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 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 be thought of as real numbers, and if they are, 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
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 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 and octal numbers can be specified similarly
Traditionally Vattezhuttu characters were used, but now Arabic numerals have become commonplace. Old Tamil possesses a special character for zero and it is read as andru. But yet Modern Tamil renounces the use of its native character, Modern Tamil words for zero include சுழியம் or பூச்சியம். Tamil has a prefix for each number from 1 to 9. For instance, the word for fifty, ஐம்பது is a combination of ஐ, the prefix for nine changes with respect to the succeeding base 10. தொ+ the unvoiced consonant of the succeeding base 10 forms the prefix for nine, for instance,90 is தொ+ண், hence, தொண்ணூறு). These are typically void in the Tamil language except for some Hindu and Christian religious references, example அட்ட இலட்சுமிகள் in a Hindu context, unlike other Indian languages, Tamil has distinct digits for 10,100, and 1000. It has characters for other number-based aspects of day-to-day life. − − − − − − − − − − − − There are two systems that can be used in the Tamil language, the Tamil system which is as follows.
The following are the numbers of the Ancient Tamil Country. Sanskrit based multiples like lakhs are followed just like other Indian languages and you can transcribe any fraction, by affixing -இல் after the denominator followed by the numerator. For instance, 1/41 can be said as நாற்பத்து ஒன்றில் ஒன்று, the suffixing of the -இல் requires you to change the last consonant of the number to its இ form. For example, மூன்று+இல் becomes மூன்றில், note the உ has been omitted, common fractions have names already allocated to them, these names are often used rather than the above method. Other fractions are, Anu was considered as lowest fraction by ancient Tamils as size of smallest physical object, this term went to Sanskrit to refer directly atom. Decimal point is called புள்ளி in Tamil, for example,1.1 would be read as ஒன்று புள்ளி ஒன்று. Percentage is known as விழுக்காடு in Tamil or சதவீதம் and these words are simply added after a number to form percentages. For instance, four percent is நான்கு சதவீதம் or நான்கு விழுக்காடு, percentage symbol is recognised and used.
Ordinal numbers are formed by adding the suffix -ஆம் after the number, as always, when blending two words into one, an unvoiced form of the consonant as the one that the second starts with, is placed in between to blend
Devanagari, called Nagari, is an abugida alphabet of India and Nepal. It is written left to right, has a strong preference for symmetrical rounded shapes within squared outlines. The Nagari script has roots in the ancient Brāhmī script family, the Nagari script was in regular use by the 7th century CE and it was fully developed by about the end of first millennium. Nagari has been the primus inter pares of the Indic scripts, the Devanagari script is used for classical Sanskrit texts. The Devanagari script is closely related to the Nandinagari script commonly found in ancient manuscripts of South India. Devanagari script has forty-seven primary characters, of which fourteen are vowels, the ancient Nagari script for Sanskrit had two additional consonantal characters. The script has no distinction similar to the capital and small letters of the Latin alphabet, generally the orthography of the script reflects the pronunciation of the language. Devanagari is part of the Brahmic family of scripts of India, Tibet and it is a descendant of the Gupta script, along with Siddham and Sharada.
Medieval inscriptions suggest widespread diffusion of the Nagari-related scripts, with biscripts presenting local script along with the adoption of Nagari scripts, the 7th-century Tibetan king Srong-tsan-gambo ordered that all foreign books be transcribed into the Tibetan language. Other closely related scripts such as Siddham Matrka was in use in Indonesia, Japan, Sharada remained in parallel use in Kashmir. Nāgarī is the Sanskrit feminine of Nāgara relating or belonging to a town or city and it is a phrasing with lipi as nāgarī lipi script relating to a city, or spoken in city. The use of the name devanāgarī is relatively recent, and the older term nāgarī is still common, the rapid spread of the term devanāgarī may be related to the almost exclusive use of this script to publish Sanskrit texts in print since the 1870s. As a Brahmic abugida, the principle of Devanagari is that each letter represents a consonant. This is usually written in Latin as a, though it is represented as in the International Phonetic Alphabet, the letter क is read ka, the two letters कन are kana, the three कनय are kanaya, etc.
This cancels the inherent vowel, so that from क्नय knaya is derived क्नय् knay, the halant is often used for consonant clusters when typesetting conjunct ligatures is not feasible. Consonant clusters are written with ligatures, for example, the three consonants क्, न्, and य्, when written consecutively without virāma form कनय, as shown above. Alternatively, they may be joined as clusters to form क्नय knaya, कन्य kanya and this system was originally created for use with the Middle Indo-Aryan languages, which have a very limited number of clusters. When applied to Sanskrit, however, it added a deal of complexity to the script
120, read as one hundred twenty, is the natural number following 119 and preceding 121. In English and other Germanic languages, it was formerly known as one hundred. This hundred of six score is now obsolete, but is described as the hundred or great hundred in historical contexts. 120 is the factorial of 5, and the sum of a twin prime pair,120 is the sum of four consecutive prime numbers, four consecutive powers of 2, and four consecutive powers of 3. It is a sparsely totient number,120 is the smallest number to appear six times in Pascals triangle. 120 is the smallest multiple of 6 with no adjacent prime number and it is the eighth hexagonal number and the fifteenth triangular number, as well as the sum of the first eight triangular numbers, making it a tetrahedral number. 120 is divisible by the first 5 triangular numbers and the first 4 tetrahedral numbers,120 is the first multiply perfect number of order three. The sum of its factors sum to 360, exactly three times 120, note that perfect numbers are order two by the same definition. 120 is divisible by the number of primes below it,30 in this case, there is no integer which has 120 as the sum of its proper divisors, making 120 an untouchable number.
The sum of Eulers totient function φ over the first nineteen integers is 120,120 figures in Pierre de Fermats modified Diophantine problem as the largest known integer of the sequence 1,3,8,120. Fermat wanted to another positive integer that multiplied with any of the other numbers in the sequence yields a number that is one less than a square. Leonhard Euler searched for this number, but failed to find it, the internal angles of a regular hexagon are all 120 degrees. 120 is a Harshad number in base 10,120 is the atomic number of Unbinilium, an element yet to be discovered. The cubits of the height of the Temple building The age at which Moses died, in astrology, when two planets in a persons chart are 120 degrees apart from each other, this is called a trine. This is supposed to bring luck in the persons life. The height in inches of a hoop in the National Basketball Association. 120 is also, The medical telephone number in China In Austria, in the US Army, a common diameter for a mortar in mm. TT scale, a scale for model trains, is 1,120.
120 film is a medium format film developed by Kodak,120, a 2008 Turkish film The Israeli national legislature, the Knesset, has 120 seats
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 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 occurred at around the same time in India 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, two classes of such formal definitions were constructed, 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
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, 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 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 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
A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or fraction of the unit. While all metric prefixes in use today are decadic, historically there have been a number of binary metric prefixes as well. Each prefix has a symbol that is prepended to the unit symbol. The prefix kilo-, for example, may be added to gram to indicate multiplication by one thousand, the prefix milli-, may be added to metre to indicate division by one thousand, one millimetre is equal to one thousandth of a metre. Decimal multiplicative prefixes have been a feature of all forms of the system with six dating back to the systems introduction in the 1790s. Metric prefixes have even been prepended to non-metric units, the SI prefixes are standardized for use in the International System of Units by the International Bureau of Weights and Measures in resolutions dating from 1960 to 1991. Since 2009, they have formed part of the International System of Quantities, the BIPM specifies twenty prefixes for the International System of Units.
Each prefix name has a symbol which is used in combination with the symbols for units of measure. For example, the symbol for kilo- is k, and is used to produce km, kg, and kW, which are the SI symbols for kilometre, prefixes corresponding to an integer power of one thousand are generally preferred. Hence 100 m is preferred over 1 hm or 10 dam, the prefixes hecto, deca and centi are commonly used for everyday purposes, and the centimetre is especially common. However, some building codes require that the millimetre be used in preference to the centimetre, because use of centimetres leads to extensive usage of decimal points. Prefixes may not be used in combination and this applies to mass, for which the SI base unit already contains a prefix. For example, milligram is used instead of microkilogram, in the arithmetic of measurements having units, the units are treated as multiplicative factors to values. If they have prefixes, all but one of the prefixes must be expanded to their numeric multiplier,1 km2 means one square kilometre, or the area of a square of 1000 m by 1000 m and not 1000 square metres.
2 Mm3 means two cubic megametres, or the volume of two cubes of 1000000 m by 1000000 m by 1000000 m or 2×1018 m3, and not 2000000 cubic metres, examples 5 cm = 5×10−2 m =5 ×0.01 m =0. The prefixes, including those introduced after 1960, are used with any metric unit, metric prefixes may be used with non-metric units. The choice of prefixes with a unit is usually dictated by convenience of use. Unit prefixes for amounts that are larger or smaller than those actually encountered are seldom used
The system of Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet. The system was adapted from that of the Greek numerals in the late 2nd century BC, the current numeral system is known as the Hebrew alphabetic numerals to contrast with earlier systems of writing numerals used in classical antiquity. The Greek system was adopted in Hellenistic Judaism and had been in use in Greece since about the 5th century BC, in this system, there is no notation for zero, and the numeric values for individual letters are added together. Each unit is assigned a letter, each tens a separate letter. The hundreds are represented by the sum of two or three letters representing the first four hundreds, to represent numbers from 1,000 to 999,999, the same letters are reused to serve as thousands, tens of thousands, and hundreds of thousands. In Israel today, the system of Arabic numerals is used in almost all cases. The Hebrew numerals are used only in cases, such as when using the Hebrew calendar, or numbering a list.
Numbers in Hebrew from zero to one million, Hebrew alphabet are used to a limited extent to represent numbers, widely used on calendars. In other situations Arabic numerals are used and ordinal numbers must agree in gender with the noun they are describing. If there is no such noun, the form is used. For ordinal numbers greater than ten the cardinal is used and numbers above the value 20 have no gender, For ordinal numbers greater than 10, cardinal numbers are used instead. Note, For numbers greater than 20, gender does not apply and ordinal numbers must agree in gender with the noun they are describing. If there is no such noun, the form is used. Ordinal numbers must agree in number and definite status like other adjectives, the cardinal number precedes the noun, except for the number one which succeeds it. The number two is special - shnayim and shtayim become shney and shtey when followed by the noun they count, for ordinal numbers greater than ten the cardinal is used. The Hebrew numeric system operates on the principle in which the numeric values of the letters are added together to form the total.
For example,177 is represented as קעז which corresponds to 100 +70 +7 =177, this type of system requires 27 letters. In practice the last letter, tav is used in combination with itself and/or other letters from kof onwards, to numbers from 500
Qoph or Qop is the nineteenth letter of the Semitic abjads. Aramaic Qop is derived from the Phoenician letter, and derivations from Aramaic include Hebrew Qof ק, Syriac Qōp̄ ܩ and its original sound value was a West Semitic emphatic stop, presumably or. In Hebrew gematria, it has the value of 100. The origin of the shape of qōp is uncertain. It is usually suggested to have originally depicted either a needle, specifically the eye of a needle, or the back of a head. According to a suggestion, it may have been a picture of a monkey. Besides Aramaic Qop, which rise to the letter in the Semitic abjads used in classical antiquity, Phoenician qōp is the origin of the Latin letter Q. The Arabic letter ق is named قاف qāf and it is written is several ways depending in its position in the word, It is usually transliterated into Latin script as q, though some scholarly works use ḳ. According to Sibawayh, author of the first book on Arabic grammar, in Hejazi Arabic, Najdi Arabic, Gulf Arabic, Libyan Arabic, rural Jordan, Southern Mesopotamian Arabic and some forms of Yemeni and Saidi Arabic and partially in Maghrebi dialects.
In Egyptian Arabic, as well as Levantine Arabic and forms of Algerian Arabic and Moroccan Arabic from around Tlemcen, in Sudanese Arabic and some forms of Yemeni Arabic. In rural Palestinian Arabic it is pronounced as a voiceless velar plosive. Optionally in Iraqi and in Gulf Arabic, it is pronounced as a voiced postalveolar affricate. Note, that most dialects of Arabic do use the sound for this letter when it is found in learned words borrowed from standard Arabic into the respective dialect. The Maghrebi style of writing qāf is different, having only a point above. The earliest Arabic manuscripts show qāf in several variants, pointed or unpointed, within Maghribi texts, there is no possibility of confusing it with the letter fāʼ, as it is instead written with a dot underneath in the Maghribi script. The Oxford Hebrew-English Dictionary transliterates the letter Qoph a transliteration as q or k, the English spellings of Biblical names containing this letter may represent it as c or k, e. g.
Cain for Hebrew Qayin, or Kenan for Qenan. In modern Israeli Hebrew the letter is called kuf, the letter represents /k/, i. e. no distinction is made between Qof and Kaph. However, many groups have made that distinction, with Qof being pronounced by Iraqi Jews and other Mizrahim
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 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, 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