written in highly disparate regional dialects and exhibit significant Latin influence, particularly in vocabulary. At this point monasteries, where most written works were produced, were dominated by Latin, and German saw only occasional use in official and ecclesiastical writing. While there is no complete agreement over the dates of the Middle High German (MHG) period, it
1680-498: A second is performed by a sequence of steps in which a value (initially the first of the two numbers) is either doubled or has the first number added back into it; the order in which these steps are to be performed is given by the binary representation of the second number. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus , which dates to around 1650 BC. The I Ching dates from
1792-732: A singular Arierschein (Aryan attestation) that was titled "Ahnentafel". German language German (German: Deutsch , pronounced [dɔʏtʃ] ) is a West Germanic language in the Indo-European language family , mainly spoken in Western and Central Europe . It is the most spoken native language within the European Union . It is the most widely spoken and official (or co-official) language in Germany , Austria , Switzerland , Liechtenstein , and
1904-488: A twofold difference only; as by Bells, by Trumpets, by Lights and Torches, by the report of Muskets, and any instruments of like nature". (See Bacon's cipher .) In 1617, John Napier described a system he called location arithmetic for doing binary calculations using a non-positional representation by letters. Thomas Harriot investigated several positional numbering systems, including binary, but did not publish his results; they were found later among his papers. Possibly
2016-659: Is a method for storing a binary tree in an array by listing the nodes (individuals) in level-order (in generation order). The ahnentafel system of numeration is also known as the Eytzinger Method , for Michaël Eytzinger , the Austrian-born historian who first published the principles of the system in 1590; the Sosa Method , named for Jerónimo (Jerome) de Sosa , the Spanish genealogist who popularized
2128-601: Is a recognized minority language in the following countries: In France, the High German varieties of Alsatian and Moselle Franconian are identified as " regional languages ", but the European Charter for Regional or Minority Languages of 1998 has not yet been ratified by the government. Namibia also was a colony of the German Empire, from 1884 to 1915. About 30,000 people still speak German as
2240-427: Is addition. Adding two single-digit binary numbers is relatively simple, using a form of carrying: Adding two "1" digits produces a digit "0", while 1 will have to be added to the next column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented: This is known as carrying . When
2352-510: Is also no.81 because she was the great-great-grandmother of William's grandmother Queen Elizabeth II. The relationships are easier to follow using the ancestry tree with ahnentafel numbering. European nobility took pride in displaying their descent. In the German language, the term Ahnentafel may refer to a list of coats of arms and names of one's ancestors, even when it does not follow the numbered tabular representation given above. In this case,
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#17327867081182464-559: Is also notable for its broad spectrum of dialects , with many varieties existing in Europe and other parts of the world. Some of these non-standard varieties have become recognized and protected by regional or national governments. Since 2004, heads of state of the German-speaking countries have met every year, and the Council for German Orthography has been the main international body regulating German orthography . German
2576-558: Is an Indo-European language that belongs to the West Germanic group of the Germanic languages . The Germanic languages are traditionally subdivided into three branches: North Germanic , East Germanic , and West Germanic . The first of these branches survives in modern Danish , Swedish , Norwegian , Faroese , and Icelandic , all of which are descended from Old Norse . The East Germanic languages are now extinct, and Gothic
2688-448: Is based on the simple premise that under the binary system, when given a stretch of digits composed entirely of n ones (where n is any integer length), adding 1 will result in the number 1 followed by a string of n zeros. That concept follows, logically, just as in the decimal system, where adding 1 to a string of n 9s will result in the number 1 followed by a string of n 0s: Such long strings are quite common in
2800-478: Is called the "German Sprachraum ". German is the official language of the following countries: German is a co-official language of the following countries: Although expulsions and (forced) assimilation after the two World wars greatly diminished them, minority communities of mostly bilingual German native speakers exist in areas both adjacent to and detached from the Sprachraum. Within Europe, German
2912-430: Is complicated by the existence of several varieties whose status as separate "languages" or "dialects" is disputed for political and linguistic reasons, including quantitatively strong varieties like certain forms of Alemannic and Low German . With the inclusion or exclusion of certain varieties, it is estimated that approximately 90–95 million people speak German as a first language , 10–25 million speak it as
3024-535: Is generally seen as ending when the 1346–53 Black Death decimated Europe's population. Modern High German begins with the Early New High German (ENHG) period, which Wilhelm Scherer dates 1350–1650, terminating with the end of the Thirty Years' War . This period saw the further displacement of Latin by German as the primary language of courtly proceedings and, increasingly, of literature in
3136-611: Is generally seen as lasting from 1050 to 1350. This was a period of significant expansion of the geographical territory occupied by Germanic tribes, and consequently of the number of German speakers. Whereas during the Old High German period the Germanic tribes extended only as far east as the Elbe and Saale rivers, the MHG period saw a number of these tribes expanding beyond this eastern boundary into Slavic territory (known as
3248-565: Is less closely related to languages based on Low Franconian dialects (e.g., Dutch and Afrikaans), Low German or Low Saxon dialects (spoken in northern Germany and southern Denmark ), neither of which underwent the High German consonant shift. As has been noted, the former of these dialect types is Istvaeonic and the latter Ingvaeonic, whereas the High German dialects are all Irminonic; the differences between these languages and standard German are therefore considerable. Also related to German are
3360-521: Is not necessarily equivalent to the numerical value of one; it depends on the architecture in use. In keeping with the customary representation of numerals using Arabic numerals , binary numbers are commonly written using the symbols 0 and 1 . When written, binary numerals are often subscripted, prefixed, or suffixed to indicate their base, or radix . The following notations are equivalent: When spoken, binary numerals are usually read digit-by-digit, to distinguish them from decimal numerals. For example,
3472-434: Is often called the first digit . When the available symbols for this position are exhausted, the least significant digit is reset to 0 , and the next digit of higher significance (one position to the left) is incremented ( overflow ), and incremental substitution of the low-order digit resumes. This method of reset and overflow is repeated for each digit of significance. Counting progresses as follows: Binary counting follows
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3584-435: Is one of the major languages of the world . German is the second-most widely spoken Germanic language , after English, both as a first and as a second language . German is also widely taught as a foreign language , especially in continental Europe (where it is the third most taught foreign language after English and French), and in the United States. Overall, German is the fourth most commonly learned second language, and
3696-597: Is one of the three biggest newspapers in Namibia and the only German-language daily in Africa. An estimated 12,000 people speak German or a German variety as a first language in South Africa, mostly originating from different waves of immigration during the 19th and 20th centuries. One of the largest communities consists of the speakers of "Nataler Deutsch", a variety of Low German concentrated in and around Wartburg . The South African constitution identifies German as
3808-401: Is partly derived from Latin and Greek , along with fewer words borrowed from French and Modern English . English, however, is the main source of more recent loanwords . German is a pluricentric language ; the three standardized variants are German , Austrian , and Swiss Standard German . Standard German is sometimes called High German , which refers to its regional origin. German
3920-440: Is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Before examining binary counting, it is useful to briefly discuss the more familiar decimal counting system as a frame of reference. Decimal counting uses the ten symbols 0 through 9 . Counting begins with the incremental substitution of the least significant digit (rightmost digit) which
4032-507: Is sometimes called a "Kekulé" after Stephan Kekulé von Stradonitz . A variant of this is known in French as Seize Quartiers . To find out what someone's number would be without compiling a list, one must first trace how they relate back to the subject or person of interest, meaning that one determines for example that some ancestor is the subject's father's mother's mother's father's father. Once one has done that, one can use two methods. Use
4144-415: Is that 1 ∨ 1 = 1 {\displaystyle 1\lor 1=1} , while 1 + 1 = 10 {\displaystyle 1+1=10} . Subtraction works in much the same way: Subtracting a "1" digit from a "0" digit produces the digit "1", while 1 will have to be subtracted from the next column. This is known as borrowing . The principle is the same as for carrying. When
4256-525: Is the Sachsenspiegel , the first book of laws written in Middle Low German ( c. 1220 ). The abundance and especially the secular character of the literature of the MHG period demonstrate the beginnings of a standardized written form of German, as well as the desire of poets and authors to be understood by individuals on supra-dialectal terms. The Middle High German period
4368-578: Is the only language in this branch which survives in written texts. The West Germanic languages, however, have undergone extensive dialectal subdivision and are now represented in modern languages such as English, German, Dutch , Yiddish , Afrikaans , and others. Within the West Germanic language dialect continuum, the Benrath and Uerdingen lines (running through Düsseldorf - Benrath and Krefeld - Uerdingen , respectively) serve to distinguish
4480-509: Is translated into English as the "Explanation of Binary Arithmetic, which uses only the characters 1 and 0, with some remarks on its usefulness, and on the light it throws on the ancient Chinese figures of Fu Xi " . Leibniz's system uses 0 and 1, like the modern binary numeral system. An example of Leibniz's binary numeral system is as follows: While corresponding with the Jesuit priest Joachim Bouvet in 1700, who had made himself an expert on
4592-485: Is understood in all areas where German is spoken. Approximate distribution of native German speakers (assuming a rounded total of 95 million) worldwide: As a result of the German diaspora , as well as the popularity of German taught as a foreign language , the geographical distribution of German speakers (or "Germanophones") spans all inhabited continents. However, an exact, global number of native German speakers
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4704-689: The Ostsiedlung ). With the increasing wealth and geographic spread of the Germanic groups came greater use of German in the courts of nobles as the standard language of official proceedings and literature. A clear example of this is the mittelhochdeutsche Dichtersprache employed in the Hohenstaufen court in Swabia as a standardized supra-dialectal written language. While these efforts were still regionally bound, German began to be used in place of Latin for certain official purposes, leading to
4816-639: The Alamanni , Bavarian, and Thuringian groups, all belonging to the Elbe Germanic group ( Irminones ), which had settled in what is now southern-central Germany and Austria between the second and sixth centuries, during the great migration. In general, the surviving texts of Old High German (OHG) show a wide range of dialectal diversity with very little written uniformity. The early written tradition of OHG survived mostly through monasteries and scriptoria as local translations of Latin originals; as
4928-537: The Early Middle Ages . German is an inflected language , with four cases for nouns, pronouns, and adjectives (nominative, accusative, genitive, dative); three genders (masculine, feminine, neuter) and two numbers (singular, plural). It has strong and weak verbs . The majority of its vocabulary derives from the ancient Germanic branch of the Indo-European language family, while a smaller share
5040-417: The European Union 's population, spoke German as their mother tongue, making it the second most widely spoken language on the continent after Russian and the second biggest language in terms of overall speakers (after English), as well as the most spoken native language. The area in central Europe where the majority of the population speaks German as a first language and has German as a (co-)official language
5152-522: The German states . While these states were still part of the Holy Roman Empire , and far from any form of unification, the desire for a cohesive written language that would be understandable across the many German-speaking principalities and kingdoms was stronger than ever. As a spoken language German remained highly fractured throughout this period, with a vast number of often mutually incomprehensible regional dialects being spoken throughout
5264-746: The Holy Roman Emperor Maximilian I , and the other being Meißner Deutsch , used in the Electorate of Saxony in the Duchy of Saxe-Wittenberg . Alongside these courtly written standards, the invention of the printing press led to the development of a number of printers' languages ( Druckersprachen ) aimed at making printed material readable and understandable across as many diverse dialects of German as possible. The greater ease of production and increased availability of written texts brought about increased standardisation in
5376-451: The I Ching have also been used in traditional African divination systems, such as Ifá among others, as well as in medieval Western geomancy . The majority of Indigenous Australian languages use a base-2 system. In the late 13th century Ramon Llull had the ambition to account for all wisdom in every branch of human knowledge of the time. For that purpose he developed a general method or "Ars generalis" based on binary combinations of
5488-577: The I Ching which has 64. The Ifá originated in 15th century West Africa among Yoruba people . In 2008, UNESCO added Ifá to its list of the " Masterpieces of the Oral and Intangible Heritage of Humanity ". The residents of the island of Mangareva in French Polynesia were using a hybrid binary- decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa and Asia. Sets of binary combinations similar to
5600-542: The I Ching while a missionary in China, Leibniz explained his binary notation, and Bouvet demonstrated in his 1701 letters that the I Ching was an independent, parallel invention of binary notation. Leibniz & Bouvet concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired. Of this parallel invention, Leibniz wrote in his "Explanation Of Binary Arithmetic" that "this restitution of their meaning, after such
5712-666: The Old High German language in several Elder Futhark inscriptions from as early as the sixth century AD (such as the Pforzen buckle ), the Old High German period is generally seen as beginning with the Abrogans (written c. 765–775 ), a Latin-German glossary supplying over 3,000 Old High German words with their Latin equivalents. After the Abrogans , the first coherent works written in Old High German appear in
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#17327867081185824-676: The Sprachraum in Europe. German is used in a wide variety of spheres throughout the country, especially in business, tourism, and public signage, as well as in education, churches (most notably the German-speaking Evangelical Lutheran Church in Namibia (GELK) ), other cultural spheres such as music, and media (such as German language radio programs by the Namibian Broadcasting Corporation ). The Allgemeine Zeitung
5936-550: The Standard German language in its written form, and the Duden Handbook was declared its standard definition. Punctuation and compound spelling (joined or isolated compounds) were not standardized in the process. The Deutsche Bühnensprache ( lit. ' German stage language ' ) by Theodor Siebs had established conventions for German pronunciation in theatres , three years earlier; however, this
6048-460: The Upper German dialects spoken in the southern German-speaking countries , such as Swiss German ( Alemannic dialects ) and the various Germanic dialects spoken in the French region of Grand Est , such as Alsatian (mainly Alemannic, but also Central–and Upper Franconian dialects) and Lorraine Franconian (Central Franconian). After these High German dialects, standard German
6160-410: The logarithm to base 2 of the ahnentafel number, and rounding down to a full integer by truncating decimal digits. For example, the number 38 is between 2=32 and 2=64, so log 2 (38) is between 5 and 6. This means that ancestor no.38 belongs to generation five, and was a great-great-great-grandparent of the reference person who is no.1 (generation zero). The example, shown below, is an ahnentafel of
6272-469: The pagan Germanic tradition. Of particular interest to scholars, however, has been the Hildebrandslied , a secular epic poem telling the tale of an estranged father and son unknowingly meeting each other in battle. Linguistically, this text is highly interesting due to the mixed use of Old Saxon and Old High German dialects in its composition. The written works of this period stem mainly from
6384-497: The 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique. It is based on taoistic duality of yin and yang . Eight trigrams (Bagua) and a set of 64 hexagrams ("sixty-four" gua) , analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou dynasty of ancient China. The Song dynasty scholar Shao Yong (1011–1077) rearranged
6496-575: The Binary Progression" , in 1679, Leibniz introduced conversion between decimal and binary, along with algorithms for performing basic arithmetic operations such as addition, subtraction, multiplication, and division using binary numbers. He also developed a form of binary algebra to calculate the square of a six-digit number and to extract square roots.. His most well known work appears in his article Explication de l'Arithmétique Binaire (published in 1703). The full title of Leibniz's article
6608-723: The Empire. Its use indicated that the speaker was a merchant or someone from an urban area, regardless of nationality. Prague (German: Prag ) and Budapest ( Buda , German: Ofen ), to name two examples, were gradually Germanized in the years after their incorporation into the Habsburg domain; others, like Pressburg ( Pozsony , now Bratislava), were originally settled during the Habsburg period and were primarily German at that time. Prague, Budapest, Bratislava, and cities like Zagreb (German: Agram ) or Ljubljana (German: Laibach ), contained significant German minorities. In
6720-748: The Frisian languages— North Frisian (spoken in Nordfriesland ), Saterland Frisian (spoken in Saterland ), and West Frisian (spoken in Friesland )—as well as the Anglic languages of English and Scots. These Anglo-Frisian dialects did not take part in the High German consonant shift, and the Anglic languages also adopted much vocabulary from both Old Norse and the Norman language . The history of
6832-622: The German "Tafel" is taken literally to be a physical "display board" instead of an abstract scheme. In Nazi Germany , the Law for the Restoration of the Professional Civil Service required a person to prove non-Jewish ancestry with an Ariernachweis (Aryan certificate). The certificate could take the form of entries in the permanent Ahnenpass (that was sorted according to the ahnentafel numbering system) or as entries in
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#17327867081186944-483: The German language begins with the High German consonant shift during the Migration Period , which separated Old High German dialects from Old Saxon . This sound shift involved a drastic change in the pronunciation of both voiced and voiceless stop consonants ( b , d , g , and p , t , k , respectively). The primary effects of the shift were the following below. While there is written evidence of
7056-451: The German states; the invention of the printing press c. 1440 and the publication of Luther's vernacular translation of the Bible in 1534, however, had an immense effect on standardizing German as a supra-dialectal written language. The ENHG period saw the rise of several important cross-regional forms of chancery German, one being gemeine tiutsch , used in the court of
7168-681: The Germanic dialects that were affected by the High German consonant shift (south of Benrath) from those that were not (north of Uerdingen). The various regional dialects spoken south of these lines are grouped as High German dialects, while those spoken to the north comprise the Low German and Low Franconian dialects. As members of the West Germanic language family, High German, Low German, and Low Franconian have been proposed to be further distinguished historically as Irminonic , Ingvaeonic , and Istvaeonic , respectively. This classification indicates their historical descent from dialects spoken by
7280-655: The Irminones (also known as the Elbe group), Ingvaeones (or North Sea Germanic group), and Istvaeones (or Weser–Rhine group). Standard German is based on a combination of Thuringian - Upper Saxon and Upper Franconian dialects, which are Central German and Upper German dialects belonging to the High German dialect group. German is therefore closely related to the other languages based on High German dialects, such as Luxembourgish (based on Central Franconian dialects ) and Yiddish . Also closely related to Standard German are
7392-1033: The Italian autonomous province of South Tyrol . It is also an official language of Luxembourg , Belgium and the Italian autonomous region of Friuli-Venezia Giulia , as well as a recognized national language in Namibia . There are also notable German-speaking communities in France ( Alsace ), the Czech Republic ( North Bohemia ), Poland ( Upper Silesia ), Slovakia ( Košice Region , Spiš , and Hauerland ), Denmark ( North Schleswig ), Romania and Hungary ( Sopron ). Overseas, sizeable communities of German-speakers are found in Brazil ( Blumenau and Pomerode ), South Africa ( Kroondal ), Namibia , among others, some communities have decidedly Austrian German or Swiss German characters (e.g. Pozuzo , Peru). German
7504-518: The MHG period. Significantly, these texts include a number of impressive secular works, such as the Nibelungenlied , an epic poem telling the story of the dragon -slayer Siegfried ( c. thirteenth century ), and the Iwein , an Arthurian verse poem by Hartmann von Aue ( c. 1203 ), lyric poems , and courtly romances such as Parzival and Tristan . Also noteworthy
7616-504: The Prince of Wales , listing all of his ancestors up to his fourth great-grandparents. The same information in a tree: 1. William, Prince of Wales (born 21 June 1982) An ancestor may have two or more numbers due to pedigree collapse . For example, in the above Ahnentafel for Prince William, Queen Victoria is both no.79 and no.81. She is no.79 because she was the great-great-grandmother of William's grandfather Prince Philip, and she
7728-519: The binary expression for 1/3 = .010101..., this means: 1/3 = 0 × 2 + 1 × 2 + 0 × 2 + 1 × 2 + ... = 0.3125 + ... An exact value cannot be found with a sum of a finite number of inverse powers of two, the zeros and ones in the binary representation of 1/3 alternate forever. Arithmetic in binary is much like arithmetic in other positional notation numeral systems . Addition, subtraction, multiplication, and division can be performed on binary numerals. The simplest arithmetic operation in binary
7840-549: The binary fractions 1/2, 1/4, 1/8, 1/16, 1/32, and 1/64. Early forms of this system can be found in documents from the Fifth Dynasty of Egypt , approximately 2400 BC, and its fully developed hieroglyphic form dates to the Nineteenth Dynasty of Egypt , approximately 1200 BC. The method used for ancient Egyptian multiplication is also closely related to binary numbers. In this method, multiplying one number by
7952-400: The binary number system) and Horus-Eye fractions (so called because many historians of mathematics believe that the symbols used for this system could be arranged to form the eye of Horus , although this has been disputed). Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of a hekat is expressed as a sum of
8064-492: The binary numeral 100 is pronounced one zero zero , rather than one hundred , to make its binary nature explicit and for purposes of correctness. Since the binary numeral 100 represents the value four, it would be confusing to refer to the numeral as one hundred (a word that represents a completely different value, or amount). Alternatively, the binary numeral 100 can be read out as "four" (the correct value ), but this does not make its binary nature explicit. Counting in binary
8176-422: The binary system. From that one finds that large binary numbers can be added using two simple steps, without excessive carry operations. In the following example, two numerals are being added together: 1 1 1 0 1 1 1 1 1 0 2 (958 10 ) and 1 0 1 0 1 1 0 0 1 1 2 (691 10 ), using the traditional carry method on the left, and the long carry method on the right: The top row shows the carry bits used. Instead of
8288-412: The carry bits used. Starting in the rightmost column, 1 + 1 = 10 2 . The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: 1 + 0 + 1 = 10 2 again; the 1 is carried, and 0 is written at the bottom. The third column: 1 + 1 + 1 = 11 2 . This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives
8400-476: The conference who witnessed the demonstration were John von Neumann , John Mauchly and Norbert Wiener , who wrote about it in his memoirs. The Z1 computer , which was designed and built by Konrad Zuse between 1935 and 1938, used Boolean logic and binary floating-point numbers . Any number can be represented by a sequence of bits (binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states. Any of
8512-699: The definition that a father's number will be twice that individual's number, or a mother's will be twice plus one, and just multiply and add 1 accordingly. For instance, someone can find out what number Sophia of Hanover would be on an ahnentafel of Peter Phillips (son of Princess Anne and grandson of Elizabeth II ). Sophia is Phillips's mother's mother's father 's father 's father 's mother 's father 's father 's father 's father 's father 's mother . So, we multiply and add: Thus, if we were to make an ahnentafel for Peter Phillips, Electress Sophia would be #7233, among other numbers due to royal intermarriage causing pedigree collapse . (See #Multiple numbers for
8624-430: The development of non-local forms of language and exposed all speakers to forms of German from outside their own area. With Luther's rendering of the Bible in the vernacular, German asserted itself against the dominance of Latin as a legitimate language for courtly, literary, and now ecclesiastical subject-matter. His Bible was ubiquitous in the German states: nearly every household possessed a copy. Nevertheless, even with
8736-456: The dialect so as to make the work as natural and accessible to German speakers as possible. Copies of Luther's Bible featured a long list of glosses for each region, translating words which were unknown in the region into the regional dialect. Luther said the following concerning his translation method: One who would talk German does not ask the Latin how he shall do it; he must ask the mother in
8848-519: The eastern provinces of Banat , Bukovina , and Transylvania (German: Banat, Buchenland, Siebenbürgen ), German was the predominant language not only in the larger towns—like Temeschburg ( Timișoara ), Hermannstadt ( Sibiu ), and Kronstadt ( Brașov )—but also in many smaller localities in the surrounding areas. In 1901, the Second Orthographic Conference ended with a (nearly) complete standardization of
8960-431: The exact same procedure, and again the incremental substitution begins with the least significant binary digit, or bit (the rightmost one, also called the first bit ), except that only the two symbols 0 and 1 are available. Thus, after a bit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next bit to the left: In the binary system, each bit represents an increasing power of 2, with
9072-408: The final answer 100100 2 (36 10 ). When computers must add two numbers, the rule that: x xor y = (x + y) mod 2 for any two bits x and y allows for very fast calculation, as well. A simplification for many binary addition problems is the "long carry method" or "Brookhouse Method of Binary Addition". This method is particularly useful when one of the numbers contains a long stretch of ones. It
9184-454: The final answer of 1 1 0 0 1 1 1 0 0 0 1 2 (1649 10 ). In our simple example using small numbers, the traditional carry method required eight carry operations, yet the long carry method required only two, representing a substantial reduction of effort. The binary addition table is similar to, but not the same as, the truth table of the logical disjunction operation ∨ {\displaystyle \lor } . The difference
9296-462: The first publication of the system in Europe was by Juan Caramuel y Lobkowitz , in 1700. Leibniz wrote in excess of a hundred manuscripts on binary, most of them remaining unpublished. Before his first dedicated work in 1679, numerous manuscripts feature early attempts to explore binary concepts, including tables of numbers and basic calculations, often scribbled in the margins of works unrelated to mathematics. His first known work on binary, “On
9408-451: The first time in history. Entitled A Symbolic Analysis of Relay and Switching Circuits , Shannon's thesis essentially founded practical digital circuit design. In November 1937, George Stibitz , then working at Bell Labs , completed a relay-based computer he dubbed the "Model K" (for " K itchen", where he had assembled it), which calculated using binary addition. Bell Labs authorized a full research program in late 1938 with Stibitz at
9520-482: The following rows of symbols can be interpreted as the binary numeric value of 667: The numeric value represented in each case depends on the value assigned to each symbol. In the earlier days of computing, switches, punched holes, and punched paper tapes were used to represent binary values. In a modern computer, the numeric values may be represented by two different voltages ; on a magnetic disk , magnetic polarities may be used. A "positive", " yes ", or "on" state
9632-586: The helm. Their Complex Number Computer, completed 8 January 1940, was able to calculate complex numbers . In a demonstration to the American Mathematical Society conference at Dartmouth College on 11 September 1940, Stibitz was able to send the Complex Number Calculator remote commands over telephone lines by a teletype . It was the first computing machine ever used remotely over a phone line. Some participants of
9744-535: The hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically. Viewing the least significant bit on top of single hexagrams in Shao Yong's square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63. Etruscans divided
9856-435: The home, the children on the streets, the common man in the market-place and note carefully how they talk, then translate accordingly. They will then understand what is said to them because it is German. When Christ says ' ex abundantia cordis os loquitur ,' I would translate, if I followed the papists, aus dem Überflusz des Herzens redet der Mund . But tell me is this talking German? What German understands such stuff? No,
9968-481: The influence of Luther's Bible as an unofficial written standard, a widely accepted standard for written German did not appear until the middle of the eighteenth century. German was the language of commerce and government in the Habsburg Empire , which encompassed a large area of Central and Eastern Europe . Until the mid-nineteenth century, it was essentially the language of townspeople throughout most of
10080-480: The largest concentrations of German speakers are in the states of Rio Grande do Sul (where Riograndenser Hunsrückisch developed), Santa Catarina , and Espírito Santo . Binary numeral system A binary number is a number expressed in the base -2 numeral system or binary numeral system , a method for representing numbers that uses only two symbols for the natural numbers : typically "0" ( zero ) and "1" ( one ). A binary number may also refer to
10192-402: The mother in the home and the plain man would say, Wesz das Herz voll ist, des gehet der Mund über . Luther's translation of the Bible into High German was also decisive for the German language and its evolution from Early New High German to modern Standard German. The publication of Luther's Bible was a decisive moment in the spread of literacy in early modern Germany , and promoted
10304-653: The ninth century, chief among them being the Muspilli , Merseburg charms , and Hildebrandslied , and other religious texts (the Georgslied , Ludwigslied , Evangelienbuch , and translated hymns and prayers). The Muspilli is a Christian poem written in a Bavarian dialect offering an account of the soul after the Last Judgment , and the Merseburg charms are transcriptions of spells and charms from
10416-518: The numbering system in his work Noticia de la gran casa de los marqueses de Villafranca in 1676; and the Sosa–Stradonitz Method , for Stephan Kekulé von Stradonitz , the genealogist and son of chemist Friedrich August Kekulé , who published his interpretation of Sosa's method in his Ahnentafel-atlas in 1898. "Ahnentafel" is a loan word from the German language, and its German equivalents are Ahnenreihe and Ahnenliste . An ahnentafel list
10528-475: The outer edge of divination livers into sixteen parts, each inscribed with the name of a divinity and its region of the sky. Each liver region produced a binary reading which was combined into a final binary for divination. Divination at Ancient Greek Dodona oracle worked by drawing from separate jars, questions tablets and "yes" and "no" pellets. The result was then combined to make a final prophecy. The Indian scholar Pingala (c. 2nd century BC) developed
10640-402: The paternal grandparents as No. 4 and No. 5 and the maternal grandparents as No. 6 and No. 7, and so on, back through the generations. Apart from No. 1, who can be male or female, all even-numbered persons are male, and all odd-numbered persons are female. In this schema , the number of any person's father is double the person's number, and a person's mother is double
10752-468: The person's number plus one. Using this definition of numeration, one can derive some basic information about individuals who are listed without additional research. This construct displays a person's genealogy compactly, without the need for a diagram such as a family tree . It is particularly useful in situations where one may be restricted to presenting a genealogy in plain text, for example, in emails or newsgroup articles. In effect, an ahnentafel
10864-462: The pronunciation of the ending -ig as [ɪk] instead of [ɪç]. In Northern Germany, High German was a foreign language to most inhabitants, whose native dialects were subsets of Low German. It was usually encountered only in writing or formal speech; in fact, most of High German was a written language, not identical to any spoken dialect, throughout the German-speaking area until well into the 19th century. However, wider standardization of pronunciation
10976-481: The relation is from the number. On an ahnentafel of Prince William , John Wark is number 116. We follow the steps: We reverse that, and we get that #116, John Wark, is Prince William's mother's mother's father's mother's father's father. 1. Convert the ahnentafel number from decimal to binary, then replace the leftmost "1" with the subject's name and replace each following "0" and "1" with "father" and "mother" respectively. The generation number can be calculated as
11088-413: The result of a subtraction is less than 0, the least possible value of a digit, the procedure is to "borrow" the deficit divided by the radix (that is, 10/10) from the left, subtracting it from the next positional value. Subtracting a positive number is equivalent to adding a negative number of equal absolute value . Computers use signed number representations to handle negative numbers—most commonly
11200-456: The result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary: In this example, two numerals are being added together: 01101 2 (13 10 ) and 10111 2 (23 10 ). The top row shows
11312-438: The right, and not to the left like in the binary numbers of the modern positional notation . In Pingala's system, the numbers start from number one, and not zero. Four short syllables "0000" is the first pattern and corresponds to the value one. The numerical value is obtained by adding one to the sum of place values . The Ifá is an African divination system . Similar to the I Ching , but has up to 256 binary signs, unlike
11424-554: The rightmost bit representing 2 , the next representing 2 , then 2 , and so on. The value of a binary number is the sum of the powers of 2 represented by each "1" bit. For example, the binary number 100101 is converted to decimal form as follows: Fractions in binary arithmetic terminate only if the denominator is a power of 2 . As a result, 1/10 does not have a finite binary representation ( 10 has prime factors 2 and 5 ). This causes 10 × 1/10 not to precisely equal 1 in binary floating-point arithmetic . As an example, to interpret
11536-465: The same person below.) Write down the digit "1", which represents the subject, then from left to right write "0" for each father and "1" for each mother in the relation, ending with the ancestor of interest. The result will be the binary representation of the ancestor's ahnentafel number. Then convert the binary number to decimal form. Using the Sophia example: We can also work in reverse to find what
11648-435: The standard carry from one column to the next, the lowest-ordered "1" with a "1" in the corresponding place value beneath it may be added and a "1" may be carried to one digit past the end of the series. The "used" numbers must be crossed off, since they are already added. Other long strings may likewise be cancelled using the same technique. Then, simply add together any remaining digits normally. Proceeding in this manner gives
11760-546: The states of North Dakota and South Dakota , German is the most common language spoken at home after English. As a legacy of significant German immigration to the country , German geographical names can be found throughout the Midwest region , such as New Ulm and Bismarck (North Dakota's state capital), plus many other regions. A number of German varieties have developed in the country and are still spoken today, such as Pennsylvania Dutch and Texas German . In Brazil,
11872-500: The third most commonly learned second language in the United States in K-12 education. The language has been influential in the fields of philosophy, theology, science, and technology. It is the second most commonly used language in science and the third most widely used language on websites . The German-speaking countries are ranked fifth in terms of annual publication of new books, with one-tenth of all books (including e-books) in
11984-555: The world being published in German. German is most closely related to other West Germanic languages, namely Afrikaans , Dutch , English , the Frisian languages , and Scots . It also contains close similarities in vocabulary to some languages in the North Germanic group , such as Danish , Norwegian , and Swedish . Modern German gradually developed from Old High German , which in turn developed from Proto-Germanic during
12096-602: The world can better present and demonstrate this power than the origin of numbers, as it is presented here through the simple and unadorned presentation of One and Zero or Nothing. In 1854, British mathematician George Boole published a landmark paper detailing an algebraic system of logic that would become known as Boolean algebra . His logical calculus was to become instrumental in the design of digital electronic circuitry. In 1937, Claude Shannon produced his master's thesis at MIT that implemented Boolean algebra and binary arithmetic using electronic relays and switches for
12208-600: The written form of German. One of the central events in the development of ENHG was the publication of Luther's translation of the Bible into High German (the New Testament was published in 1522; the Old Testament was published in parts and completed in 1534). Luther based his translation primarily on the Meißner Deutsch of Saxony , spending much time among the population of Saxony researching
12320-429: Was a "neutral" language as there were virtually no English native speakers in Namibia at that time. German, Afrikaans, and several indigenous languages thus became "national languages" by law, identifying them as elements of the cultural heritage of the nation and ensuring that the state acknowledged and supported their presence in the country. Today, Namibia is considered to be the only German-speaking country outside of
12432-547: Was an artificial standard that did not correspond to any traditional spoken dialect. Rather, it was based on the pronunciation of German in Northern Germany, although it was subsequently regarded often as a general prescriptive norm, despite differing pronunciation traditions especially in the Upper-German-speaking regions that still characterise the dialect of the area today – especially
12544-492: Was established on the basis of public speaking in theatres and the media during the 20th century and documented in pronouncing dictionaries. Official revisions of some of the rules from 1901 were not issued until the controversial German orthography reform of 1996 was made the official standard by governments of all German-speaking countries. Media and written works are now almost all produced in Standard German which
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