Bureaucracy

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Bureaucracy

Bureaucracy (/bjʊəˈrɒkrəsi/ bure-OK-rə-see) is a system of organization where laws or regulatory authority are implemented by civil servants, non-elected officials. Historically, a bureaucracy was a government administration managed by departments staffed with non-elected officials. Today, bureaucracy is the administrative system governing any large institution, whether publicly owned or privately owned. The public administration in many jurisdictions is an example of bureaucracy, as is any centralized hierarchical structure of an institution, including corporations, societies, nonprofit organizations, and clubs.

There are two key dilemmas in bureaucracy. The first dilemma relates to whether bureaucrats should be autonomous or directly accountable to their political masters. The second dilemma relates to bureaucrats' responsibility to follow preset rules, and what degree of latitude they may have to determine appropriate solutions for circumstances that are unaccounted for in advance.

Various commentators have argued for the necessity of bureaucracies in modern society. The German sociologist Max Weber argued that bureaucracy constitutes the most efficient and rational way in which human activity can be organized and that systematic processes and organized hierarchies are necessary to maintain order, maximize efficiency, and eliminate favoritism. On the other hand, Weber also saw unfettered bureaucracy as a threat to individual freedom, with the potential of trapping individuals in an impersonal "iron cage" of rule-based, rational control.

Etymology and usage

The term bureaucracy originated in the French language: it combines the French word bureau – 'desk' or 'office' – with the Greek word κράτος (kratos) – 'rule' or 'political power'. The French economist Jacques Claude Marie Vincent de Gournay coined the word in the mid-18th century. Gournay never wrote the term down but a letter from a contemporary later quoted him:

The late M. de Gournay... sometimes used to say: "We have an illness in France which bids fair to play havoc with us; this illness is called bureaumania." Sometimes he used to invent a fourth or fifth form of government under the heading of "bureaucracy."

— Baron von Grimm (1723–1807)

The first known English-language use dates to 1818 with Irish novelist Lady Morgan referring to the apparatus used by the British government to subjugate Ireland as "the Bureaucratie, or office tyranny, by which Ireland has so long been governed". By the mid-19th century the word appeared in a more neutral sense, referring to a system of public administration in which offices were held by unelected career officials. In this context bureaucracy was seen as a distinct form of management, often subservient to a monarchy.

In the 1920s the German sociologist Max Weber expanded the definition to include any system of administration conducted by trained professionals according to fixed rules. Weber saw bureaucracy as a relatively positive development; however, by 1944 the Austrian economist Ludwig von Mises opined in the context of his experience in the Nazi regime that the term bureaucracy was "always applied with an opprobrious connotation", and by 1957 the American sociologist Robert Merton suggested that the term bureaucrat had become an "epithet, a Schimpfwort" in some circumstances.

The word bureaucracy is also used in politics and government with a disapproving tone to disparage official rules that appear to make it difficult—by insistence on procedure and compliance to rule, regulation, and law—to get things done. In workplaces, the word is used very often to blame complicated rules, processes, and written work that are interpreted as obstacles rather than safeguards and accountability assurances. Socio-bureaucracy would then refer to certain social influences that may affect the function of a society.

In modern usage, modern bureaucracy has been defined as comprising four features:

1. hierarchy (clearly defined spheres of competence and divisions of labor)
2. continuity (a structure where administrators have a full-time salary and advance within the structure)
3. impersonality (prescribed rules and operating rules rather than arbitrary actions)
4. expertise (officials are chosen according to merit, have been trained, and hold access to knowledge)

History

Ancient

Students competed in imperial examinations to receive a position in the bureaucracy of Imperial China.

Although the term bureaucracy first originated in the mid-18th century, organized and consistent administrative systems existed much earlier. The development of writing (c. 3500 BC) and the use of documents was a critical component of such systems. The first definitive example of bureaucracy occurred in ancient Sumer, where an emergent class of scribes used clay tablets to document and carry out various administrative functions, such as the management of taxes, workers, and public goods/resources like granaries. Similarly, Ancient Egypt had a hereditary class of scribes that administered a civil-service bureaucracy.

In China, when the Qin dynasty (221–206 BC) unified China under the Legalist system, the emperor assigned administration to dedicated officials rather than nobility, ending feudalism in China, replacing it with a centralized, bureaucratic government. The form of government created by the first emperor and his advisors was used by later dynasties to structure their own government. Under this system, the government thrived, as talented individuals could be more easily identified in the transformed society. The Han dynasty (202 BC – 220 AD) established a complicated bureaucracy based on the teachings of Confucius, who emphasized the importance of ritual in family, relationships, and politics. With each subsequent dynasty, the bureaucracy evolved. In 165 BC, Emperor Wen introduced the first method of recruitment to civil service through examinations. Emperor Wu (r. 141–87 BC) cemented the ideology of Confucius into mainstream governance by installing a system of recommendation and nomination in government service known as xiaolian, and a national academy where officials would select candidates to take part in an examination of the Confucian classics, from which Emperor Wu would select officials.

In the Sui dynasty (581–618) and the subsequent Tang dynasty (618–907) the shi class would begin to present itself by means of the fully standardized civil service examination system, of partial recruitment of those who passed standard exams and earned an official degree. Yet recruitment by recommendations to office was still prominent in both dynasties. It was not until the Song dynasty (960–1279) that the recruitment of those who passed the exams and earned degrees was given greater emphasis and significantly expanded. During the Song dynasty (960–1279) the bureaucracy became meritocratic. Following the Song reforms, competitive examinations took place to determine which candidates qualified to hold given positions. The imperial examination system lasted until 1905, six years before the Qing dynasty collapsed, marking the end of China's traditional bureaucratic system.

A hierarchy of regional proconsuls and their deputies administered the Roman Empire. The reforms of Diocletian (Emperor from 284 to 305) doubled the number of administrative districts and led to a large-scale expansion of Roman bureaucracy. The early Christian author Lactantius (c. 250 – c. 325) claimed that Diocletian's reforms led to widespread economic stagnation, since "the provinces were divided into minute portions, and many presidents and a multitude of inferior officers lay heavy on each territory." After the Empire split, the Byzantine Empire developed a notoriously complicated administrative hierarchy, and in the 20th century the term Byzantine came to refer to any complex bureaucratic structure.

Modern

Persia

Uzun Hasan's conquest of most of mainland Iran shifted the seat of power to the east, where the Aq Qoyunlu adopted Iranian customs for administration and culture. In the Iranian areas, Uzun Hasan preserved the previous bureaucratic structure along with its secretaries, who belonged to families that had in a number of instances served under different dynasties for several generations. The four top civil posts of the Aq Qoyunlu were all occupied by Iranians, which under Uzun Hasan included: the vizier, who led the great council (divan); the mostawfi al-mamalek, high-ranking financial accountants; the mohrdar, who affixed the state seal; and the marakur 'stable master', who supervised the royal court. Through the use of his increasing revenue, Uzun Hasan was able to buy the approval of the ulama (clergy) and the mainly Iranian urban elite, while also taking care of the impoverished rural inhabitants.

The Safavid state was one of checks and balance, both within the government and on a local level. At the apex of this system was the Shah, with total power over the state, legitimized by his bloodline as a sayyid, or descendant of Muhammad. To ensure transparency and avoid decisions being made that circumvented the Shah, a complex system of bureaucracy and departmental procedures had been put in place that prevented fraud. Every office had a deputy or superintendent, whose job was to keep records of all actions of the state officials and report directly to the Shah. The Shah himself exercised his own measures for keeping his ministers under control by fostering an atmosphere of rivalry and competitive surveillance. And since the Safavid society was meritocratic, and successions seldom were made on the basis of heritage, this meant that government offices constantly felt the pressure of being under surveillance and had to make sure they governed in the best interest of their leader, and not merely their own.

The Ottomans adopted Persian bureaucratic traditions and culture.

Russia

The Russian autocracy survived the Time of Troubles and the rule of weak or corrupt tsars because of the strength of the government's central bureaucracy. Government functionaries continued to serve, regardless of the ruler's legitimacy or the boyar faction controlling the throne. In the 17th century, the bureaucracy expanded dramatically. The number of government departments (prikazy; sing., prikaz ) increased from twenty-two in 1613 to eighty by mid-century. Although the departments often had overlapping and conflicting jurisdictions, the central government, through provincial governors, was able to control and regulate all social groups, as well as trade, manufacturing, and even the Eastern Orthodox Church.

The tsarist bureaucracy, alongside the military, the judiciary and the Russian Orthodox Church, played a major role in solidifying and maintaining the rule of the Tsars in the Tsardom of Russia (1547–1721) and in the Russian Empire (1721–1917). In the 19th century, the forces of change brought on by the Industrial Revolution propelled many countries, especially in Europe, to significant social changes. However, due to the conservative nature of the Tsarist regime and its desire to maintain power and control, social change in Russia lagged behind that of Europe.

Russian-speakers referred to bureaucrats as chinovniki (чиновники) because of the rank or chin (чин) which they held.

Ashanti Empire

The government of the Ashanti Empire was built upon a sophisticated bureaucracy in Kumasi, with separate ministries which saw to the handling of state affairs. Ashanti's Foreign Office was based in Kumasi. Despite the small size of the office, it allowed the state to pursue complex negotiations with foreign powers. The Office was divided into departments that handled Ashanti relations separately with the British, French, Dutch, and Arabs. Scholars of Ashanti history, such as Larry Yarak and Ivor Wilkes, disagree over the power of this sophisticated bureaucracy in comparison to the Asantehene. However, both scholars agree that it was a sign of a highly developed government with a complex system of checks and balances.

United Kingdom

The 18th century Department of Excise developed a sophisticated bureaucracy. Pictured, the Custom House in the City of London.

Instead of the inefficient and often corrupt system of tax farming that prevailed in absolutist states such as France, the Exchequer was able to exert control over the entire system of tax revenue and government expenditure. By the late 18th century, the ratio of fiscal bureaucracy to population in Britain was approximately 1 in 1300, almost four times larger than the second most heavily bureaucratized nation, France. Thomas Taylor Meadows, Britain's consul in Guangzhou, argued in his Desultory Notes on the Government and People of China (1847) that "the long duration of the Chinese empire is solely and altogether owing to the good government which consists in the advancement of men of talent and merit only", and that the British must reform their civil service by making the institution meritocratic. Influenced by the ancient Chinese imperial examination, the Northcote–Trevelyan Report of 1854 recommended that recruitment should be on the basis of merit determined through competitive examination, candidates should have a solid general education to enable inter-departmental transfers, and promotion should be through achievement rather than "preferment, patronage, or purchase". This led to implementation of His Majesty's Civil Service as a systematic, meritocratic civil service bureaucracy.

In the British civil service, just as it was in China, entrance to the civil service was usually based on a general education in ancient classics, which similarly gave bureaucrats greater prestige. The Cambridge-Oxford ideal of the civil service was identical to the Confucian ideal of a general education in world affairs through humanism. Well into the 20th century, classics, literature, history and language remained heavily favoured in British civil service examinations. In the period of 1925–1935, 67 percent of British civil service entrants consisted of such graduates. Like the Chinese model's consideration of personal values, the British model also took personal physique and character into account.

France

Like the British, the development of French bureaucracy was influenced by the Chinese system. Under Louis XIV of France, the old nobility had neither power nor political influence, their only privilege being exemption from taxes. The dissatisfied noblemen complained about this "unnatural" state of affairs, and discovered similarities between absolute monarchy and bureaucratic despotism. With the translation of Confucian texts during the Enlightenment, the concept of a meritocracy reached intellectuals in the West, who saw it as an alternative to the traditional ancien regime of Europe. Western perception of China even in the 18th century admired the Chinese bureaucratic system as favourable over European governments for its seeming meritocracy; Voltaire claimed that the Chinese had "perfected moral science" and François Quesnay advocated an economic and political system modeled after that of the Chinese. The governments of China, Egypt, Peru and Empress Catherine II were regarded as models of Enlightened Despotism, admired by such figures as Diderot, D'Alembert and Voltaire.

Napoleonic France adopted this meritocracy system and soon saw a rapid and dramatic expansion of government, accompanied by the rise of the French civil service and its complex systems of bureaucracy. This phenomenon became known as "bureaumania". In the early 19th century, Napoleon attempted to reform the bureaucracies of France and other territories under his control by the imposition of the standardized Napoleonic Code. But paradoxically, that led to even further growth of the bureaucracy.

French civil service examinations adopted in the late 19th century were also heavily based on general cultural studies. These features have been likened to the earlier Chinese model.

Other industrialized nations

By the mid-19th century, bureaucratic forms of administration were firmly in place across the industrialized world. Thinkers like John Stuart Mill and Karl Marx began to theorize about the economic functions and power-structures of bureaucracy in contemporary life. Max Weber was the first to endorse bureaucracy as a necessary feature of modernity, and by the late 19th century bureaucratic forms had begun their spread from government to other large-scale institutions.

Within capitalist systems, informal bureaucratic structures began to appear in the form of corporate power hierarchies, as detailed in mid-century works like The Organization Man and The Man in the Gray Flannel Suit. Meanwhile, in the Soviet Union and Eastern Bloc nations, a powerful class of bureaucratic administrators termed nomenklatura governed nearly all aspects of public life.

The 1980s brought a backlash against perceptions of "big government" and the associated bureaucracy. Politicians like Margaret Thatcher and Ronald Reagan gained power by promising to eliminate government regulatory bureaucracies, which they saw as overbearing, and return economic production to a more purely capitalistic mode, which they saw as more efficient. In the business world, managers like Jack Welch gained fortune and renown by eliminating bureaucratic structures inside corporations. Still, in the modern world, most organized institutions rely on bureaucratic systems to manage information, process records, and administer complex systems, although the decline of paperwork and the widespread use of electronic databases is transforming the way bureaucracies function.

Theories

Karl Marx

Karl Marx theorized about the role and function of bureaucracy in his Critique of Hegel's Philosophy of Right, published in 1843. In Philosophy of Right, Hegel had supported the role of specialized officials in public administration, although he never used the term bureaucracy himself. By contrast, Marx was opposed to bureaucracy. Marx posited that while corporate and government bureaucracy seem to operate in opposition, in actuality they mutually rely on one another to exist. He wrote that "The Corporation is civil society's attempt to become state; but the bureaucracy is the state which has really made itself into civil society."

Leon Trotsky

Leon Trotsky developed a critical theory of the emerging Soviet bureaucracy during the early years of the Soviet Union. According to political scientist Thomas M. Twiss, Trotsky associated bureaucratism with authoritarianism, excessive centralism and conservatism. Social theorist Martin Krygier had noted the impact of Trotsky's post-1923 writings in shaping receptive views of bureaucracy among later Marxists and many non-Marxists. Twiss argued that Trotsky's theory of Soviet bureaucracy was essential for a study of Soviet history and understanding the process of capitalist restoration in Russia and Eastern Europe. Political scientist, Baruch Knei-Paz argued Trotsky had, above all others, written "to show the historical and social roots of Stalinism" as a bureaucratic system.

One of the predictions made by Trotsky in his 1936 work, The Revolution Betrayed, was that the USSR would come before a disjuncture: either the toppling of the ruling bureaucracy by means of a political revolution, or capitalist restoration led by the bureaucracy:

The fall of the present bureaucratic dictatorship, if it were not replaced by a new socialist power, would thus mean a return to capitalist relations with a catastrophic decline of industry and culture.

John Stuart Mill

Writing in the early 1860s, political scientist John Stuart Mill theorized that successful monarchies were essentially bureaucracies, and found evidence of their existence in Imperial China, the Russian Empire, and the regimes of Europe. Mill referred to bureaucracy as a distinct form of government, separate from representative democracy. He believed bureaucracies had certain advantages, most importantly the accumulation of experience in those who actually conduct the affairs. Nevertheless, he believed this form of governance compared poorly to representative government, as it relied on appointment rather than direct election. Mill wrote that ultimately the bureaucracy stifles the mind, and that "a bureaucracy always tends to become a pedantocracy."

Max Weber

The fully developed bureaucratic apparatus compares with other organisations exactly as does the machine with the non-mechanical modes of production.

–Max Weber

The German sociologist Max Weber (1864-1920) was the first to study bureaucracy formally, and his works led to the popularization of this term. In his essay Bureaucracy, published in his magnum opus, Economy and Society in 1921, Weber described many ideal-typical forms of public administration, government, and business. His ideal-typical bureaucracy, whether public or private, is characterized by:

• hierarchical organization
• formal lines of authority (chain of command)
• a fixed area of activity
• rigid division of labor
• regular and continuous execution of assigned tasks
• all decisions and powers specified and restricted by regulations
• officials with expert training in their fields
• career advancement dependent on technical qualifications
• qualifications evaluated by organizational rules, not by individuals

Weber listed several preconditions for the emergence of bureaucracy, including an increase in the amount of space and population being administered, an increase in the complexity of the administrative tasks being carried out, and the existence of a monetary economy requiring a more efficient administrative system. Development of communication and transportation technologies make more efficient administration possible, and democratization and rationalization of culture results in demands for equal treatment.

Although he was not necessarily an admirer of bureaucracy, Weber saw bureaucratization as the most efficient and rational way of organizing human activity and therefore as the key to rational-legal authority, indispensable to the modern world. Furthermore, he saw it as the key process in the ongoing rationalization of Western society. Weber also saw bureaucracy, however, as a threat to individual freedoms, and ongoing bureaucratization as leading to a "polar night of icy darkness", in which increasing rationalization of human life traps individuals in a soulless "iron cage" of bureaucratic, rule-based, rational control. Weber's critical study of the bureaucratization of society became one of the most enduring parts of his work. Many aspects of modern public administration are based on his work, and a classic, hierarchically organized civil service of the Continental type is called a "Weberian civil service" or a "Weberian bureaucracy". Social scientists debate whether Weberian bureaucracy contributes to economic growth.

Political scientist Jan Vogler challenges Max Weber's characterization of modern bureaucracies. Whereas Weber describes bureaucracies as entailing strict merit recruitment, clearly delineated career-paths for bureaucrats, the full separation of bureaucratic operations from politics, and mutually exclusive spheres of competence for government agencies, Vogler argues that the overwhelming majority of existing public administrative systems are not like this. Instead, modern bureaucracies require only "minimal competence" from candidates for bureaucratic offices, leaving space for biases in recruitment processes that give preferential treatment to members of specific social, economic, or ethnic groups, which are observed in many real-world bureaucratic systems. Bureaucracies are also not strictly separated from politics.

Woodrow Wilson

Writing as an academic while a professor at Bryn Mawr College, Woodrow Wilson's essay The Study of Administration argued for bureaucracy as a professional cadre, devoid of allegiance to fleeting politics. Wilson advocated a bureaucracy that:

...is a part of political life only as the methods of the counting house are a part of the life of society; only as machinery is part of the manufactured product. But it is, at the same time, raised very far above the dull level of mere technical detail by the fact that through its greater principles it is directly connected with the lasting maxims of political wisdom, the permanent truths of political progress.

Wilson did not advocate a replacement of rule by the governed, he simply advised that, "Administrative questions are not political questions. Although politics sets the tasks for administration, it should not be suffered to manipulate its offices". This essay became a foundation for the study of public administration in America.

Ludwig von Mises

In his 1944 work Bureaucracy, the Austrian economist Ludwig von Mises compared bureaucratic management to profit management. Profit management, he argued, is the most effective method of organization when the services rendered may be checked by economic calculation of profit and loss. When, however, the service in question cannot be subjected to economic calculation, bureaucratic management is necessary. He did not oppose universally bureaucratic management; on the contrary, he argued that bureaucracy is an indispensable method for social organization, for it is the only method by which the law can be made supreme, and is the protector of the individual against despotic arbitrariness. Using the example of the Catholic Church, he pointed out that bureaucracy is only appropriate for an organization whose code of conduct is not subject to change. He then went on to argue that complaints about bureaucratization usually refer not to the criticism of the bureaucratic methods themselves, but to "the intrusion of bureaucracy into all spheres of human life." Mises saw bureaucratic processes at work in both the private and public spheres; however, he believed that bureaucratization in the private sphere could only occur as a consequence of government interference. According to him, "What must be realized is only that the strait jacket of bureaucratic organization paralyzes the individual's initiative, while within the capitalist market society an innovator still has a chance to succeed. The former makes for stagnation and preservation of inveterate methods, the latter makes for progress and improvement."

Robert K. Merton

American sociologist Robert K. Merton expanded on Weber's theories of bureaucracy in his work Social Theory and Social Structure, published in 1957. While Merton agreed with certain aspects of Weber's analysis, he also noted the dysfunctional aspects of bureaucracy, which he attributed to a "trained incapacity" resulting from "over conformity". He believed that bureaucrats are more likely to defend their own entrenched interests than to act to benefit the organization as a whole but that pride in their craft makes them resistant to changes in established routines. Merton stated that bureaucrats emphasize formality over interpersonal relationships, and have been trained to ignore the special circumstances of particular cases, causing them to come across as "arrogant" and "haughty".

Elliott Jaques

In his book A General Theory of Bureaucracy, first published in 1976, Elliott Jaques describes the discovery of a universal and uniform underlying structure of managerial or work levels in the bureaucratic hierarchy for any type of employment systems.

Jaques argues and presents evidence that for the bureaucracy to provide a valuable contribution to the open society some of the following conditions must be met:

• The number of levels in the hierarchy of a bureaucracy must match the complexity level of the employment system for which the bureaucratic hierarchy is created. (Jaques identified a maximum of eight levels of complexity for bureaucratic hierarchies.)
• Roles within a bureaucratic hierarchy differ in the level of work complexity.
• The level of work complexity in the roles must be matched by the level of human capability of the role holders. (Jaques identified maximum of eight levels of human capability.)
• The level of work complexity in any managerial role within a bureaucratic hierarchy must be one level higher than the level of work complexity of the subordinate roles.
• Any managerial role in a bureaucratic hierarchy must have full managerial accountabilities and authorities (veto selection to the team, decide task types and specific task assignments, decide personal effectiveness and recognition, decide initiation of removal from the team within due process).
• Lateral working accountabilities and authorities must be defined for all the roles in the hierarchy (seven types of lateral working accountabilities and authorities: collateral, advisory, service-getting and -giving, coordinative, monitoring, auditing, prescribing).

Bureaucracy and democracy

See also: Bureaucratic drift, Bureaucratic inertia, and Representative bureaucracy

Like every modern state, a liberal democracy is highly bureaucratized, with numerous sizable organizations filled with career civil servants. Some of those bureaucracies have a substantial amount of influence to preserve the current political system because they are primarily focused on defending the country and the state from threats from both within and beyond. Since these institutions frequently operate independently and are mostly shielded from politics, they frequently have no affiliation with any particular political party or group. For instance, loyal British civil officials work for both the Conservative and Labour parties. However, on occasion a group might seize control of a bureaucratic state, as the Nazis did in Germany in the 1930s.

Although numerous ideals associated with democracy, such as equality, participation, and individuality, are in stark contrast to those associated with modern bureaucracy, specifically hierarchy, specialization, and impersonality, political theorists did not recognize bureaucracy as a threat to democracy. Yet, democratic theorists still have not developed an adequate response to the challenge posed by bureaucratic power within democratic governance.

One approach to addressing this issue rejects the idea that bureaucracy has any role at all in a true democracy. Theorists who adopt this perspective typically understand that they must demonstrate that bureaucracy does not necessarily occur in every contemporary society; only in those they perceive to be non-democratic. Thus, 19th century British writers frequently referred to bureaucracy as the "Continental nuisance," because their democracy was resistant to it, in their point of view.

According to Marx and other socialist thinkers, the most advanced bureaucracies were those in France and Germany. However, they argued that bureaucracy was a symptom of the bourgeois state and would vanish along with capitalism, which gave rise to the bourgeois state. Though clearly not the democracies Marx had in mind, socialist societies ended up being more bureaucratic than the governments they replaced. Similarly, after capitalist economies developed the administrative systems required to support their extensive welfare states, the idea that bureaucracy exclusively exists in socialist governments could scarcely be maintained.

Hexadecimal

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Hexadecimal

Hexadecimal (also known as base-16 or simply hex) is a positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbols, hexadecimal uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9 and "A"–"F" to represent values from ten to fifteen.

Software developers and system designers widely use hexadecimal numbers because they provide a convenient representation of binary-coded values. Each hexadecimal digit represents four bits (binary digits), also known as a nibble (or nybble). For example, an 8-bit byte is two hexadecimal digits and its value can be written as 00 to FF in hexadecimal.

In mathematics, a subscript is typically used to specify the base. For example, the decimal value 711 would be expressed in hexadecimal as 2C7₁₆. In programming, several notations denote hexadecimal numbers, usually involving a prefix. The prefix 0x is used in C, which would denote this value as 0x2C7.

Hexadecimal is used in the transfer encoding Base 16, in which each byte of the plain text is broken into two 4-bit values and represented by two hexadecimal digits.

Representation

Written representation

In most current use cases, the letters A–F or a–f represent the values 10–15, while the numerals 0–9 are used to represent their decimal values.

There is no universal convention to use lowercase or uppercase, so each is prevalent or preferred in particular environments by community standards or convention; even mixed case is used. Some seven-segment displays use mixed-case 'A b C d E F' to distinguish the digits A–F from one another and from 0–9.

There is some standardization of using spaces (rather than commas or another punctuation mark) to separate hex values in a long list. For instance, in the following hex dump, each 8-bit byte is a 2-digit hex number, with spaces between them, while the 32-bit offset at the start is an 8-digit hex number.

00000000  57 69 6b 69 70 65 64 69  61 2c 20 74 68 65 20 66  
00000010  72 65 65 20 65 6e 63 79  63 6c 6f 70 65 64 69 61  
00000020  20 74 68 61 74 20 61 6e  79 6f 6e 65 20 63 61 6e 
00000030  20 65 64 69 74 0a

Distinguishing from decimal

In contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript (itself written in decimal) can give the base explicitly: 159₁₀ is decimal 159; 159₁₆ is hexadecimal 159, which equals 345₁₀. Some authors prefer a text subscript, such as 159_decimal and 159_hex, or 159_d and 159_h.

Donald Knuth introduced the use of a particular typeface to represent a particular radix in his book The TeXbook. Hexadecimal representations are written there in a typewriter typeface: 5A3, C1F27ED

In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen:

• Although best known from the C programming language (and the many languages influenced by C), the prefix 0x to indicate a hex constant may have had origins in the IBM Stretch systems. It is derived from the 0 prefix already in use for octal constants. Byte values can be expressed in hexadecimal with the prefix \x followed by two hex digits: '\x1B' represents the Esc control character; "\x1B[0m\x1B[25;1H" is a string containing 11 characters with two embedded Esc characters. To output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used.
• In XML and XHTML, characters can be expressed as hexadecimal numeric character references using the notation &#xcode;, for instance T represents the character U+0054 (the uppercase letter "T"). If there is no x the number is decimal (thus T is the same character).
• In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h: FFh or 05A3H. Some implementations require a leading zero when the first hexadecimal digit character is not a decimal digit, so one would write 0FFh instead of FFh. Some other implementations (such as NASM) allow C-style numbers (0x42).
• Other assembly languages (6502, Motorola), Pascal, Delphi, some versions of BASIC (Commodore), GameMaker Language, Godot and Forth use $ as a prefix: $5A3, $C1F27ED.
• Some assembly languages (Microchip) use the notation H'ABCD' (for ABCD₁₆). Similarly, Fortran 95 uses Z'ABCD'.
• Ada and VHDL enclose hexadecimal numerals in based "numeric quotes": 16#5A3#, 16#C1F27ED#. For bit vector constants VHDL uses the notation x"5A3", x"C1F27ED".
• Verilog represents hexadecimal constants in the form 8'hFF, where 8 is the number of bits in the value and FF is the hexadecimal constant.
• The Icon and Smalltalk languages use the prefix 16r: 16r5A3
• PostScript and the Bourne shell and its derivatives denote hex with prefix 16#: 16#5A3, 16#C1F27ED.
• Common Lisp uses the prefixes #x and #16r. Setting the variables *read-base* and *print-base* to 16 can also be used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, when the input or output base has been changed to 16.
• MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H: &H5A3
• BBC BASIC and Locomotive BASIC use & for hex.
• TI-89 and 92 series uses a 0h prefix: 0h5A3, 0hC1F27ED
• ALGOL 68 uses the prefix 16r to denote hexadecimal numbers: 16r5a3, 16rC1F27ED. Binary, quaternary (base-4), and octal numbers can be specified similarly.
• The most common format for hexadecimal on IBM mainframes (zSeries) and midrange computers (IBM i) running the traditional OS's (zOS, zVSE, zVM, TPF, IBM i) is X'5A3' or X'C1F27ED', and is used in Assembler, PL/I, COBOL, JCL, scripts, commands and other places. This format was common on other (and now obsolete) IBM systems as well. Occasionally quotation marks were used instead of apostrophes.

Syntax that is always Hex

Sometimes the numbers are known to be Hex.

• In URIs (including URLs), character codes are written as hexadecimal pairs prefixed with %: http://www.example.com/name%20with%20spaces where %20 is the code for the space (blank) character, ASCII code point 20 in hex, 32 in decimal.
• In the Unicode standard, a character value is represented with U+ followed by the hex value, e.g. U+00A1 is the inverted exclamation point (¡).
• Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits (two each for the red, green and blue components, in that order) prefixed with #: magenta, for example, is represented as #FF00FF. CSS also allows 3-hexdigit abbreviations with one hexdigit per component: #FA3 abbreviates #FFAA33 (a golden orange:  ).
• In MIME (e-mail extensions) quoted-printable encoding, character codes are written as hexadecimal pairs prefixed with =: Espa=F1a is "España" (F1_hex is the code for ñ in the ISO/IEC 8859-1 character set).)
• PostScript binary data (such as image pixels) can be expressed as unprefixed consecutive hexadecimal pairs: AA213FD51B3801043FBC ...
• Any IPv6 address can be written as eight groups of four hexadecimal digits (sometimes called hextets), where each group is separated by a colon (:). This, for example, is a valid IPv6 address: 2001:0db8:85a3:0000:0000:8a2e:0370:7334 or abbreviated by removing leading zeros as 2001:db8:85a3::8a2e:370:7334 (IPv4 addresses are usually written in decimal).
• Globally unique identifiers are written as thirty-two hexadecimal digits, often in unequal hyphen-separated groupings, for example 3F2504E0-4F89-41D3-9A0C-0305E82C3301.

Other symbols for 10–15 and mostly different symbol sets

The use of the letters A through F to represent the digits above 9 was not universal in the early history of computers.

• During the 1950s, some installations, such as Bendix-14, favored using the digits 0 through 5 with an overline to denote the values 10–15 as 0, 1, 2, 3, 4 and 5.
• The SWAC (1950) and Bendix G-15 (1956) computers used the lowercase letters u, v, w, x, y and z for the values 10 to 15.
• The ORDVAC and ILLIAC I (1952) computers (and some derived designs, e.g. BRLESC) used the uppercase letters K, S, N, J, F and L for the values 10 to 15.
• The Librascope LGP-30 (1956) used the letters F, G, J, K, Q and W for the values 10 to 15.
• On the PERM (1956) computer, hexadecimal numbers were written as letters O for zero, A to N and P for 1 to 15. Many machine instructions had mnemonic hex-codes (A=add, M=multiply, L=load, F=fixed-point etc.); programs were written without instruction names.
• The Honeywell Datamatic D-1000 (1957) used the lowercase letters b, c, d, e, f, and g whereas the Elbit 100 (1967) used the uppercase letters B, C, D, E, F and G for the values 10 to 15.
• The Monrobot XI (1960) used the letters S, T, U, V, W and X for the values 10 to 15.
• The NEC parametron computer NEAC 1103 (1960) used the letters D, G, H, J, K (and possibly V) for values 10–15.
• The Pacific Data Systems 1020 (1964) used the letters L, C, A, S, M and D for the values 10 to 15.

Bibi-binary

• New numeric symbols and names were introduced in the Bibi-binary notation by Boby Lapointe in 1968.

Bruce Alan Martin's hexadecimal notation proposal

• Bruce Alan Martin of Brookhaven National Laboratory considered the choice of A–F "ridiculous". In a 1968 letter to the editor of the CACM, he proposed an entirely new set of symbols based on the bit locations.

Ronald O. Whitaker's hexadecimal notation proposal.

• In 1972, Ronald O. Whitaker of Rowco Engineering Co. proposed a triangular font that allows "direct binary reading" to "permit both input and output from computers without respect to encoding matrices."
• Some seven-segment display decoder chips (i.e., 74LS47) show unexpected output due to logic designed only to produce 0–9 correctly.

Verbal and digital representations

Hexadecimal finger-counting scheme

Since there were no traditional numerals to represent the quantities from ten to fifteen, alphabetic letters were re-employed as a substitute. Most European languages lack non-decimal-based words for some of the numerals eleven to fifteen. Some people read hexadecimal numbers digit by digit, like a phone number, or using the NATO phonetic alphabet, the Joint Army/Navy Phonetic Alphabet, or a similar ad-hoc system. In the wake of the adoption of hexadecimal among IBM System/360 programmers, Magnuson (1968) suggested a pronunciation guide that gave short names to the letters of hexadecimal – for instance, "A" was pronounced "ann", B "bet", C "chris", etc. Another naming-system was published online by Rogers (2007) that tries to make the verbal representation distinguishable in any case, even when the actual number does not contain numbers A–F. Examples are listed in the tables below. Yet another naming system was elaborated by Babb (2015), based on a joke in Silicon Valley. The system proposed by Babb was further improved by Atkins-Bittner in 2015-2016.

Others have proposed using the verbal Morse Code conventions to express four-bit hexadecimal digits, with "dit" and "dah" representing zero and one, respectively, so that "0000" is voiced as "dit-dit-dit-dit" (....), dah-dit-dit-dah (-..-) voices the digit with a value of nine, and "dah-dah-dah-dah" (----) voices the hexadecimal digit for decimal 15.

Systems of counting on digits have been devised for both binary and hexadecimal. Arthur C. Clarke suggested using each finger as an on/off bit, allowing finger counting from zero to 1023₁₀ on ten fingers. Another system for counting up to FF₁₆ (255₁₀) is illustrated on the right.

Magnuson (1968)
naming method

Number	Pronunciation
A	ann
B	bet
C	chris
D	dot
E	ernest
F	frost
1A	annteen
A0	annty
5B	fifty-bet
A,01C	annty christeen
1,AD0	annteen dotty
3,A7D	thirty-ann seventy-dot

Rogers (2007)
naming method

Number	Pronunciation
A	ten
B	eleven
C	twelve
D	draze
E	eptwin
F	fim
10	tex
11	oneteek
1F	fimteek
50	fiftek
C0	twelftek
100	hundrek
1,000	thousek
3E	thirtek-eptwin
E1	eptek-one
C4A	twelve-hundrek-fourtek-ten
1,743	one-thousek-seven- -hundrek-fourtek-three

Atkins-Bittner (2015)
naming method

Number	Pronunciation
A	ae
B	bee
C	cee
D	dee
E	ee
F	eff
A0	atta
B0	bitta
C0	citta
D0	dickety
E0	eckity
F0	fleventy
1A	abteen
1B	bibteen
1C	cibteen
1D	dibbleteen
1E	ebbleteen
1F	fleventeen
100	one bitey
10,000	one millby
100,000,000	one billby

Signs

The hexadecimal system can express negative numbers the same way as in decimal: −2A to represent −42₁₀, −B01D9 to represent −721369₁₀ and so on.

Hexadecimal can also be used to express the exact bit patterns used in the processor, so a sequence of hexadecimal digits may represent a signed or even a floating-point value. This way, the negative number −42₁₀ can be written as FFFF FFD6 in a 32-bit CPU register (in two's complement), as C228 0000 in a 32-bit FPU register or C045 0000 0000 0000 in a 64-bit FPU register (in the IEEE floating-point standard).

Hexadecimal exponential notation

Just as decimal numbers can be represented in exponential notation, so too can hexadecimal numbers. P notation uses the letter P (or p, for "power"), whereas E (or e) serves a similar purpose in decimal E notation. The number after the P is decimal and represents the binary exponent. Increasing the exponent by 1 multiplies by 2, not 16: 20p0 = 10p1 = 8p2 = 4p3 = 2p4 = 1p5. Usually, the number is normalized so that the hexadecimal digits start with 1. (zero is usually 0 with no P).

Example: 1.3DEp42 represents 1.3DE₁₆ × 2^42₁₀.

P notation is required by the IEEE 754-2008 binary floating-point standard and can be used for floating-point literals in the C99 edition of the C programming language. Using the %a or %A conversion specifiers, this notation can be produced by implementations of the printf family of functions following the C99 specification and Single Unix Specification (IEEE Std 1003.1) POSIX standard.

Conversion

Binary conversion

The programmable RPN-calculator HP-16C Computer Scientist from 1982 was designed for programmers. One of its key features was the conversion between different numeral systems (note hex number in display).

Most computers manipulate binary data, but it is difficult for humans to work with a large number of digits for even a relatively small binary number. Although most humans are familiar with the base 10 system, it is much easier to map binary to hexadecimal than to decimal because each hexadecimal digit maps to a whole number of bits (4₁₀). This example converts 1111₂ to base ten. Since each position in a binary numeral can contain either a 1 or a 0, its value may be easily determined by its position from the right:

• 0001₂ = 1₁₀
• 0010₂ = 2₁₀
• 0100₂ = 4₁₀
• 1000₂ = 8₁₀

Therefore:

11112	= 810 + 410 + 210 + 110
	= 1510

With little practice, mapping 1111₂ to F₁₆ in one step becomes easy (see table in written representation). The advantage of using hexadecimal rather than decimal increases rapidly with the size of the number. When the number becomes large, conversion to decimal is very tedious. However, when mapping to hexadecimal, it is trivial to regard the binary string as 4-digit groups and map each to a single hexadecimal digit.

This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results.

(1001011100)2	= 51210 + 6410 + 1610 + 810 + 410
	= 60410

Compare this to the conversion to hexadecimal, where each group of four digits can be considered independently and converted directly:

(1001011100)2	=	0010	0101	11002
	=	2	5	C16
	=	25C16

The conversion from hexadecimal to binary is equally direct.

Other simple conversions

Although quaternary (base 4) is little used, it can easily be converted to and from hexadecimal or binary. Each hexadecimal digit corresponds to a pair of quaternary digits, and each quaternary digit corresponds to a pair of binary digits. In the above example 2 5 C₁₆ = 02 11 30₄.

The octal (base 8) system can also be converted with relative ease, although not quite as trivially as with bases 2 and 4. Each octal digit corresponds to three binary digits, rather than four. Therefore, we can convert between octal and hexadecimal via an intermediate conversion to binary followed by regrouping the binary digits in groups of either three or four.

Division-remainder in source base

As with all bases there is a simple algorithm for converting a representation of a number to hexadecimal by doing integer division and remainder operations in the source base. In theory, this is possible from any base, but for most humans, only decimal and for most computers, only binary (which can be converted by far more efficient methods) can be easily handled with this method.

Let d be the number to represent in hexadecimal, and the series h_ih_i−1...h₂h₁ be the hexadecimal digits representing the number.

1. i ← 1
2. h_i ← d mod 16
3. d ← (d − h_i) / 16
4. If d = 0 (return series h_i) else increment i and go to step 2

"16" may be replaced with any other base that may be desired.

The following is a JavaScript implementation of the above algorithm for converting any number to a hexadecimal in String representation. Its purpose is to illustrate the above algorithm. To work with data seriously, however, it is much more advisable to work with bitwise operators.

function toHex(d) {
  var r = d % 16;
  if (d - r == 0) {
    return toChar(r);
  }
  return toHex((d - r) / 16) + toChar(r);
}

function toChar(n) {
  const alpha = "0123456789ABCDEF";
  return alpha.charAt(n);
}

Conversion through addition and multiplication

A hexadecimal multiplication table

It is also possible to make the conversion by assigning each place in the source base the hexadecimal representation of its place value — before carrying out multiplication and addition to get the final representation. For example, to convert the number B3AD to decimal, one can split the hexadecimal number into its digits: B (11₁₀), 3 (3₁₀), A (10₁₀) and D (13₁₀), and then get the final result by multiplying each decimal representation by 16^p (p being the corresponding hex digit position, counting from right to left, beginning with 0). In this case, we have that:

B3AD = (11 × 16³) + (3 × 16²) + (10 × 16¹) + (13 × 16⁰)

which is 45997 in base 10.

Tools for conversion

Many computer systems provide a calculator utility capable of performing conversions between the various radices frequently including hexadecimal.

In Microsoft Windows, the Calculator, on its Programmer mode, allows conversions between hexadecimal and other common programming bases.

Elementary arithmetic

Elementary operations such as division can be carried out indirectly through conversion to an alternate numeral system, such as the commonly used decimal system or the binary system where each hex digit corresponds to four binary digits.

Alternatively, one can also perform elementary operations directly within the hex system itself — by relying on its addition/multiplication tables and its corresponding standard algorithms such as long division and the traditional subtraction algorithm.

Real numbers

Rational numbers

As with other numeral systems, the hexadecimal system can be used to represent rational numbers, although repeating expansions are common since sixteen (10₁₆) has only a single prime factor: two.

For any base, 0.1 (or "1/10") is always equivalent to one divided by the representation of that base value in its own number system. Thus, whether dividing one by two for binary or dividing one by sixteen for hexadecimal, both of these fractions are written as 0.1. Because the radix 16 is a perfect square (4²), fractions expressed in hexadecimal have an odd period much more often than decimal ones, and there are no cyclic numbers (other than trivial single digits). Recurring digits are exhibited when the denominator in lowest terms has a prime factor not found in the radix; thus, when using hexadecimal notation, all fractions with denominators that are not a power of two result in an infinite string of recurring digits (such as thirds and fifths). This makes hexadecimal (and binary) less convenient than decimal for representing rational numbers since a larger proportion lies outside its range of finite representation.

All rational numbers finitely representable in hexadecimal are also finitely representable in decimal, duodecimal and sexagesimal: that is, any hexadecimal number with a finite number of digits also has a finite number of digits when expressed in those other bases. Conversely, only a fraction of those finitely representable in the latter bases are finitely representable in hexadecimal. For example, decimal 0.1 corresponds to the infinite recurring representation 0.19 in hexadecimal. However, hexadecimal is more efficient than duodecimal and sexagesimal for representing fractions with powers of two in the denominator. For example, 0.0625₁₀ (one-sixteenth) is equivalent to 0.1₁₆, 0.09₁₂, and 0;3,45₆₀.

n	Decimal Prime factors of: base, b = 10: 2, 5; b − 1 = 9: 3; b + 1 = 11: 11			Hexadecimal Prime factors of: base, b = 1610 = 10: 2; b − 1 = 1510 = F: 3, 5; b + 1 = 1710 = 11: 11
	Reciprocal	Prime factors	Positional representation (decimal)	Positional representation (hexadecimal)	Prime factors	Reciprocal
2	1/2	2	0.5	0.8	2	1/2
3	1/3	3	0.3333... = 0.3	0.5555... = 0.5	3	1/3
4	1/4	2	0.25	0.4	2	1/4
5	1/5	5	0.2	0.3	5	1/5
6	1/6	2, 3	0.16	0.2A	2, 3	1/6
7	1/7	7	0.142857	0.249	7	1/7
8	1/8	2	0.125	0.2	2	1/8
9	1/9	3	0.1	0.1C7	3	1/9
10	1/10	2, 5	0.1	0.19	2, 5	1/A
11	1/11	11	0.09	0.1745D	B	1/B
12	1/12	2, 3	0.083	0.15	2, 3	1/C
13	1/13	13	0.076923	0.13B	D	1/D
14	1/14	2, 7	0.0714285	0.1249	2, 7	1/E
15	1/15	3, 5	0.06	0.1	3, 5	1/F
16	1/16	2	0.0625	0.1	2	1/10
17	1/17	17	0.0588235294117647	0.0F	11	1/11
18	1/18	2, 3	0.05	0.0E38	2, 3	1/12
19	1/19	19	0.052631578947368421	0.0D79435E5	13	1/13
20	1/20	2, 5	0.05	0.0C	2, 5	1/14
21	1/21	3, 7	0.047619	0.0C3	3, 7	1/15
22	1/22	2, 11	0.045	0.0BA2E8	2, B	1/16
23	1/23	23	0.0434782608695652173913	0.0B21642C859	17	1/17
24	1/24	2, 3	0.0416	0.0A	2, 3	1/18
25	1/25	5	0.04	0.0A3D7	5	1/19
26	1/26	2, 13	0.0384615	0.09D8	2, D	1/1A
27	1/27	3	0.037	0.097B425ED	3	1/1B
28	1/28	2, 7	0.03571428	0.0924	2, 7	1/1C
29	1/29	29	0.0344827586206896551724137931	0.08D3DCB	1D	1/1D
30	1/30	2, 3, 5	0.03	0.08	2, 3, 5	1/1E
31	1/31	31	0.032258064516129	0.08421	1F	1/1F
32	1/32	2	0.03125	0.08	2	1/20
33	1/33	3, 11	0.03	0.07C1F	3, B	1/21
34	1/34	2, 17	0.02941176470588235	0.078	2, 11	1/22
35	1/35	5, 7	0.0285714	0.075	5, 7	1/23
36	1/36	2, 3	0.027	0.071C	2, 3	1/24
37	1/37	37	0.027	0.06EB3E453	25	1/25
38	1/38	2, 19	0.02631578947368421	0.0435E50D79435E4AC62B4	2, 13	1/26
39	1/39	3, 13	0.0256410	0.069	3, D	1/27
40	1/40	2, 5	0.025	0.06	2, 5	1/28
41	1/41	41	0.02439	0.063E9538D283B5B62FB8	29	1/29
42	1/42	2, 3, 7	0.0238095	0.0618	2, 3, 7	1/2A
43	1/43	43	0.023255813953488372093	0.05F417D	2B	1/2B
44	1/44	2, 11	0.0227	0.05D1745	2, B	1/2C
45	1/45	3, 5	0.02	0.05B	3, 5	1/2D
46	1/46	2, 23	0.02173913043478765869567	0.0590B21642C9C4EF44A9	2, 17	1/2E
47	1/47	47	0.0212765957446808510638297872340425531914893617	0.0572620AE4C415C9882B931	2F	1/2F
48	1/48	2, 3	0.02083	0.05	2, 3	1/30

Irrational numbers

The table below gives the expansions of some common irrational numbers in decimal and hexadecimal.

Number	Positional representation
	Decimal	Hexadecimal
√2 (the length of the diagonal of a unit square)	1.414213562373095048...	1.6A09E667F3BCD...
√3 (the length of the diagonal of a unit cube)	1.732050807568877293...	1.BB67AE8584CAA...
√5 (the length of the diagonal of a 1×2 rectangle)	2.236067977499789696...	2.3C6EF372FE95...
φ (phi, the golden ratio = (1+√5)/2)	1.618033988749894848...	1.9E3779B97F4A...
π (pi, the ratio of circumference to diameter of a circle)	3.141592653589793238462643 383279502884197169399375105...	3.243F6A8885A308D313198A2E0 3707344A4093822299F31D008...
e (the base of the natural logarithm)	2.718281828459045235...	2.B7E151628AED2A6B...
τ (the Thue–Morse constant)	0.412454033640107597...	0.6996 9669 9669 6996...
γ (the limiting difference between the harmonic series and the natural logarithm)	0.577215664901532860...	0.93C467E37DB0C7A4D1B...

Powers

Powers of two have very simple expansions in hexadecimal. The first sixteen powers of two are shown below.

2x	Value	Value (Decimal)
20	1	1
21	2	2
22	4	4
23	8	8
24	10hex	16dec
25	20hex	32dec
26	40hex	64dec
27	80hex	128dec
28	100hex	256dec
29	200hex	512dec
2A (210dec)	400hex	1,024dec
2B (211dec)	800hex	2,048dec
2C (212dec)	1,000hex	4,096dec
2D (213dec)	2,000hex	8,192dec
2E (214dec)	4,000hex	16,384dec
2F (215dec)	8,000hex	32,768dec
210 (216dec)	10,000hex	65,536dec

Cultural history

The traditional Chinese units of measurement were base-16. For example, one jīn (斤) in the old system equals sixteen taels. The suanpan (Chinese abacus) can be used to perform hexadecimal calculations such as additions and subtractions.

As with the duodecimal system, there have been occasional attempts to promote hexadecimal as the preferred numeral system. These attempts often propose specific pronunciation and symbols for the individual numerals. Some proposals unify standard measures so that they are multiples of 16. An early such proposal was put forward by John W. Nystrom in Project of a New System of Arithmetic, Weight, Measure and Coins: Proposed to be called the Tonal System, with Sixteen to the Base, published in 1862. Nystrom among other things suggested hexadecimal time, which subdivides a day by 16, so that there are 16 "hours" (or "10 tims", pronounced tontim) in a day.

Look up hexadecimal in Wiktionary, the free dictionary.

The word hexadecimal is first recorded in 1952. It is macaronic in the sense that it combines Greek ἕξ (hex) "six" with Latinate -decimal. The all-Latin alternative sexadecimal (compare the word sexagesimal for base 60) is older, and sees at least occasional use from the late 19th century. It is still in use in the 1950s in Bendix documentation. Schwartzman (1994) argues that use of sexadecimal may have been avoided because of its suggestive abbreviation to sex. Many western languages since the 1960s have adopted terms equivalent in formation to hexadecimal (e.g. French hexadécimal, Italian esadecimale, Romanian hexazecimal, Serbian хексадецимални, etc.) but others have introduced terms which substitute native words for "sixteen" (e.g. Greek δεκαεξαδικός, Icelandic sextándakerfi, Russian шестнадцатеричной etc.)

Terminology and notation did not become settled until the end of the 1960s. In 1969, Donald Knuth argued that the etymologically correct term would be senidenary, or possibly sedenary, a Latinate term intended to convey "grouped by 16" modelled on binary, ternary, quaternary, etc. According to Knuth's argument, the correct terms for decimal and octal arithmetic would be denary and octonary, respectively. Alfred B. Taylor used senidenary in his mid-1800s work on alternative number bases, although he rejected base 16 because of its "incommodious number of digits".

The now-current notation using the letters A to F establishes itself as the de facto standard beginning in 1966, in the wake of the publication of the Fortran IV manual for IBM System/360, which (unlike earlier variants of Fortran) recognizes a standard for entering hexadecimal constants. As noted above, alternative notations were used by NEC (1960) and The Pacific Data Systems 1020 (1964). The standard adopted by IBM seems to have become widely adopted by 1968, when Bruce Alan Martin in his letter to the editor of the CACM complains that

With the ridiculous choice of letters A, B, C, D, E, F as hexadecimal number symbols adding to already troublesome problems of distinguishing octal (or hex) numbers from decimal numbers (or variable names), the time is overripe for reconsideration of our number symbols. This should have been done before poor choices gelled into a de facto standard!

Martin's argument was that use of numerals 0 to 9 in nondecimal numbers "imply to us a base-ten place-value scheme": "Why not use entirely new symbols (and names) for the seven or fifteen nonzero digits needed in octal or hex. Even use of the letters A through P would be an improvement, but entirely new symbols could reflect the binary nature of the system". He also argued that "re-using alphabetic letters for numerical digits represents a gigantic backward step from the invention of distinct, non-alphabetic glyphs for numerals sixteen centuries ago" (as Brahmi numerals, and later in a Hindu–Arabic numeral system), and that the recent ASCII standards (ASA X3.4-1963 and USAS X3.4-1968) "should have preserved six code table positions following the ten decimal digits -- rather than needlessly filling these with punctuation characters" (":;<=>?") that might have been placed elsewhere among the 128 available positions.

Base16 (transfer encoding)

Base16 (as a proper name without a space) can also refer to a binary to text encoding belonging to the same family as Base32, Base58, and Base64.

In this case, data is broken into 4-bit sequences, and each value (between 0 and 15 inclusively) is encoded using one of 16 symbols from the ASCII character set. Although any 16 symbols from the ASCII character set can be used, in practice, the ASCII digits "0"–"9" and the letters "A"–"F" (or the lowercase "a"–"f") are always chosen in order to align with standard written notation for hexadecimal numbers.

There are several advantages of Base16 encoding:

• Most programming languages already have facilities to parse ASCII-encoded hexadecimal
• Being exactly half a byte, 4-bits is easier to process than the 5 or 6 bits of Base32 and Base64 respectively
• The symbols 0–9 and A–F are universal in hexadecimal notation, so it is easily understood at a glance without needing to rely on a symbol lookup table.
• Many CPU architectures have dedicated instructions that allow access to a half-byte (otherwise known as a "nibble"), making it more efficient in hardware than Base32 and Base64

The main disadvantages of Base16 encoding are:

• Space efficiency is only 50%, since each 4-bit value from the original data will be encoded as an 8-bit byte. In contrast, Base32 and Base64 encodings have a space efficiency of 63% and 75% respectively.
• Possible added complexity of having to accept both uppercase and lowercase letters

Support for Base16 encoding is ubiquitous in modern computing. It is the basis for the W3C standard for URL percent encoding, where a character is replaced with a percent sign "%" and its Base16-encoded form. Most modern programming languages directly include support for formatting and parsing Base16-encoded numbers.