The Khawarij, Kharijites, or the ash-Shurah were a sect that appeared in the first century of Islam during the First Fitna, the crisis of leadership after the death of Muhammad. It broke into revolt against the authority of the Caliph Ali after he agreed to arbitration with his rival, Muawiyah I, to decide the succession to the Caliphate following the Battle of Siffin. A Khariji assassinated Ali, for hundreds of years, the Khawarij were a source of insurrection against the Caliphate; the Khawarij opposed arbitration as a means to choose a new ruler on the grounds that "judgement belongs to God alone". They considered arbitration a means for people to make decisions while the victor in a battle was determined by God, they believed that any Muslim—even one, not a Quraysh or an Arab—could be the Imam, the leader of the community, if he was morally irreproachable. If the leader sinned, it was the duty of Muslims to depose him; the only surviving sect of Khawarij are the Ibadis. The term al-Khariji was used as an exonym by their opponents from the fact that they left Ali's army.

The name comes from the Arabic root خ ر ج, which has the primary meaning "to leave" or "to get out", as in the basic word خرج "to go out", "to walk out", "to come out", etc. However, these groups called themselves ash-Shurah "the Exchangers", which they understood within the context of Islamic scripture and philosophy to mean "those who have traded the mortal life for the other life "; the origin of Kharijism lies in the First Fitna, the struggle for political supremacy over the Muslim community in the years following the death of Muhammad. After the death of the third Rashidun Caliph, Uthman, a struggle for succession ensued between Ali and Muawiyah I, the governor of Syria and cousin of Uthman, in league with a variety of other opponents. In 657, Ali's forces met Muawiyah's at the Battle of Siffin; the battle went against Muawiyah but on the brink of defeat, Muawiyah directed his army to hoist Qurans on their lances. Mu'awiya proposed to Ali to settle their dispute through arbitration, with each side appointing referees who would pronounce judgment according to the Quran.

While most of Ali's army accepted the proposal, one group from the tribe of Tamim, vehemently objected to the arbitration and left the ranks of Ali's army. These dissenters, who initiated what would become known as the Kharijite movement, wished to secede from Ali's army in order to uphold their principles, they held that the third caliph Uthman had deserved his death because of his faults, that Ali was the legitimate caliph, while Mu'awiya was a rebel. They believed that the Quran stated that as a rebel Mu'awiya was not entitled to arbitration, but rather should be fought until he repented, pointing to the verse: If two parties of the faithful fight each other conciliate them, yet if one is rebellious to the other fight the insolent one until it returns to God 's command. The dissenters held that in agreeing to arbitration Ali committed the grave sin of rejecting God's judgment and attempted to substitute human judgment for God's clear injunction, which prompted their motto la hukma illa li-llah.

From this expression, which they were the first to use as a motto, they became known as Muhakkima. They believed that Muslims owe allegiance only to the Quran and the sunna of Muhammad, Abu Bakr, Umar, denied that the right to the imamate should be based on close kinship with Muhammad; the initial group of dissenters went to the village of Harura' near Kufa, where they elected an obscure soldier named Ibn Wahb al-Rasibi as their leader. This gave rise to their alternative name, al-Haruriyya. Other defectors from Kufa, where Ali's army had returned awaiting the outcome of arbitration joined the dissenters, while Ali persuaded some dissenters to return to Kufa. However, when the arbitration ended in a verdict unfavorable to Ali, a large number of his followers left Kufa to join Ibn Wahb, who had meanwhile moved his camp to another location along the Nahrawan canal. At this point, the Kharijites proclaimed Ali's caliphate to be null and void and began to denounce as infidels anyone who did not accept their point of view.

From Nahrawan they began to raid his territories. When attempts at conciliation failed, Ali's forces attacked the Kharijites in their camp, inflicting a heavy defeat on them at the Battle of Nahrawan in 658, killing Ibn Wahb and most of his supporters; this bloodshed sealed the split of Kharijites from Ali's followers, Kharijite calls for revenge led to Ali's assassination in 661 by a Kharijite. For hundreds of years the Khawarij continued to be a source of insurrection against the Caliphate. and they aroused condemnation by mainstream scholars such as 14th-century Muslim Ismail ibn Kathir who wrote, "If they gained strength, they would corrupt the whole of the Earth and Shaam – they would not leave a baby, male or female, neither a man or a woman, because as far as they are concerned the people have caused corruption, a corruption that cannot be rectified except by mass killing." In a similar vein, the 10th century Islamic scholar Abu Bakr al-Ajurri said, "None of the scholars, in either past or recent times disagreed that the Khawarij are an evil group, disobedient to Allah Almighty and to His Messenger - Peace Be Upon Him.

If they pray, fast, or strive in worship, it does not benefit them, if they enjoin good and forbid evil it does not benefit them, as they are a people who interpret the Quran according to their desire." One modern historia

Leucospermum oleifolium

Leucospermum oleifolium is an erect shrub of about 1 m high and 1½ m across, assigned to the family Proteaceae. It has spreading branches, densely set with felty, oval, olive-colored leaves of about 3½ cm long and 1½ cm wide, with a bony tip that sometimes has two to five blunt teeth, with a blunt base and conspicuous veins; the flowers and their long thread-like styles are sulfur yellow, but soon become orange and turn brilliant crimson. The flower heads are about 4 cm in diameter, crowded at the tip of the branches with a maximum of five that start flowering in turn; this provides for a colour spectacle from August till December. It is called by various names in South Africa such as Overberg pincushion, flame pincushion, mix pincushion and tuft pincushion, it occurs in fynbos in the Western Cape province of South Africa. Leucospermum oleifolium is an erect compact and rigid shrub of about 1 m high and 1½ m across with a single stem at its base, branches bending upwards. Stems that are ready to flower are 3–6 mm in diameter hairy when young, but losing these fine hairs.

The leaves have no stalks or bracts at their base, are softly hairy but losing its indumentum when aging, 4–6 cm long and 8–25 mm wide, with an entire, sometimes wavy margin, a bony tip that may have one to five blunt teeth. The cup-shaped flower heads are each 2½–4 cm in diameter without a peduncle, with two to five crowded at the end of the branches on its own; the floral base is flat and 12 mm in diameter. It is covered below by felty to hairless, egg-shaped, long pointed, overlapping involucral bracts of 9–36 mm long and 5–7 mm wide, sometimes with a tuft of long hairs at its tip; the papery bracts at the base of the individual flowers are narrowly lance-shaped, 1–3 cm long, wooly near the base and sofly hairy towards the tip. The individual flower bud is a straight tube of about 2 cm long transparante whitish-transparent to pale yellowish green, yellow when opening becoming orange and turning bright crimson with age; when the flower opens, a cylindrical, hairless tube of 8 mm remains that widens towards the top, four thread-shaped lobes that are curled on itself.

The styles are strongly arched like a swan's neck, but straighten and grow to a thread of 2½–3 cm long, at first pale yellow but turning crimson when developed. The pollen-presenter, a slight thickening at the tip of the style, is cylindrical, thread-like, only at its base thicker than the style, 1 mm long, the stigma a groove across the tip of the pollen-presenter. At the base of the ovary are for blunt thread-like opaque scales of about 2 mm long; the fruit is ellipsoid in shape, 7½ mm long, the surface thinly covered with a fine powder. The species flowers between August and January, peaking in October; the flowers are bright yellow, but soon turn orange to end in a brilliant crimson, they may remain vibrant for two months. The subtribe Proteinae, to which the genus Leucospermum has been assigned has a basic chromosome number of twelve; this species was first described as Leucadendron oleaefolium in the Kungliga Vetenskaps Academiens Handlingar in 1766 by Peter Jonas Bergius. He based this description on a dried specimen, donated to him by the director of the Swedish East India Company Michael Grubb, who had purchased a collection of dried plants in 1764 in the Dutch Cape Colony from Johann Andreas Auge on a return trip from China.

Carl Peter Thunberg described in 1781 another specimen as Protea crinita, which Carl Linnaeus the Younger renamed to Protea criniflora during that same year. Jean-Baptiste Lamarck described a third specimen as Protea venosa in 1792. In 1809, Joseph Knight published a book titled On the cultivation of the plants belonging to the natural order of Proteeae, that contained an extensive revision of the Proteaceae attributed to Richard Anthony Salisbury. Salisbury assigned Thunberg's specimen to his new genus Leucadendrum and called it Leucadendrum criniflorum, it is assumed that Salisbury had based his review on a draft he seen of a paper called On the natural order of plants called Proteaceae that Robert Brown was to publish in 1810. Brown assigned the specimens of Bergius and Thunberg to the genus Leucospermum as L. oleaefolium and L. crinitum, added a description of a further specimen as L. molle. The French botanist Jean Poiret assigned that last specimen to the genus Protea in 1816, making the new combination P. mollis.

Ernst Gottlieb von Steudel assigned crinitum in 1841 to the genus Leucadendron. Heinrich Wilhelm Buek created the names Leucospermum penicillatum and L. cryptanthum in a book by Johann Franz Drège from 1843, Carl Meissner provided in 1856 a description for L. penicillatum. Otto Kuntze moved penicillatum in 1891 to Leucadendron. Michel Gandoger added Leucospermum schinzianum in 1913. John Patrick Rourke in 1970, regards all these names as synonymous. During the 1970s, the International Botanical Congress decided to replace all instances of compounding by the genitive "ae" by "i", thus changing the spelling of the name of this species to Leucospermum oleifolium. L. oleifolium has been assigned to the section Crinitae. The species name oleifolium means olive-leaf. Leucospermum oleifolium ranges from Pringle Bay in the south, via the Kogelberg Nature Reserve, Hottentots Holland Mountains, Franschhoek

Quaternary numeral system

Quaternary is the base-4 numeral system. It uses the digits 1, 2 and 3 to represent any real number. Four is the largest number within the subitizing range and one of two numbers, both a square and a composite number, making quaternary a convenient choice for a base at this scale. Despite being twice as large, its radix economy is equal to that of binary. However, it fares no better in the localization of prime numbers. Quaternary shares with all fixed-radix numeral systems many properties, such as the ability to represent any real number with a canonical representation and the characteristics of the representations of rational numbers and irrational numbers. See decimal and binary for a discussion of these properties; as with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix 4, 8 and 16 is a power of 2, so the conversion to and from binary is implemented by matching each digit with 2, 3 or 4 binary digits, or bits. For example, in base 4, 2302104 = 10 11 00 10 01 002.

Since 16 is a power of 4, conversion between these bases can be implemented by matching each hexadecimal digit with 2 quaternary digits. In the above example, 23 02 104 = B2416Although octal and hexadecimal are used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, quaternary does not enjoy the same status. Although quaternary has limited practical use, it can be helpful if it is necessary to perform hexadecimal arithmetic without a calculator; each hexadecimal digit can be turned into a pair of quaternary digits, arithmetic can be performed easily before converting the end result back to hexadecimal. Quaternary is convenient for this purpose, since numbers have only half the digit length compared to binary, while still having simple multiplication and addition tables with only three unique non-trivial elements. By analogy with byte and nybble, a quaternary digit is sometimes called a crumb. Due to having only factors of two, many quaternary fractions have repeating digits, although these tend to be simple: Many or all of the Chumashan languages used a base 4 counting system, in which the names for numbers were structured according to multiples of 4 and 16.

There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca. 1819. The Kharosthi numerals have a partial base 4 counting system from 1 to decimal 10. Quaternary numbers are used in the representation of 2D Hilbert curves. Here a real number between 0 and 1 is converted into the quaternary system; every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected. Parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA; the four DNA nucleotides in alphabetical order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0, 1, 2, 3. With this encoding, the complementary digit pairs 0↔3, 1↔2 match the complementation of the base pairs: A↔T and C↔G and can be stored as data in DNA sequence. For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010. Quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits.

Some computers have used quaternary floating point arithmetic including the Illinois ILLIAC II and the Digital Field System DFS IV and DFS V high-resolution site survey systems. Conversion between bases Moser–de Bruijn sequence, the numbers that have only 0 or 1 as their base-4 digits Quaternary Base Conversion, includes fractional part, from Math Is Fun Base42 Proposes unique symbols for Quaternary and Hexadecimal digits