1.
Integer
–
An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain

2.
Negative number
–
In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers

3.
100 (number)
–
100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the hundred or five score in order to differentiate the English. The standard SI prefix for a hundred is hecto-,100 is the basis of percentages, with 100% being a full amount. 100 is the sum of the first nine prime numbers, as well as the sum of pairs of prime numbers e. g.3 +97,11 +89,17 +83,29 +71,41 +59. 100 is the sum of the cubes of the first four integers and this is related by Nicomachuss theorem to the fact that 100 also equals the square of the sum of the first four integers,100 =102 =2. 26 +62 =100, thus 100 is a Leyland number and it is divisible by the number of primes below it,25 in this case. It can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient and it can be expressed as a sum of some of its divisors, making it a semiperfect number. 100 is a Harshad number in base 10, and also in base 4, there are exactly 100 prime numbers whose digits are in strictly ascending order. 100 is the smallest number whose common logarithm is a prime number,100 senators are in the U. S One hundred is the atomic number of fermium, an actinide. On the Celsius scale,100 degrees is the temperature of pure water at sea level. The Kármán line lies at an altitude of 100 kilometres above the Earths sea level and is used to define the boundary between Earths atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of Rosh Hashana, a religious Jew is expected to utter at least 100 blessings daily. In Hindu Religion - Mythology Book Mahabharata - Dhritarashtra had 100 sons known as kauravas, the United States Senate has 100 Senators. Most of the currencies are divided into 100 subunits, for example, one euro is one hundred cents. The 100 Euro banknotes feature a picture of a Rococo gateway on the obverse, the U. S. hundred-dollar bill has Benjamin Franklins portrait, the Benjamin is the largest U. S. bill in print. American savings bonds of $100 have Thomas Jeffersons portrait, while American $100 treasury bonds have Andrew Jacksons portrait, One hundred is also, The number of years in a century. The number of pounds in an American short hundredweight, in Greece, India, Israel and Nepal,100 is the police telephone number. In Belgium,100 is the ambulance and firefighter telephone number, in United Kingdom,100 is the operator telephone number

4.
Factorization
–
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 ×5, in all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viètes formulas relate the coefficients of a polynomial to its roots, the opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms. Integer factorization for large integers appears to be a difficult problem, there is no known method to carry it out quickly. Its complexity is the basis of the security of some public key cryptography algorithms. A matrix can also be factorized into a product of matrices of special types, One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types, QR decomposition, LQ, QL, RQ and this situation is generalized by factorization systems. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm, for very large numbers, no efficient classical algorithm is known. Modern techniques for factoring polynomials are fast and efficient, but use sophisticated mathematical ideas and these techniques are used in the construction of computer routines for carrying out polynomial factorization in Computer algebra systems. This article is concerned with classical techniques. While the general notion of factoring just means writing an expression as a product of simpler expressions, when factoring polynomials this means that the factors are to be polynomials of smaller degree. Thus, while x 2 − y = is a factorization of the expression, another issue concerns the coefficients of the factors. It is not always possible to do this, and a polynomial that can not be factored in this way is said to be irreducible over this type of coefficient, thus, x2 -2 is irreducible over the integers and x2 +4 is irreducible over the reals. In the first example, the integers 1 and -2 can also be thought of as real numbers, and if they are, then x 2 −2 = shows that this polynomial factors over the reals. Similarly, since the integers 1 and 4 can be thought of as real and hence complex numbers, x2 +4 splits over the complex numbers, i. e. x 2 +4 =. The fundamental theorem of algebra can be stated as, Every polynomial of n with complex number coefficients splits completely into n linear factors

5.
Greek numerals
–
Greek numerals are a system of writing numbers using the letters of the Greek alphabet. These alphabetic numerals are known as Ionic or Ionian numerals, Milesian numerals. In modern Greece, they are used for ordinal numbers. For ordinary cardinal numbers, however, Greece uses Arabic numerals, attic numerals, which were later adopted as the basis for Roman numerals, were the first alphabetic set. They were acrophonic, derived from the first letters of the names of the numbers represented and they ran =1, =5, =10, =100, =1000, and =10000. 50,500,5000, and 50000 were represented by the letter with minuscule powers of ten written in the top right corner, the same system was used outside of Attica, but the symbols varied with the local alphabets, in Boeotia, was 1000. The present system probably developed around Miletus in Ionia, 19th-century classicists placed its development in the 3rd century BC, the occasion of its first widespread use. The present system uses the 24 letters adopted by Euclid as well as three Phoenician and Ionic ones that were not carried over, digamma, koppa, and sampi. The position of characters within the numbering system imply that the first two were still in use while the third was not. Greek numerals are decimal, based on powers of 10, the units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well and this alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total. For example,241 was represented as, in ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars, α, β, γ, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as χξϛ, although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early. These new letter forms sometimes replaced the ones, especially in the case of the obscure numerals. The old Q-shaped koppa began to be broken up and simplified, the numeral for 6 changed several times. During antiquity, the letter form of digamma came to be avoided in favor of a special numerical one. By the Byzantine era, the letter was known as episemon and this eventually merged with the sigma-tau ligature stigma. In modern Greek, a number of changes have been made

6.
Roman numerals
–
The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX. Both methods appear in documents from the Roman era, even within the same document, double subtractives also occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, however, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used

7.
Binary number
–
The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, 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 rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, 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, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra

8.
Ternary numeral system
–
The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary. The value of a number with n bits that are all 1 is 2n −1. Then N = M, N = /, and N = bd −1, for a three-digit ternary number, N =33 −1 =26 =2 ×32 +2 ×31 +2 ×30 =18 +6 +2. Nonary or septemvigesimal can be used for representation of ternary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary and this is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open, in this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a reference, or at a certain voltage level. Thus, the voltage level is sometimes unpredictable. A rare ternary point is used to denote fractional parts of an inning in baseball, since each inning consists of three outs, each out is considered one third of an inning and is denoted as.1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like the Sierpinski triangle or the Cantor set conveniently, additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have an expression that does not contain any instance of the digit 1

9.
Quaternary numeral system
–
Quaternary is the base-4 numeral system. It uses the digits 0,1,2 and 3 to represent any real number. Four is the largest number within the range and one of two numbers that is both a square and a highly composite number, making quaternary a convenient choice for a base at this scale. Despite being twice as large, its economy is equal to that of binary. However, it no better in the localization of prime 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, for example, in base 4,302104 =11001001002. Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, by analogy with byte and nybble, a quaternary digit is sometimes called a crumb. There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca, 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 order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0,1,2. With this encoding, the complementary digit pairs 0↔3, and 1↔2 match the complementation of the pairs, A↔T and C↔G. 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

10.
Quinary
–
Quinary is a numeral system with five as the base. A possible origination of a system is that there are five fingers on either hand. The base five is stated from 0–4, in the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100, today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a system, is sexagesimal, base 60. Each quinary digit has log25 bits of information, many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true 5–25 language, in which 25 is the group of 5. The Gumatj numerals are shown below, In the video game Riven and subsequent games of the Myst franchise, a decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are a biquinary system, the numbers 1,5,10, and 50 are written as I, V, X, and L respectively. Eight is VIII and seventy is LXX, most versions of the abacus use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary, units of currencies are commonly partially or wholly biquinary. A vigesimal system with 4 and 5 as a sub-bases is found in Nahuatl, pentimal system Quibinary Yan Tan Tethera References, Quinary Base Conversion, includes fractional part, from Math Is Fun Media related to Quinary numeral system at Wikimedia Commons

11.
Senary
–
The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion

12.
Octal
–
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping binary digits into groups of three. For example, the representation for decimal 74 is 1001010. Two zeroes can be added at the left,1001010, corresponding the octal digits 112, in the decimal system each decimal place is a power of ten. For example,7410 =7 ×101 +4 ×100 In the octal system each place is a power of eight. The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves and it has been suggested that the reconstructed Proto-Indo-European word for nine might be related to the PIE word for new. Based on this, some have speculated that proto-Indo-Europeans used a number system. In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult, in 1718 Swedenborg wrote a manuscript, En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10. The numbers 1-7 are there denoted by the l, s, n, m, t, f, u. Thus 8 = lo,16 = so,24 = no,64 = loo,512 = looo etc, numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule. Writing under the pseudonym Hirossa Ap-Iccim in The Gentlemans Magazine, July 1745, Hugh Jones proposed a system for British coins, weights. In 1801, James Anderson criticized the French for basing the Metric system on decimal arithmetic and he suggested base 8 for which he coined the term octal. In the mid 19th century, Alfred B. Taylor concluded that Our octonary radix is, therefore, so, for example, the number 65 would be spoken in octonary as under-un. Taylor also republished some of Swedenborgs work on octonary as an appendix to the above-cited publications, in the 2009 film Avatar, the language of the extraterrestrial Navi race employs an octal numeral system, probably due to the fact that they have four fingers on each hand. In the TV series Stargate SG-1, the Ancients, a race of beings responsible for the invention of the Stargates, in the tabletop game series Warhammer 40,000, the Tau race use an octal number system. Octal became widely used in computing systems such as the PDP-8, ICL1900. Octal was an abbreviation of binary for these machines because their word size is divisible by three

13.
Duodecimal
–
The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten and B or E for eleven. The number twelve is written as 10 in duodecimal, whereas the digit string 12 means 1 dozen and 2 units. Similarly, in duodecimal 100 means 1 gross,1000 means 1 great gross, the number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range. As a result, duodecimal has been described as the number system. Of its factors,2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a representation in duodecimal. In particular, the five most elementary fractions all have a terminating representation in duodecimal. This all makes it a convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary. Although the trigesimal and sexagesimal systems do even better in respect, this is at the cost of unwieldy multiplication tables. In this section, numerals are based on decimal places, for example,10 means ten,12 means twelve. Languages using duodecimal number systems are uncommon, germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif, historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in a foot,12 troy ounces in a troy pound,12 old British pence in a shilling,24 hours in a day. The Romans used a system based on 12, including the uncia which became both the English words ounce and inch. The importance of 12 has been attributed to the number of cycles in a year. It is possible to count to 12 with the acting as a pointer

14.
Hexadecimal
–
In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a more human-friendly representation of binary-coded values, one hexadecimal digit represents a nibble, which is half of an octet or byte. For example, a byte can have values ranging from 00000000 to 11111111 in binary form. In a non-programming context, a subscript is typically used to give the radix, several notations are used to support hexadecimal representation of constants in programming languages, usually involving a prefix or suffix. The prefix 0x is used in C and related languages, where this value might be denoted as 0x2AF3, 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 can give the base explicitly,15910 is decimal 159,15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. example. com/name%20with%20spaces where %20 is the space character, thus ’, represents the right single quotation mark, Unicode code point number 2019 in hex,8217. In the Unicode standard, a value is represented with U+ followed by the hex value. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #, white, CSS allows 3-hexdigit abbreviations with one hexdigit per component, #FA3 abbreviates #FFAA33. *nix shells, AT&T assembly language and likewise the C programming language, to output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h, some assembly languages use the notation HABCD. Ada and VHDL enclose hexadecimal numerals in based numeric quotes, 16#5A3#, for bit vector constants VHDL uses the notation x5A3. Verilog represents hexadecimal constants in the form 8hFF, where 8 is the number of bits in the value, the Smalltalk language uses the prefix 16r, 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#, 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs, in early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong. Common Lisp uses the prefixes #x and #16r, setting the variables *read-base* and *print-base* to 16 can also 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, 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 ALGOL68 uses the prefix 16r to denote hexadecimal numbers, binary, quaternary and octal numbers can be specified similarly

15.
Vigesimal
–
The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used. One modern method of finding the extra needed symbols is to write ten as the letter A20, to write nineteen as J20, and this is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters A–F. Another method skips over the letter I, in order to avoid confusion between I20 as eighteen and one, so that the number eighteen is written as J20, the number twenty is written as 1020. According to this notation,2020 means forty in decimal = + D020 means two hundred and sixty in decimal = +10020 means four hundred in decimal = + +, in the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example,10 means ten,20 means twenty, in decimal, dividing by three twice only gives one digit periods because 9 is the number below ten. 21, however, the adjacent to 20 that is divisible by 3, is not divisible by 9. Ninths in vigesimal have six-digit periods, the prime factorization of twenty is 22 ×5, so it is not a perfect power. However, its part,5, is congruent to 1. Thus, according to Artins conjecture on primitive roots, vigesimal has infinitely many cyclic primes, but the fraction of primes that are cyclic is not necessarily ~37. 395%. An UnrealScript program that computes the lengths of recurring periods of various fractions in a set of bases found that, of the first 15,456 primes. In many European languages,20 is used as a base, vigesimal systems are common in Africa, for example in Yoruba. Ogún,20, is the basic numeric block, ogójì,40, =20 multiplied by 2. Ogota,60, =20 multiplied by 3, ogorin,80, =20 multiplied by 4. Ogorun,100, =20 multiplied by 5, twenty was a base in the Maya and Aztec number systems. The Maya used the names for the powers of twenty, kal, bak, pic, calab, kinchil. See also Maya numerals and Maya calendar, Mayan languages, Yucatec, the Aztec called them, cempoalli, centzontli, cenxiquipilli, cempoalxiquipilli, centzonxiquipilli and cempoaltzonxiquipilli. Note that the ce prefix at the beginning means one and is replaced with the number to get the names of other multiples of the power

16.
Base 36
–
The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion

17.
Natural number
–
In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory

18.
Sphenic number
–
In number theory, a sphenic number is a positive integer that is the product of three distinct prime numbers. A sphenic number is a product pqr where p, q and this definition is more stringent than simply requiring the integer to have exactly three prime factors. For instance,60 =22 ×3 ×5 has exactly 3 prime factors, the smallest sphenic number is 30 =2 ×3 ×5, the product of the smallest three primes. The first few numbers are 30,42,66,70,78,102,105,110,114,130,138,154,165. As of January 2016 the largest known number is × ×. It is the product of the three largest known primes, all sphenic numbers have exactly eight divisors. If we express the number as n = p ⋅ q ⋅ r, where p, q. For example,24 is not a number, but it has exactly eight divisors. All sphenic numbers are by definition squarefree, because the factors must be distinct. The Möbius function of any number is −1. The cyclotomic polynomials Φ n, taken over all sphenic numbers n, the first case of two consecutive sphenic integers is 230 = 2×5×23 and 231 = 3×7×11. The first case of three is 1309 = 7×11×17,1310 = 2×5×131, and 1311 = 3×19×23, there is no case of more than three, because every fourth consecutive positive integer is divisible by 4 = 2×2 and therefore not squarefree. The numbers 2013,2014, and 2015 are all sphenic, the next three consecutive sphenic years will be 2665,2666 and 2667. Semiprimes, products of two prime numbers

19.
The Simpsons
–
The Simpsons is an American animated sitcom created by Matt Groening for the Fox Broadcasting Company. The series is a depiction of working-class life epitomized by the Simpson family, which consists of Homer, Marge, Bart, Lisa. The show is set in the town of Springfield and parodies American culture, society, television. The family was conceived by Groening shortly before a solicitation for a series of animated shorts with producer James L. Brooks, Groening created a dysfunctional family and named the characters after members of his own family, substituting Bart for his own name. The shorts became a part of The Tracey Ullman Show on April 19,1987. After a three-season run, the sketch was developed into a prime time show and became an early hit for Fox. Since its debut on December 17,1989,615 episodes of The Simpsons have been broadcast and its 28th season began on September 25,2016. It is the longest-running American sitcom and the longest-running American animated program, the Simpsons Movie, a feature-length film, was released in theaters worldwide on July 27,2007, and grossed over $527 million. On May 4,2015, the series was renewed for seasons 27 and 28, on November 4,2016, the series was renewed for seasons 29 and 30, consisting of 22 episodes each. The Simpsons received widespread critical acclaim throughout its first nine or ten seasons, Time named it the 20th centurys best television series, and Erik Adams of The A. V. Club named it televisions crowning achievement regardless of format, on January 14,2000, the Simpson family was awarded a star on the Hollywood Walk of Fame. It has won dozens of awards since it debuted as a series, including 31 Primetime Emmy Awards,30 Annie Awards, Homers exclamatory catchphrase Doh. has been adopted into the English language, while The Simpsons has influenced many adult-oriented animated sitcoms. Despite this, the show has also criticized for what many perceive as a decline in quality over the years. The Simpsons are a family who live in a fictional Middle American town of Springfield, Homer, the father, works as a safety inspector at the Springfield Nuclear Power Plant, a position at odds with his careless, buffoonish personality. He is married to Marge Simpson, a stereotypical American housewife, although the family is dysfunctional, many episodes examine their relationships and bonds with each other and they are often shown to care about one another. The family owns a dog, Santas Little Helper, and a cat, Snowball V, renamed Snowball II in I, both pets have had starring roles in several episodes. The show includes an array of supporting characters, co-workers, teachers, family friends, extended relatives, townspeople. The creators originally intended many of these characters as jokes or for fulfilling needed functions in the town

20.
British Rail Class 318
–
The British Rail Class 318 is an electric multiple unit train, which operates exclusively in the Strathclyde Partnership for Transport network in West Central Scotland. The units were introduced fully on 29 September 1986 as part of the electrification of the Ayrshire Coast Line between Glasgow Central and Ayr/Ardrossan with alternating current overhead lines, usage was extended to Largs in January 1987. They were also used on the Inverclyde Line in small numbers, the trains currently operate Argyle and North Clyde Line services. The technical description of the units are DTSO+MSO+DTSO, consisting of a motor car with a driving trailer at either end. The units run on the standard 25 kV AC overhead line system, up to four sets can be worked in multiple to form a 12-car set, although platforms are only capable of handling 8-car trains. The units have a speed of 90 mph, and can complete the 41-mile journey between Glasgow Central and Ayr in around 52 minutes. With the introduction of the Class 334s, in 2001 -2002, onto Ayrshire/Inverclyde routes, the 318 can also be coupled in tandem with the slightly newer Class 320 in a six car formation, this practice is regularly used on the North Clyde and Argyle Lines. The train crashed through shops at the front of the station, being very early in the morning, there were no serious injuries. The cab of vehicle 77244 needed to be completely rebuilt,318262 managed to move by rail back to Shields depot. The reconstruction of the Largs railway station building took almost ten years to complete, on 3 September 2007, unit 318254 derailed near Exhibition Centre railway station in Glasgow. Two of the four were injured and the others walked away unharmed and this is the same unit that was involved in the Largs rail crash. 318254 is still in service today, on 16 January 2008, number 318267 was involved in a minor collision at Glasgow Central station whilst stationary. British Rail Class 334 number 334017 was working the 08,24 passenger service from Gourock when it collided with the empty Class 318 train whilst preparing to terminate at Glasgow Central. The Class 334 train was in the process of braking and was travelling at less than 4 mph when the collision occurred, four passengers were slightly injured, however only one required hospital attention, and was transferred to Glasgow Southern General hospital in a taxi. Between 2005 and 2007, all Class 318s underwent a process which involves the removal of the corridor connection on the drivers cars. Although one may see this as detrimental to passenger operations, there have no great problems with the Class 334 or Class 320 units which also have no corridor connections. Apart from cleaning up the appearance of the cars, this allows the driver to have a far more spacious operating environment. The Class 318 units were refurbished at Hunslet-Barclay, the passenger accommodation has also been improved, with new passenger door controls

21.
Electric multiple unit
–
An electric multiple unit or EMU is a multiple unit train consisting of self-propelled carriages, using electricity as the motive power. An EMU requires no separate locomotive, as traction motors are incorporated within one or a number of the carriages. An EMU is usually formed of two or more semi-permanently coupled carriages, but electrically powered single-unit railcars are also classed as EMUs. EMUs are popular on commuter and suburban rail networks around the world due to their fast acceleration, being quieter than DMUs and locomotive-drawn trains, EMUs can operate later at night and more frequently without disturbing residents living near the railway lines. The first EMUs were used on the elevated Liverpool Overhead Railway in 1893, the southern terminal of the railway was underground, giving the LOR the distinction of also being the first to use EMUs underground. Each carriage had a traction motor and was specifically designed and constructed to be light in weight while running on elevated steel sections. The first EMUs were two-carriage trains later graduating to three carriages, with the front and rear carriages powered, Liverpool Museum retains an example of the Liverpool Overhead Railway EMU carriage. An early proponent of EMUs was the American engineer Frank J. Sprague, the cars that form a complete EMU set can usually be separated by function into four types, power car, motor car, driving car, and trailer car. Each car can have more than one function, such as a car or power-driving car. A power car carries the equipment to draw power from the electrified infrastructure, such as pickup shoes for third rail systems and pantographs for overhead systems. Motor cars carry the traction motors to move the train, and are combined with the power car to avoid high-voltage inter-car connections. Driving cars are similar to a cab car, containing a drivers cab for controlling the train, an EMU will usually have two driving cars at its outer ends. Trailer cars are any cars that carry little or no traction or power related equipment, on third rail systems the outer vehicles usually carry the pick up shoes, with the motor vehicles receiving the current via intra-unit connections. Many modern 2-car EMU sets are set up as married pair units, while both units in a married pair are typically driving motors, the ancillary equipment are shared between the two cars in the set. Since neither car can operate without its partner, such sets are permanently coupled, advantages of married pair units include weight and cost savings over single-unit cars while allowing all cars to be powered, unlike a motor-trailer combination. Disadvantages include a loss of flexibility, as trains must be multiples of two cars, and a failure on a single car could force removing both it and its partner from service. The retired New York–Washington Metroliner service, first operated by the Pennsylvania Railroad and later by Amtrak, also featured high-speed electric multiple-unit cars, diesel multiple unit Battery electric multiple unit British electric multiple units Media related to Electric multiple unit at Wikimedia Commons

22.
Train
–
A train is a form of rail transport consisting of a series of vehicles that usually runs along a rail track to transport cargo or passengers. Motive power is provided by a locomotive or individual motors in self-propelled multiple units. Although historically steam propulsion dominated, the most common forms are diesel and electric locomotives. Other energy sources include horses, engine or water-driven rope or wire winch, gravity, pneumatics, batteries, the word train comes from the Old French trahiner, from the Latin trahere pull, draw. There are various types of trains that are designed for particular purposes, a train may consist of a combination of one or more locomotives and attached railroad cars, or a self-propelled multiple unit. The first trains were rope-hauled, gravity powered or pulled by horses, from the early 19th century almost all were powered by steam locomotives. A passenger train is one which includes passenger-carrying vehicles which can often be very long, one notable and growing long-distance train category is high-speed rail. In order to much faster operation over 500 km/h, innovative Maglev technology has been researched for years. In most countries, such as the United Kingdom, the distinction between a tramway and a railway is precise and defined in law, a freight train uses freight cars to transport goods or materials. Freight and passengers may be carried in the train in a mixed consist. Rail cars and machinery used for maintenance and repair of tracks, etc. are termed maintenance of way equipment, similarly, dedicated trains may be used to provide support services to stations along a train line, such as garbage or revenue collection. There are various types of trains that are designed for particular purposes, a train can consist of a combination of one or more locomotives and attached railroad cars, or a self-propelled multiple unit. Trains can also be hauled by horses, pulled by a cable, special kinds of trains running on corresponding special railways are atmospheric railways, monorails, high-speed railways, maglev, rubber-tired underground, funicular and cog railways. A passenger train may consist of one or several locomotives and coaches, alternatively, a train may consist entirely of passenger carrying coaches, some or all of which are powered as a multiple unit. In many parts of the world, particularly the Far East and Europe, freight trains are composed of wagons or trucks rather than carriages, though some parcel and mail trains are outwardly more like passenger trains. Trains can also be mixed, comprising both passenger accommodation and freight vehicles, special trains are also used for track maintenance, in some places, this is called maintenance of way. A train with a locomotive attached at each end is described as top and tailed, where a second locomotive is attached temporarily to assist a train up steep banks or grades it is referred to as banking in the UK, or helper service in North America. Recently, many loaded trains in the United States have been made up one or more locomotives in the middle or at the rear of the train

23.
Scotland
–
Scotland is a country that is part of the United Kingdom and covers the northern third of the island of Great Britain. It shares a border with England to the south, and is surrounded by the Atlantic Ocean, with the North Sea to the east. In addition to the mainland, the country is made up of more than 790 islands, including the Northern Isles, the Kingdom of Scotland emerged as an independent sovereign state in the Early Middle Ages and continued to exist until 1707. By inheritance in 1603, James VI, King of Scots, became King of England and King of Ireland, Scotland subsequently entered into a political union with the Kingdom of England on 1 May 1707 to create the new Kingdom of Great Britain. The union also created a new Parliament of Great Britain, which succeeded both the Parliament of Scotland and the Parliament of England. Within Scotland, the monarchy of the United Kingdom has continued to use a variety of styles, titles, the legal system within Scotland has also remained separate from those of England and Wales and Northern Ireland, Scotland constitutes a distinct jurisdiction in both public and private law. Glasgow, Scotlands largest city, was one of the worlds leading industrial cities. Other major urban areas are Aberdeen and Dundee, Scottish waters consist of a large sector of the North Atlantic and the North Sea, containing the largest oil reserves in the European Union. This has given Aberdeen, the third-largest city in Scotland, the title of Europes oil capital, following a referendum in 1997, a Scottish Parliament was re-established, in the form of a devolved unicameral legislature comprising 129 members, having authority over many areas of domestic policy. Scotland is represented in the UK Parliament by 59 MPs and in the European Parliament by 6 MEPs, Scotland is also a member nation of the British–Irish Council, and the British–Irish Parliamentary Assembly. Scotland comes from Scoti, the Latin name for the Gaels, the Late Latin word Scotia was initially used to refer to Ireland. By the 11th century at the latest, Scotia was being used to refer to Scotland north of the River Forth, alongside Albania or Albany, the use of the words Scots and Scotland to encompass all of what is now Scotland became common in the Late Middle Ages. Repeated glaciations, which covered the land mass of modern Scotland. It is believed the first post-glacial groups of hunter-gatherers arrived in Scotland around 12,800 years ago, the groups of settlers began building the first known permanent houses on Scottish soil around 9,500 years ago, and the first villages around 6,000 years ago. The well-preserved village of Skara Brae on the mainland of Orkney dates from this period and it contains the remains of an early Bronze Age ruler laid out on white quartz pebbles and birch bark. It was also discovered for the first time that early Bronze Age people placed flowers in their graves, in the winter of 1850, a severe storm hit Scotland, causing widespread damage and over 200 deaths. In the Bay of Skaill, the storm stripped the earth from a large irregular knoll, when the storm cleared, local villagers found the outline of a village, consisting of a number of small houses without roofs. William Watt of Skaill, the laird, began an amateur excavation of the site, but after uncovering four houses

24.
BMW
–
Bayerische Motoren Werke AG, usually known under its abbreviation BMW, is a German luxury vehicle, motorcycle, and engine manufacturing company founded in 1916. It is one of the luxury automakers in the world. The company is a component of the Euro Stoxx 50 stock market index, headquartered in Munich, Bavaria, BMW owns Mini cars and is the parent company of Rolls-Royce Motor Cars. BMW was established as a business entity following a restructuring of the Rapp Motorenwerke aircraft manufacturing firm in 1912 named Aerowerke Gustav Otto, after the end of World War I in 1918, BMW was forced to cease aircraft-engine production by the terms of the Versailles Armistice Treaty. The company consequently shifted to production as the restrictions of the treaty started to be lifted in 1923. BMWs first significant aircraft engine, and commercial product of any sort, was the BMW IIIa inline-six liquid-cooled engine of 1918, known for good fuel economy, with German rearmament in the 1930s, the company again began producing aircraft engines for the Luftwaffe. The factory in Munich made ample use of forced labour, foreign civilians, prisoners of war, the few Me 262 A-1b test examples built used the more developed version of the 003 jet, recording an official top speed of 800 km/h. The first-ever four-engine jet aircraft flown were the sixth and eighth prototypes of the Arado Ar 234 jet reconnaissance-bomber. Through 1944 the 003s reliability improved, making it a power plant for air frame designs competing for the Jägernotprogramms light fighter production contract. Which was won by the Heinkel He 162 Spatz design, the BMW003 aviation turbojet was also under consideration as the basic starting point for a pioneering turboshaft powerplant for German armored fighting vehicles in 1944–45, as the GT101. Towards the end of the Third Reich, BMW developed some military aircraft projects for the Luftwaffe, the BMW Strahlbomber, the BMW Schnellbomber and the BMW Strahljäger, but none of them were built. By the year 1958, the division of BMW was in financial difficulties. It was decided to carry on by trying to cash in on the current economy car boom exploited so successfully by German ex-aircraft manufacturers such as Messerschmitt, BMW bought the rights to manufacture the Italian Iso Isetta. BMWs version of the cars were to be powered by a modified form of BMWs motorcycle engine. This was moderately successful and helped the company get back on its feet, since 1959, the controlling majority shareholder of the BMW Aktiengesellschaft has been the Quandt family, which owns about 46% of the stock. The rest is in public float, BMW acquired the Hans Glas company based in Dingolfing, Germany, in 1966. Glas vehicles were badged as BMW until the company was fully absorbed. However, this factory was outmoded and BMWs biggest immediate gain was, according to themselves, the Glas factories continued to build a limited number of their existing models, while adding the manufacture of BMW front and rear axles until they could be closer incorporated into BMW

25.
Automobile
–
A car is a wheeled, self-powered motor vehicle used for transportation and a product of the automotive industry. The year 1886 is regarded as the year of the modern car. In that year, German inventor Karl Benz built the Benz Patent-Motorwagen, cars did not become widely available until the early 20th century. One of the first cars that was accessible to the masses was the 1908 Model T, an American car manufactured by the Ford Motor Company. Cars were rapidly adopted in the United States of America, where they replaced animal-drawn carriages and carts, cars are equipped with controls used for driving, parking, passenger comfort and safety, and controlling a variety of lights. Over the decades, additional features and controls have been added to vehicles, examples include rear reversing cameras, air conditioning, navigation systems, and in car entertainment. Most cars in use in the 2010s are propelled by a combustion engine. Both fuels cause air pollution and are blamed for contributing to climate change. Vehicles using alternative fuels such as ethanol flexible-fuel vehicles and natural gas vehicles are also gaining popularity in some countries, electric cars, which were invented early in the history of the car, began to become commercially available in 2008. There are costs and benefits to car use, the costs of car usage include the cost of, acquiring the vehicle, interest payments, repairs and auto maintenance, fuel, depreciation, driving time, parking fees, taxes, and insurance. The costs to society of car use include, maintaining roads, land use, road congestion, air pollution, public health, health care, road traffic accidents are the largest cause of injury-related deaths worldwide. The benefits may include transportation, mobility, independence. The ability for humans to move flexibly from place to place has far-reaching implications for the nature of societies and it was estimated in 2010 that the number of cars had risen to over 1 billion vehicles, up from the 500 million of 1986. The numbers are increasing rapidly, especially in China, India, the word car is believed to originate from the Latin word carrus or carrum, or the Middle English word carre. In turn, these originated from the Gaulish word karros, the Gaulish language was a branch of the Brythoic language which also used the word Karr, the Brythonig language evolved into Welsh where Car llusg and car rhyfel still survive. It originally referred to any wheeled vehicle, such as a cart, carriage. Motor car is attested from 1895, and is the formal name for cars in British English. Autocar is a variant that is attested from 1895

26.
BMW 3 Series
–
The BMW3 Series is an entry-level luxury car manufactured by the German automaker BMW since May 1975. It is the successor to the 02 Series and has produced in six different generations. In 2013, the coupé and convertible models started to be badged as 4 Series, the 3 Series is BMWs best-selling model, accounting for around 30% of the BMW brands annual total sales. The BMW3 Series has won awards throughout its history. The M version of the 3 series, M3, debuted with the E30 M3 in 1988, e93, 2007–2013 Convertible F30, 2012–present Sedan. F34, 2013–present Gran Turismo The E21 replaced the 02 Series and was available as a 2-door sedan. At launch, all models used carburetted 4-cylinder engines, however fuel injected models were introduced in late 1975, a cabriolet body style - manufactured by Baur - was available from 1978 to 1981. The E30 was initially produced in the 2-door sedan body style, in following years, the 4-door sedan, convertible and estate body styles were added. The E30 models were powered by a range of 4-cylinder and straight-6 engines, the BMW M3 was first introduced on the E30 platform. The E30 M3 was fitted with the high-revving S14 4-cylinder engine, the 325iX model was the first 3 Series to use all-wheel drive. The saloon version of the E30 was replaced in late 1990, the E36 was produced in sedan, coupe, convertible, station wagon and hatchback body styles. The hatchback body style, known as the BMW Compact, was introduced in 1994 starting with the 318ti, being popular in Europe. The Touring estate was sold in Europe from 1995, but was not available in the United States, even though the BMW factory was producing its own E36 convertibles, the Baur Top Cabriolet conversion was also available. BMWs first sequential manual transmission debuted in the E36 M3, the E36 was a successful model which laid strong foundations for the E46s success in subsequent years. For every year of its production, the E36 was named in Car, additionally, the E36 M3 was named Car and Drivers best handling car for over $30k in 1997. The E46 was produced in sedan, coupé, convertible, wagon, all-wheel drive was re-introduced in the 3 Series, after a break of 18 years. The AWD system of the E46 was split with a 38/62 percent front/rear bias and it was available for the six-cylinder 325xi and 330xi sedan/wagon models. The E46 was considered the benchmark of its class and experienced success in many markets

27.
Chrysler LA engine
–
The LA engines are a family of pushrod OHV 90° V-configured gasoline engines built by Chrysler Corporation. It was factory-installed in passenger vehicles, trucks and vans, commercial vehicles, marine, the combustion chambers are wedge-shaped, rather than the polyspherical combustion chambers in the predecessor A engine or the hemispherical chambers in the Chrysler Hemi engine. All versions are made of cast iron, except for the Viper V10 which is aluminum, LA engines have the same 4. 46-inch bore spacing as the A engines. LA engines were made at Chryslers Mound Road Engine plant in Detroit, Michigan, as well as plants in Canada, the LA stands for Light A, as the older A engine it was closely based on was nearly 50 pounds heavier. Willem Weertman, who later became Chief Engineer – Engine Design, the 273 was the first LA engine, beginning model year 1964 and offered through 1969, rated at 180 BHP. It had a 3.625 in bore and 3.31 in stroke and it had a mechanical solid lifter valvetrain until 1968 when hydraulic lifters were introduced, hydraulic lifters generally make for a quieter valvetrain. The reciprocating assembly included a cast or forged steel crankshaft, drop forged steel connecting rods, the valvetrain consisted of a cast nodular iron camshaft, solid or hydraulic lifters, solid pushrods and shaft-mounted, malleable iron rocker arms. These actuated the overhead steel intake and exhaust valves, the cylinder heads featured wedge-shaped combustion chambers with a single intake and a single exhaust valve for each cylinder. Spark plugs were located in the side of the cylinder head, a high performance 235 hp was offered 1965-67, this was standard in the Barracuda Formula S model and optional in all other compact models exc. station wagons. In 1965 the muffler was of straight through construction, a special version was also available in 1966 only - it used a 0. 500-inch lift solid-lifter camshaft, fabricated-steel-tube exhaust, and a Holley 4-barrel carburetor, producing 275 horsepower. It was available in the Dodge Dart only, and the car so equipped was called the D-Dart, a reference to its classification in NHRA D-stock for drag racing, the LA318 was a 318 cu in relative of the A318. Like the A318, it has a cylinder bore at 3.91 in as well as a stroke of 3.31 in. It appeared shortly after the 273, in 1967, and proved tremendously successful, a crop duster version of this engine was available until 1991 when it was superseded by the Magnum version. The 318 received roller lifters and a cylinder head in 1985. Throttle-body electronic fuel injection was factory equipment on the 1981-1983 Imperial, from 1988 to 1991, another throttle-body fuel injection system was used for truck and van applications. In the mid-1960s, Chrysler decided to adapt the 318 cu. in. small block V8 into a lightweight and its block was bored out to 4. 04-inch but 3. 31-inch stroke left unchanged, resulting in the 340 cu in. Anticipating higher loads resulting from racing operation, the engineers fitted a forged shot peened steel crankshaft instead of the cast nodular iron unit used in the 318 and this also included shot peened and forged pushrods, connecting rods and pistons. A 4-barrel carburetor was mated to a high-rise, dual plane intake manifold feeding high-flow cylinder heads that are considered among the best of that era

28.
Small-block
–
The Chevrolet small-block engine is a series of automobile V8 engines built by the Chevrolet Division of General Motors using the same basic small engine block. The small blocks small physical size also set apart from their massive physical size cousins. Retroactively referred to as the Generation I small-block, it is distinct from subsequent Generation II LT and Generation III/IV LS, engineer Ed Cole, who would later become GM President, is credited with leading the design for this engine. Production of the original small-block began in the fall of 1954 for the 1955 model year with a displacement of 265 cu in, growing incrementally over time to 400 cu in by 1970. Several intermediate displacements appeared over the years, such as the 283 cu in that was available with fuel injection. Introduced as an engine in 1967, the 350 went on to be employed in both high- and low-output variants across the entire Chevrolet product line. Although all of Chevrolets siblings of the period designed their own V8s, over the years, every American General Motors division except Saturn used it and its descendants in their vehicles. In all, over 100,000,000 small-blocks have been built in carbureted, the small-block family line was honored as one of the 10 Best Engines of the 20th Century by automotive magazine Wards AutoWorld. The first generation of Chevrolet small-blocks began with the 1955 Chevrolet 265 cu in V8 offered in the Corvette, by 1957 it had grown to 283 cu in. Fitted with the optional Rochester mechanical fuel injection, it was one of the first production engines to make 1 hp per 1 cu in, the 283 would later be extended to other Chevrolet models, replacing the old style 265 V8s. A high-performance 327 cu in variant followed, turning out as much as 375 hp and it was, however, the 350 cu in series that came to be the best known Chevrolet small-block. Installed in everything from station wagons to sports cars, in commercial vehicles, from 1955 to 1974, the small-block engine was known as the Turbo-Fire V8. All Chevrolet V8s, from the big-blocks to todays LS7 and LS9, of the three engines in this family, two of them, the 265 and the 283, have gone down in automotive history. The first of this family was the 265, introduced in 1955, the 265 had a 3.75 in bore. The stroke of the 265 was 3.00 in, like the 283, the last of this family was the 307 cu in, which was a stroked 283 with a medium journal. The 265 cu in Turbo-Fire V8 was the first Chevrolet small-block, coles design borrowed the valve train design scheduled to be used at the time in the Pontiac V8. Internal GM rules at that time were that once an automotive division had introduced a technological innovation no other GM division could use it for a period of two years. The stud mounted independent ball rocker arm design patented by Pontiac engineer Clayton Leach was scheduled for introduction in the Pontiac 1955 V8

29.
V8 engine
–
A V8 engine is an eight-cylinder V configuration engine with the cylinders mounted on the crankcase in two sets of four, with all eight pistons driving a common crankshaft. Most banks are set at an angle to each other, some at a narrower angle, with 45°, 60°. In its simplest form, the V8 is basically two parallel inline-four engines sharing a common crankshaft, since the 1920s, most V8s have used the somewhat more complex crossplane crankshaft with heavy counterweights to eliminate the vibrations. This results in an engine that is smoother than a V6, most racing V8s continue to use the single plane crankshaft because it allows faster acceleration and more efficient exhaust system designs. In 1902, Léon Levavasseur took out a patent on a light and he called it the Antoinette after the young daughter of his financial backer. From 1904 he installed this engine in a number of competition speedboats, the aviation pioneer Alberto Santos-Dumont saw one of these boats in Côte dAzur and decided to try it on his 14-bis aircraft. Its early 24 hp at 1400 rpm version with only 55 kg of weight was interesting, Santos-Dumont ordered a larger and more powerful version from Levavasseur. He changed its dimensions from the original 80 mm stroke and 80 mm bore to 105 mm stroke and 110 mm bore, obtaining 50 hp with 86 kg of weight and its power-to-weight ratio was not surpassed for 25 years. Levavasseur eventually produced its own line of V8 equipped aircraft, named Antoinette I to VIII, hubert Latham piloted the V8 powered Antoinette IV and Antoinette VII in July 1909 on two failed attempts to cross the English Channel. However, in 1910, Latham used the VII with the engine to become the first in the world to reach an altitude of 3600 feet. Voisin constructed pusher biplanes with Antoinette engines, also, notably the one first flown successfully by Henry Farman in 1908, the V8 engine configuration became popular in France from 1904 onward, and was used in a number of aircraft engines introduced by Renault, and Buchet among others. Some of these found their way into automobiles in small quantities. In 1905, Darracq built a car to beat the world speed record. They came up with two racing car built on a common crankcase and camshaft. The result was monstrous engine with a displacement of 1,551 cu in, victor Hemery fixed that record on 30 December 1905 with a speed of 109.65 mph. Rolls-Royce built a 3,535 cc V8 car from 1905 to 1906, in 1907 The Hewitt Motor Company built a large 5 passenger Touring Car. It was equipped with a hefty V8 engine that developed 50/60 horsepower and had a bore of 4 inches, the Hewitt was the first American Automobile to be equipped with a V8 engine. De Dion-Bouton introduced a 7,773 cc automobile V8 in 1910 and it was produced only in small quantities, but inspired a number of manufacturers to follow suit

30.
Chrysler Corporation
–
FCA US is one of the Big Three American automobile manufacturers. FCA US has its headquarters in Auburn Hills, Michigan and sells vehicles worldwide under its flagship Chrysler brand, as well as the Dodge, Jeep, other major divisions include Mopar, its automotive parts and accessories division, and SRT, its performance automobile division. The Chrysler Corporation was founded by Walter Chrysler in 1925, out of what remained of the Maxwell Motor Company, Chrysler greatly expanded in 1928, when Mr. The brand diversification efforts were inspired by Mr. Chryslers time working for General Motors, in the 1960s the company expanded into Europe, by taking control of French, British and Spanish auto companies, Chrysler Europe was sold in 1978 to PSA Peugeot Citroën for $1. Chrysler struggled through the 1970s to adapt to changing markets, increased US import competition, the company began an engineering partnership with Mitsubishi Motors, and began selling Mitsubishi vehicles branded as Dodge and Plymouth in North America. By the late 1970s, Chrysler was on the verge of bankruptcy, New CEO Lee Iacocca was credited with returning the company to profitability in the 1980s. In 1985, Diamond-Star Motors was created, further expanding the Chrysler-Mitsubishi relationship, in 1987, Chrysler acquired American Motors Corporation, which brought the profitable Jeep brand under the Chrysler umbrella. Like the other Big Three automobile manufacturers, Chrysler was hit hard by the industry crisis of 2008–2010. On June 10,2009, Chrysler emerged from the proceedings with the United Auto Workers pension fund, Fiat S. p. A. The bankruptcy resulted in Chrysler defaulting on over $4 billion in debts, by May 24,2011, Chrysler finished repaying its obligations to the U. S. government five years early, although the cost to the American taxpayer was $1.3 billion. Over the next few years Fiat gradually acquired the other parties shares while removing much of the weight of the loans in a short period. On January 1,2014, Fiat S. p. A announced a deal to purchase the rest of Chrysler from the United Auto Workers retiree health trust. The deal was completed on January 21,2014, making Chrysler Group a subsidiary of Fiat S. p. A, in May 2014, Fiat Chrysler Automobiles, NV was established by merging Fiat S. p. A. into the company. This was completed in August 2014, Chrysler Group LLC remained a subsidiary until December 15,2014, when it was renamed FCA US LLC, to reflect the Fiat-Chrysler merger. The Chrysler company was founded by Walter Chrysler on June 6,1925, Walter Chrysler arrived at the ailing Maxwell-Chalmers company in the early 1920s. He was hired to overhaul the companys troubled operations, in late 1923 production of the Chalmers automobile was ended. In January 1924, Walter Chrysler launched the well-received Chrysler automobile, the Chrysler was a 6-cylinder automobile, designed to provide customers with an advanced, well-engineered car, but at a more affordable price than they might expect. The original 1924 Chrysler included an air filter, high compression engine, full pressure lubrication

31.
Truck
–
A truck is a motor vehicle designed to transport cargo. Trucks vary greatly in size, power, and configuration, smaller varieties may be similar to some automobiles. Commercial trucks can be large and powerful, and may be configured to mount specialized equipment, such as in the case of fire trucks and concrete mixers. Modern trucks are powered by diesel engines, although small to medium size trucks with gasoline engines exist in the US. In the European Union, vehicles with a gross mass of up to 3.5 t are known as light commercial vehicles. Trucks and cars have an ancestor, the steam-powered fardier Nicolas-Joseph Cugnot built in 1769. However, steam wagons were not common until the mid-1800s, the roads of the time, built for horse and carriages, limited these vehicles to very short hauls, usually from a factory to the nearest railway station. The first semi-trailer appeared in 1881, towed by a tractor manufactured by De Dion-Bouton. In 1895 Karl Benz designed and built the first truck in history using the internal combustion engine, later that year some of Benzs trucks were modified to become the first bus by the Netphener, the first motorbus company in history. A year later, in 1896, another internal combustion engine truck was built by Gottlieb Daimler, other companies, such as Peugeot, Renault and Büssing, also built their own versions. The first truck in the United States was built by Autocar in 1899 and was available with optional 5 or 8 horsepower motors, trucks of the era mostly used two-cylinder engines and had a carrying capacity of 3,300 to 4,400 lb. In 1904,700 heavy trucks were built in the United States,1000 in 1907,6000 in 1910, after World War I, several advances were made, pneumatic tires replaced the previously common full rubber versions. Electric starters, power brakes,4,6, and 8 cylinder engines, closed cabs, the first modern semi-trailer trucks also appeared. Touring car builders such as Ford and Renault entered the heavy truck market, although it had been invented in 1890, the diesel engine was not common in trucks in Europe until the 1930s. In the United States, it took longer for diesel engines to be accepted. The word truck might come from a back-formation of truckle, meaning small wheel or pulley, from Middle English trokell, another possible source is the Latin trochus, meaning iron hoop. In turn, both sources emanate from the Greek trokhos, meaning wheel, from trekhein, the first known usage of truck was in 1611, when it referred to the small strong wheels on ships cannon carriages. In its extended usage it came to refer to carts for carrying heavy loads and its expanded application to motor-powered load carrier has been in usage since 1930, shortened from motor truck, which dates back to 1901

32.
Telephone
–
A telephone, or phone, is a telecommunications device that permits two or more users to conduct a conversation when they are too far apart to be heard directly. In 1876, Scottish emigrant Alexander Graham Bell was the first to be granted a United States patent for a device that produced clearly intelligible replication of the human voice and this instrument was further developed by many others. The telephone was the first device in history that people to talk directly with each other across large distances. Telephones rapidly became indispensable to businesses, government, and households, the essential elements of a telephone are a microphone to speak into and an earphone which reproduces the voice in a distant location. Until approximately the 1970s most telephones used a dial, which was superseded by the modern DTMF push-button dial. The receiver and transmitter are usually built into a handset which is held up to the ear, the dial may be located either on the handset, or on a base unit to which the handset is connected. The transmitter converts the sound waves to electrical signals which are sent through the network to the receiving phone. The receiving telephone converts the signals into audible sound in the receiver, telephones permit duplex communication, meaning they allow the people on both ends to talk simultaneously. The first telephones were connected to each other from one customers office or residence to another customers location. Being impractical beyond just a few customers, these systems were replaced by manually operated centrally located switchboards. For greater mobility, various systems were developed for transmission between mobile stations on ships and automobiles in the middle 20th century. Hand-held mobile phone]s was introduced for personal service starting in 1973, by the late 1970s several mobile telephone networks operated around the world. In 1983, the Advanced Mobile Phone System was launched, offering a standardized technology providing portability for users far beyond the residence or office. These analog cellular system evolved into digital networks with better security, greater capacity, better regional coverage, the public switched telephone network, with its hierarchical system of many switching centers, interconnects telephones around the world for communication with each other. With the standardized international numbering system, E.164, each line has an identifying telephone number. Although originally designed for voice communications, convergence has enabled most modern cell phones to have many additional capabilities. Since 1999, the trend for mobile phones is smartphones that integrate all mobile communication, a traditional landline telephone system, also known as plain old telephone service, commonly carries both control and audio signals on the same twisted pair of insulated wires, the telephone line. The control and signaling equipment consists of three components, the ringer, the hookswitch, and a dial, the ringer, or beeper, light or other device, alerts the user to incoming calls