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

Integer
–

Algebraic structure → Group theory

Group theory
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

Negative number
–
This thermometer is indicating a negative

Fahrenheit temperature (−4°F).

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

100 (number)
–
The

U.S. hundred-dollar bill, Series 2009.

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

Factorization
–
A visual representation of the factorization of cubes using volumes. For a sum of cubes, simply substitute z=-y.

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

Greek numerals
–

Numeral systems
Greek numerals
–
A

Constantinopolitan map of the British Isles from

Ptolemy 's

*Geography* (c. 1300), using Greek numerals for its

graticule: 52–63°N of the

equator and 6–33°E from Ptolemy's

Prime Meridian at the

Fortunate Isles.

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

Roman numerals
–
Entrance to section LII (52) of the

Colosseum, with numerals still visible

Roman numerals
–

Numeral systems
Roman numerals
–
A typical

clock face with Roman numerals in

Bad Salzdetfurth, Germany

Roman numerals
–
An inscription on

Admiralty Arch, London. The number is 1910, for which MCMX would be more usual

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

Binary number
–

Numeral systems
Binary number
–
Arithmetic values represented by parts of the Eye of Horus

Binary number
–
Gottfried Leibniz

Binary number
–
George Boole

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

Ternary numeral system
–

Numeral systems
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

Quaternary numeral system
–

Numeral systems
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

Quinary
–

Numeral systems
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

Senary
–

Numeral systems
Senary
–
34 senary = 22 decimal, in senary finger counting

Senary

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

Octal
–

Numeral systems
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

Duodecimal
–

Numeral systems
Duodecimal
–
A duodecimal multiplication table

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

Hexadecimal
–

Numeral systems
Hexadecimal
–
Bruce Alan Martin's hexadecimal notation proposal

Hexadecimal
–
Hexadecimal finger-counting scheme.

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

Vigesimal
–

Numeral systems
Vigesimal
–
The

Maya numerals are a base-20 system.

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

Base 36
–

Numeral systems
Base 36
–
34 senary = 22 decimal, in senary finger counting

Base 36

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

Natural number
–
The

Ishango bone (on exhibition at the

Royal Belgian Institute of Natural Sciences) is believed to have been used 20,000 years ago for natural number arithmetic.

Natural number
–
Natural numbers can be used for counting (one

apple, two apples, three apples, …)

18.
Harmonic divisor number
–
In mathematics, a harmonic divisor number, or Ore number, is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are 1,6,28,140,270,496,672,1638,2970,6200,8128,8190, for example, the harmonic divisor number 6 has the four divisors 1,2,3, and 6. Their harmonic mean is an integer,411 +12 +13 +16 =2, the number 140 has divisors 1,2,4,5,7,10,14,20,28,35,70, and 140. All of the terms in this formula are multiplicative, but not completely multiplicative, therefore, the harmonic mean H is also multiplicative. This means that, for any integer n, the harmonic mean H can be expressed as the product of the harmonic means for the prime powers in the factorization of n. For any integer M, as Ore observed, the product of the mean and arithmetic mean of its divisors equals M itself. Therefore, M is harmonic, with mean of divisors k, if. Ore showed that every number is harmonic. To see this, observe that the sum of the divisors of a perfect number M is exactly 2M, therefore, the average of the divisors is M, where τ denotes the number of divisors of M. For any M, τ is odd if and only if M is a square number, for otherwise each divisor d of M can be paired with a different divisor M/d. But, no perfect number can be a square, this follows from the form of even perfect numbers. Therefore, for a perfect number M, τ is even, Ore conjectured that no odd harmonic divisor numbers exist other than 1. If the conjecture is true, this would imply the nonexistence of odd perfect numbers. W. H. Mills showed that any odd harmonic divisor number above 1 must have a power factor greater than 107. Cohen & Sorli showed that there are no odd harmonic divisor numbers smaller than 1024, Cohen, Goto, and others starting with Ore himself have performed computer searches listing all small harmonic divisor numbers. From these results, lists are known of all harmonic divisor numbers up to 2×109, an Identity Concerning Averages of Divisors of a Given Integer. Numbers Whose Positive Divisors Have Small Integral Harmonic Mean, Cohen, Graeme L. Sorli, Ronald M. Odd harmonic numbers exceed 1024. On Divisors of Odd Perfect Numbers, on the averages of the divisors of a number

Harmonic divisor number
–
Overview

19.
On-Line Encyclopedia of Integer Sequences
–
The On-Line Encyclopedia of Integer Sequences, also cited simply as Sloanes, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs, Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to professional mathematicians and amateurs, and is widely cited. As of 30 December 2016 it contains nearly 280,000 sequences, the database is searchable by keyword and by subsequence. Neil Sloane started collecting integer sequences as a student in 1965 to support his work in combinatorics. The database was at first stored on punched cards and he published selections from the database in book form twice, A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and these books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, as a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. The database continues to grow at a rate of some 10,000 entries a year, Sloane has personally managed his sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100, 000th sequence to the database, A100000, in 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS. org was created to simplify the collaboration of the OEIS editors and contributors, besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences, the sequence of numerators and the sequence of denominators, important irrational numbers such as π =3.1415926535897. are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, yet it still uses a form of conventional mathematical notation. Greek letters are represented by their full names, e. g. mu for μ. Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces, a represents the nth term of the sequence. Zero is often used to represent non-existent sequence elements, for example, A104157 enumerates the smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists. The value of a is 2, a is 1480028129, but there is no such 2×2 magic square, so a is 0

On-Line Encyclopedia of Integer Sequences
–
On-Line Encyclopedia of Integer Sequences

20.
Practical number
–
In number theory, a practical number or panarithmic number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n. Practical numbers were used by Fibonacci in his Liber Abaci in connection with the problem of representing rational numbers as Egyptian fractions, Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators. The name practical number is due to Srinivasan and he noted that the subdivision of money, weights and measures involved numbers like 4,12,16,20 and 28 which are usually supposed to be so inconvenient as to deserve replacement by powers of 10. He rediscovered the number theoretical property of such numbers and was the first to attempt a classification of numbers that was completed by Stewart. This characterization makes it possible to determine whether a number is practical by examining its prime factorization, every even perfect number and every power of two is also a practical number. Practical numbers have also shown to be analogous with prime numbers in many of their properties. If the ordered set of all divisors of the number n is d 1, d 2. D j with d 1 =1 and d j = n, in other words the ordered sequence of all divisors d 1 < d 2 <. < d j of a number has to be a complete sub-sequence. This partial characterization was extended and completed by Stewart and Sierpiński who showed that it is straightforward to determine whether a number is practical from its prime factorization, a positive integer greater than one with prime factorization n = p 1 α1. P k α k is if and only if each of its prime factors p i is small enough for p i −1 to have a representation as a sum of smaller divisors. The condition stated above is necessary and sufficient for a number to be practical, in the other direction, the condition is sufficient, as can be shown by induction. Since q ≤ σ and n / p k α k can be shown by induction to be practical, we can find a representation of q as a sum of divisors of n / p k α k. The divisors representing r, together with p k α k times each of the divisors representing q, the only odd practical number is 1, because if n >2 is an odd number, then 2 cannot be expressed as the sum of distinct divisors of n. More strongly, Srinivasan observes that other than 1 and 2, the product of two practical numbers is also a practical number. More strongly the least common multiple of any two numbers is also a practical number. Equivalently, the set of all numbers is closed under multiplication. From the above characterization by Stewart and Sierpiński it can be seen that if n is a practical number, in the set of all practical numbers there is a primitive set of practical numbers

Practical number
–
Overview

21.
Coprime
–
In number theory, two integers a and b are said to be relatively prime, mutually prime, or coprime if the only positive integer that divides both of them is 1. That is, the common positive factor of the two numbers is 1. This is equivalent to their greatest common divisor being 1, the numerator and denominator of a reduced fraction are coprime. In addition to gcd =1 and =1, the notation a ⊥ b is used to indicate that a and b are relatively prime. For example,14 and 15 are coprime, being divisible by only 1. The numbers 1 and −1 are the only integers coprime to every integer, a fast way to determine whether two numbers are coprime is given by the Euclidean algorithm. The number of integers coprime to an integer n, between 1 and n, is given by Eulers totient function φ. A set of integers can also be called if its elements share no common positive factor except 1. A set of integers is said to be pairwise coprime if a and b are coprime for every pair of different integers in it, a number of conditions are individually equivalent to a and b being coprime, No prime number divides both a and b. There exist integers x and y such that ax + by =1, the integer b has a multiplicative inverse modulo a, there exists an integer y such that by ≡1. In other words, b is a unit in the ring Z/aZ of integers modulo a, the least common multiple of a and b is equal to their product ab, i. e. LCM = ab. As a consequence of the point, if a and b are coprime and br ≡ bs. That is, we may divide by b when working modulo a, as a consequence of the first point, if a and b are coprime, then so are any powers ak and bl. If a and b are coprime and a divides the product bc and this can be viewed as a generalization of Euclids lemma. In a sense that can be made precise, the probability that two randomly chosen integers are coprime is 6/π2, which is about 61%, two natural numbers a and b are coprime if and only if the numbers 2a −1 and 2b −1 are coprime. As a generalization of this, following easily from the Euclidean algorithm in base n >1, a set of integers S = can also be called coprime or setwise coprime if the greatest common divisor of all the elements of the set is 1. For example, the integers 6,10,15 are coprime because 1 is the positive integer that divides all of them. If every pair in a set of integers is coprime, then the set is said to be pairwise coprime, pairwise coprimality is a stronger condition than setwise coprimality, every pairwise coprime finite set is also setwise coprime, but the reverse is not true

Coprime
–
Figure 1. The numbers 4 and 9 are coprime. Therefore, the diagonal of a 4 x 9 lattice does not intersect any other

lattice points
22.
Divisor
–
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number

Divisor
–
The divisors of 10 illustrated with

Cuisenaire rods: 1, 2, 5, and 10

23.
.270 Winchester
–
The.270 Winchester was developed by Winchester Repeating Arms Company in 1923 and unveiled in 1925 as a chambering for their bolt-action Model 54. The cartridge is a necked down. 30-03, which is the length as the.280 Remington. The.270.280, and. 30-06 were all derived from the. 30-03 parent case, when loaded with a bullet that expands rapidly or fragments in tissue, this cartridge delivers devastating terminal performance. The.270 Winchester became a popular cartridge due to the widespread praises of gunwriter Jack OConnor who used the cartridge for 40 years. It drives an 8.4 grams bullet at approximately 960 m/s, the cartridge demonstrated high performance at the time of its introduction and was marketed as being suitable for big game shooting in the 370 to 910 metres range. Two additional bullet weights were introduced, a 6.5 grams hollow-point bullet for varmint shooting, and a 9.7 grams bullet for larger deer, elk. Internationally, firearms manufacturers now offer this chambering in all varieties, bolt-actions, single-shots, lever-actions, pump-actions, autoloaders. Loads are commonly available from 6.5 to 10.4 grams sizes with 8. 4-and-9. 7-gram loads being by far the most popular. Handloaders have a range of options with the availability of bullets in a number of weights from 5.8 to 11.7 grams. Common bullet weight recommendations for shooting different game are as follows,5. 8–7.1 grams bullets,8.4 grams bullets, antelope or other animals up to the size of mule deer. 9. 1–10.4 grams bullets, deer, elk, moose and it is recommended that.270 Winchester brass be formed from.35 Whelen or.280 Remington cases. List of rifle cartridges Table of handgun and rifle cartridges 7 mm caliber.270 WSM sectional density. 30-06. 270-06

.270 Winchester
–
Left to Right 130-grain (8.4 g) - Hollow Point, 100-grain (6.5 g) FMJBT, 130-grain (8.4 g) Soft point, 160-grain (10 g) Lead round nose.

.270 Winchester
–
.270 Winchester cartridge

.270 Winchester
–

Rifles
24.
Electoral College (United States)
–
Citizens of the United States vote in each state at a general election to choose a slate of electors pledged to vote for a partys candidate. The Twelfth Amendment requires each elector to cast one vote for president, each state chooses electors, amounting in number to that states combined total of senators and representatives. There are a total of 538 electors, corresponding to the 435 representatives and 100 senators, the Constitution bars any federal official, elected or appointed, from being an elector. The Office of the Federal Register is charged with administering the Electoral College, all states except Maine and Nebraska and California before 1913 have chosen electors on a winner-take-all basis since the 1880s. Under the winner-take-all system, the electors are awarded to the candidate with the most votes in that state. Maine and Nebraska use the congressional district method, selecting one elector within each district by popular vote. Although no elector is required by law to honor their pledge. If no person receives a majority of votes for vice president, then the Senate will select the vice president. The Constitutional Convention in 1787 used the Virginia Plan as the basis for discussions, the Virginia Plan called for the Congress to elect the president. Delegates from a majority of states agreed to this mode of election, however once the Electoral College had been decided on, several delegates openly recognized its ability to protect the election process from cabal, corruption, intrigue, and faction. Some delegates, including James Wilson and James Madison, preferred popular election of the executive, the right of suffrage was much more diffusive in the Northern than the Southern States, and the latter could have no influence in the election on the score of Negroes. The substitution of electors obviated this difficulty and seemed on the whole to be liable to the fewest objections, the Convention approved the Committees Electoral College proposal, with minor modifications, on September 6,1787. Delegates from states with smaller populations or limited land area such as Connecticut, New Jersey, in The Federalist Papers, James Madison explained his views on the selection of the president and the Constitution. 39, Madison argued the Constitution was designed to be a mixture of state-based and population-based government, Congress would have two houses, the state-based Senate and the population-based House of Representatives. Meanwhile, the president would be elected by a mixture of the two modes, alexander Hamilton in Federalist No.68 laid out what he believed were the key advantages to the Electoral College. The electors come directly from the people and them alone for that purpose only and this avoided a party-run legislature, or a permanent body that could be influenced by foreign interests before each election. Hamilton explained the election was to place among all the states. The choice was to be made by a majority of the Electoral College, Hamilton argued, electors meeting in the state capitals were able to have information unavailable to the general public

Electoral College (United States)
–
Certificate for the electoral vote for

Rutherford B. Hayes and

William A. Wheeler for the State of Louisiana

Electoral College (United States)
–
Electoral College map showing the results of the

2012 U.S. presidential election. President

Barack Obama (

D -

IL) won the popular vote in 26 states and the

District of Columbia (denoted in blue) to capture 332 electoral votes. Former Governor

Mitt Romney (

R -

MA) won the popular vote in 24 states (denoted in red) to capture 206 electoral votes.

25.
President of the United States
–
The President of the United States is the head of state and head of government of the United States. The president directs the executive branch of the government and is the commander-in-chief of the United States Armed Forces. The president is considered to be one of the worlds most powerful political figures, the role includes being the commander-in-chief of the worlds most expensive military with the second largest nuclear arsenal and leading the nation with the largest economy by nominal GDP. The office of President holds significant hard and soft power both in the United States and abroad, Constitution vests the executive power of the United States in the president. The president is empowered to grant federal pardons and reprieves. The president is responsible for dictating the legislative agenda of the party to which the president is a member. The president also directs the foreign and domestic policy of the United States, since the office of President was established in 1789, its power has grown substantially, as has the power of the federal government as a whole. However, nine vice presidents have assumed the presidency without having elected to the office. The Twenty-second Amendment prohibits anyone from being elected president for a third term, in all,44 individuals have served 45 presidencies spanning 57 full four-year terms. On January 20,2017, Donald Trump was sworn in as the 45th, in 1776, the Thirteen Colonies, acting through the Second Continental Congress, declared political independence from Great Britain during the American Revolution. The new states, though independent of each other as nation states, desiring to avoid anything that remotely resembled a monarchy, Congress negotiated the Articles of Confederation to establish a weak alliance between the states. Out from under any monarchy, the states assigned some formerly royal prerogatives to Congress, only after all the states agreed to a resolution settling competing western land claims did the Articles take effect on March 1,1781, when Maryland became the final state to ratify them. In 1783, the Treaty of Paris secured independence for each of the former colonies, with peace at hand, the states each turned toward their own internal affairs. Prospects for the convention appeared bleak until James Madison and Edmund Randolph succeeded in securing George Washingtons attendance to Philadelphia as a delegate for Virginia. It was through the negotiations at Philadelphia that the presidency framed in the U. S. The first power the Constitution confers upon the president is the veto, the Presentment Clause requires any bill passed by Congress to be presented to the president before it can become law. Once the legislation has been presented, the president has three options, Sign the legislation, the bill becomes law. Veto the legislation and return it to Congress, expressing any objections, in this instance, the president neither signs nor vetoes the legislation

President of the United States
–
Incumbent

Barack Obama since January 20, 2009 (2009-01-20)

President of the United States
–

Presidential Seal
President of the United States
–
Obama signing legislation at the

*Resolute* desk
President of the United States
–

Abraham Lincoln, the 16th President of the United States, successfully preserved the

Union during the

American Civil War
26.
Interstate 70
–
Interstate 70 is a major Interstate Highway in the United States that runs from I-15 near Cove Fort, Utah to I-695 near Baltimore, Maryland. I-70 approximately traces the path of U. S. Route 40 east of the Rocky Mountains, West of the Rockies, the route of I-70 was derived from multiple sources. The Interstate runs through or near major cities, including Denver, Kansas City, St. Louis, Indianapolis, Columbus, Pittsburgh. The sections of the interstate in Missouri and Kansas have laid claim to be the first interstate in the United States. The construction of I-70 in Colorado and Utah is considered an engineering marvel as the passes through the Eisenhower Tunnel, Glenwood Canyon. The Eisenhower Tunnel is the highest point along the Interstate Highway system with an elevation of 11,158 ft. Interstate 70 begins at an interchange with Interstate 15 near Cove Fort. Upon leaving the valley near Salina, I-70 crosses 7,923 ft Salina Summit, prior to the construction of I-70, the swell was inaccessible via paved roads and relatively undiscovered. It also became the longest piece of highway to be opened at one time. Although opened in 1970, this section was not formally complete until 1990, since I-70s construction, the swell has been noted for its desolate beauty. The swell has since been nominated for National Park and/or National Monument status on multiple occasions, if the swell is granted this status it arguably would be the first time a National Park owes its existence to an interstate highway. Most of the exits in this span are rest areas, brake check areas, I-70 exits the swell near Green River. From Green River to the Colorado state line, I-70 follows the edge of the Book Cliffs. Entering from Utah, I-70 descends into the Grand Valley where it meets the Colorado River, here I-70 serves the Grand Junction metro area before traversing more mountainous terrain. The last section of I-70 to be completed was the 15-mile Glenwood Canyon, construction was delayed for many years due to environmental concerns. Much of the highway is elevated above the Colorado River, the speed limit in this section is 50 mph due to the limited sight distance and sharp corners. The Eisenhower–Johnson Memorial Tunnel, the highest vehicular tunnel in North America, because of the rugged and narrow terrain of the Rocky Mountains, I-70 is one of few roads connecting Colorados ski resorts with Denver. Descending through the foothills of the Rocky Mountains, the Denver skyline can be seen on a clear day. This can fool truckers and other unsuspecting drivers because there is still over 10 miles of steep grade road before reaching the city, a series of signs warns truckers of the steep grade

Interstate 70
–
I-70 passes through Spotted Wolf Canyon inside the

San Rafael Swell
Interstate 70
–
Interstate 70

Interstate 70
–
I-70 at

Glenwood Canyon
Interstate 70
–
I-70 at the portal of the

Eisenhower–Johnson Memorial Tunnel. The traffic signal is controlled from a command center and used for incident management.

27.
Centered hexagonal number
–
The nth centered hexagonal number is given by the formula n 3 −3 =3 n +1. Expressing the formula as 1 +6 shows that the centered hexagonal number for n is 1 more than 6 times the th triangular number. The first few centered hexagonal numbers are,1,7,19,37,61,91,127,169,217,271,331,397,469,547,631,721,817,919. In base 10 one can notice that the hexagonal numbers rightmost digits follow the pattern 1–7–9–7–1, the sum of the first n centered hexagonal numbers is n3. That is, centered hexagonal pyramidal numbers and cubes are the same numbers, viewed from the opposite perspective, centered hexagonal numbers are differences of two consecutive cubes, so that the centered hexagonal numbers are the gnomon of the cubes. In particular, prime centered hexagonal numbers are cuban primes, the difference between 2 and the nth centered hexagonal number is a number of the form 3n2 + 3n −1, while the difference between 2 and the nth centered hexagonal number is a pronic number. Hexagonal number Magic hexagon Star number

Centered hexagonal number

28.
Primitive semiperfect number
–
In number theory, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its divisors is a perfect number. The first few numbers are 6,12,18,20,24,28,30,36,40. Every multiple of a number is semiperfect. A semiperfect number that is not divisible by any smaller number is primitive. Every number of the form 2mp for a number m. In particular, every number of the form 2m is semiperfect, the smallest odd semiperfect number is 945. A semiperfect number is necessarily either perfect or abundant, an abundant number that is not semiperfect is called a weird number. With the exception of 2, all primary pseudoperfect numbers are semiperfect, every practical number that is not a power of two is semiperfect. The natural density of the set of semiperfect numbers exists, a primitive semiperfect number is a semiperfect number that has no semiperfect proper divisor. The first few semiperfect numbers are 6,20,28,88,104,272,304,350. There are infinitely many such numbers, all numbers of the form 2mp, with p a prime between 2m and 2m+1, are primitive semiperfect, but this is not the only form, for example,770. Hemiperfect number Erdős–Nicolas number Friedman, Charles N, sums of divisors and Egyptian fractions. Weisstein, Eric W. Primitive semiperfect number

Primitive semiperfect number
–
Overview

29.
Pronic number
–
A pronic number is a number which is the product of two consecutive integers, that is, a number of the form n. The study of these dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers, however, the rectangular number name has also been applied to the composite numbers. The first few numbers are,0,2,6,12,20,30,42,56,72,90,110,132,156,182,210,240,272,306,342,380,420,462 …. The nth pronic number is also the difference between the odd square 2 and the st centered hexagonal number. The sum of the reciprocals of the numbers is a telescoping series that sums to 1,1 =12 +16 +112 ⋯ = ∑ i =1 ∞1 i. The partial sum of the first n terms in this series is ∑ i =1 n 1 i = n n +1, the nth pronic number is the sum of the first n even integers. It follows that all numbers are even, and that 2 is the only prime pronic number. It is also the only number in the Fibonacci sequence. The number of entries in a square matrix is always a pronic number. The fact that consecutive integers are coprime and that a number is the product of two consecutive integers leads to a number of properties. Each distinct prime factor of a number is present in only one of the factors n or n+1. Thus a pronic number is squarefree if and only if n and n +1 are also squarefree, the number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors of n and n +1. If 25 is appended to the representation of any pronic number. This is because 2 =100 n 2 +100 n +25 =100 n +25

Pronic number
–
Overview

30.
Centered triangular number
–
A centered triangular number is a centered figurate number that represents a triangle with a dot in the center and all other dots surrounding the center in successive triangular layers. The centered triangular number for n is given by the formula 3 n 2 +3 n +22, each centered triangular number from 10 onwards is the sum of three consecutive regular triangular numbers. Also each centered triangular number has a remainder of 1 when divided by three and the quotient is the regular triangular number. The sum of the first n centered triangular numbers is the constant for an n by n normal magic square for n >2. A centered triangular prime is a triangular number that is prime. The first few centered triangular primes are 19,31,109,199,409, lancelot Hogben, Mathematics for the Million. Republished by W. W. Norton & Company, ISBN 978-0-393-31071-9 Weisstein, Eric W. Centered Triangular Number

Centered triangular number
–
This article has an unclear citation style. The references used may be made clearer with a different or consistent style of

citation,

footnoting, or

external linking. (June 2014)

31.
12 (number)
–
12 is the natural number following 11 and preceding 13. The product of the first three factorials, twelve is a highly composite number, divisible by 2,3,4. It is central to systems of counting, including the Western calendar and units of time. The word twelve is the largest number with a name in English. Such uses gradually disappeared with the introduction of Arabic numerals during the 12th-century Renaissance and it derives from the Old English twelf and tuelf, first attested in the 10th-century Lindisfarne Gospels Book of John. It has cognates in every Germanic language, whose Proto-Germanic ancestor has been reconstructed as *twaliƀi, from *twa and suffix *-lif- or *-liƀ- of uncertain meaning. It is sometimes compared with the Lithuanian dvýlika, although -lika is used as the suffix for all numbers from 11 to 19, every other Indo-European language instead uses a form of two+ten, such as the Latin duōdecim. The usual ordinal form is twelfth but dozenth or duodecimal is also used in some contexts, similarly, a group of twelve things is usually a dozen but may also be referred to as a duodecad. The adjective referring to a group of twelve is duodecuple, as with eleven, the earliest forms of twelve are often considered to be connected with Proto-Germanic *liƀan or *liƀan, with the implicit meaning that two is left after having already counted to ten. The Lithuanian suffix is also considered to share a similar development, the suffix *-lif- has also been connected with reconstructions of the Proto-Germanic for ten. While, as mentioned above,12 has its own name in Germanic languages such as English and German, it is a number in many other languages, e. g. Italian dodici. In Germany, according to an old tradition, the numbers 0 through 12 were spelt out, the Duden now calls this tradition outdated and no longer valid, but many writers still follow it. Another system spells out all numbers written in one or two words, Twelve is a composite number, the smallest number with exactly six divisors, its divisors being 1,2,3,4,6 and 12. Twelve is also a composite number, the next one being twenty-four. Twelve is also a highly composite number, the next one being sixty. It is the first composite number of the form p2q, a square-prime,12 has an aliquot sum of 16. Accordingly,12 is the first abundant number and demonstrates an 8-member aliquot sequence,12 is the 3rd composite number in the 3-aliquot tree, the only number which has 12 as its aliquot sum is the square 121. Only 2 other square primes are abundant, Twelve is a sublime number, a number that has a perfect number of divisors, and the sum of its divisors is also a perfect number

12 (number)
–
12 stars are featured on the

Flag of Europe
32.
15 (number)
–
15 is the natural number following 14 and preceding 16. In English, it is the smallest natural number with seven letters in its spelled name, in spoken English, the numbers 15 and 50 are often confused because they sound similar. When carefully enunciated, they differ in which syllable is stressed,15 /fɪfˈtiːn/ vs 50 /ˈfɪfti/, however, in dates such as 1500 or when contrasting numbers in the teens, the stress generally shifts to the first syllable,15 /ˈfɪftiːn/. In a 24-hour clock, the hour is in conventional language called three or three oclock. A composite number, its divisors being 1,3 and 5. A repdigit in binary and quaternary, in hexadecimal, as well as all higher bases,15 is represented as F. the 4th discrete semiprime and the first member of the discrete semiprime family. It is thus the first odd discrete semiprime, the number proceeding 15,14 is itself a discrete semiprime and this is the first such pair of discrete semiprimes. The next example is the pair commencing 21, the smallest number that can be factorized using Shors quantum algorithm. With only two exceptions, all prime quadruplets enclose a multiple of 15, with 15 itself being enclosed by the quadruplet, the aliquot sum of 15 is 9, a square prime 15 has an aliquot sequence of 6 members. 15 is the composite number in the 3-aliquot tree. The abundant 12 is also a member of this tree, fifteen is the aliquot sum of the consecutive 4-power 16, and the discrete semiprime 33. 15 and 16 form a Ruth-Aaron pair under the definition in which repeated prime factors are counted as often as they occur. There are 15 solutions to Známs problem of length 7, if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers via the 15 and 290 theorems. Group 15 of the table are sometimes known as the pnictogens. 15 Madadgar is designated as a number in Pakistan, for mobile phones, similar to the international GSM emergency number 112, if 112 is used in Pakistan. 112 can be used in an emergency if the phone is locked. The Hanbali Sunni madhab states that the age of fifteen of a solar or lunar calendar is when ones taklif begins and is the stage whereby one has his deeds recorded. In the Hebrew numbering system, the number 15 is not written according to the method, with the letters that represent 10 and 5

15 (number)
–
Fifteen total individuals in the

mollusk species

Donax variabilis comprise the entire coloration and patterning in their

phenotypes.

15 (number)
–
The 15 perfect matchings of K 6

33.
17 (number)
–
17 is the natural number following 16 and preceding 18. In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar, when carefully enunciated, they differ in which syllable is stressed,17 /sɛvənˈtiːn/ vs 70 /ˈsɛvənti/. However, in such as 1789 or when contrasting numbers in the teens, such as 16,17,18. The number 17 has wide significance in pure mathematics, as well as in applied sciences, law, music, religion, sports,17 is the sum of the first 4 prime numbers. In a 24-hour clock, the hour is in conventional language called five or five oclock. Seventeen is the 7th prime number, the next prime is nineteen, with which it forms a twin prime. 17 is the sixth Mersenne prime exponent, yielding 131071,17 is an Eisenstein prime with no imaginary part and real part of the form 3n −1. 17 is the third Fermat prime, as it is of the form 22n +1, specifically with n =2, since 17 is a Fermat prime, regular heptadecagons can be constructed with compass and unmarked ruler. This was proven by Carl Friedrich Gauss,17 is the only positive Genocchi number that is prime, the only negative one being −3. It is also the third Stern prime,17 is the average of the first two Perfect numbers. 17 is the term of the Euclid–Mullin sequence. Seventeen is the sum of the semiprime 39, and is the aliquot sum of the semiprime 55. There are exactly 17 two-dimensional space groups and these are sometimes called wallpaper groups, as they represent the seventeen possible symmetry types that can be used for wallpaper. Like 41, the number 17 is a prime that yields primes in the polynomial n2 + n + p, the maximum possible length of such a sequence is 17. Either 16 or 18 unit squares can be formed into rectangles with equal to the area. 17 is the tenth Perrin number, preceded in the sequence by 7,10,12, in base 9, the smallest prime with a composite sum of digits is 17. 17 is the least random number, according to the Hackers Jargon File and it is a repunit prime in hexadecimal. 17 is the possible number of givens for a sudoku puzzle with a unique solution

17 (number)
–
No row 17 in

Alitalia planes.

34.
19 (number)
–
19 is the natural number following 18 and preceding 20. In a 24-hour clock, the hour is in conventional language called seven or seven oclock. 19 is the 8th prime number, the sequence continues 23,29,31,37. 19 is the seventh Mersenne prime exponent,19 is the fifth happy number and the third happy prime. 19 is the sum of two odd discrete semiprimes,65 and 77 and is the base of the 19-aliquot tree. 19 is the number of fourth powers needed to sum up to any natural number. It is the value of g.19 is the lowest prime centered triangular number, a centered hexagonal number. The only non-trivial normal magic hexagon contains 19 hexagons,19 is the first number with more than one digit that can be written from base 2 to base 19 using only the digits 0 to 9, the other number is 20. 19 is The TCP/IP port used for chargen, astronomy, Every 19 years, the solar year and the lunar year align in whats known as the metonic cycle. Quran code, There have been claims that patterns of the number 19 are present a number of times in the Quran. The Number of Verse and Sura together in the Quran which announces Jesus son of Maryams birth, in the Bábí and Baháí faiths, a group of 19 is called a Váhid, a Unity. The numerical value of this word in the Abjad numeral system is 19, the Baháí calendar is structured such that a year contains 19 months of 19 days each, as well as a 19-year cycle and a 361-year supercycle. The Báb and his disciples formed a group of 19, There were 19 Apostles of Baháulláh. With a similar name and anti-Vietnam War theme, I Was Only Nineteen by the Australian group Redgum reached number one on the Australian charts in 1983, in 2005 a hip hop version of the song was produced by The Herd. 19 is the name of Adeles 2008 debut album, so named since she was 19 years old at the time, hey Nineteen is a song by American jazz rock band Steely Dan, written by members Walter Becker and Donald Fagen, and released on their 1980 album Gaucho. Nineteen has been used as an alternative to twelve for a division of the octave into equal parts and this idea goes back to Salinas in the sixteenth century, and is interesting in part because it gives a system of meantone tuning, being close to 1/3 comma meantone. Some organs use the 19th harmonic to approximate a minor third and they refer to the ka-tet of 19, Directive Nineteen, many names add up to 19,19 seems to permeate every aspect of Roland and his travelers lives. In addition, the ends up being a powerful key

19 (number)
–
A 19x19

Go board

19 (number)
–
19 is a

centered triangular number
35.
20 (number)
–
20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score,20 is a tetrahedral number as 1,4,10,20. 20 is the basis for vigesimal number systems,20 is the third composite number comprising the product of a squared prime and a prime, and also the second member of the q family in this form. 20 has a sum of 22. Accordingly,20 is the abundant number and demonstrates an 8-member aliquot sequence. 20 is the smallest primitive abundant number,20 is the 4th composite number in the 7-aliquot tree. Two numbers have 20 as their sum, the discrete semiprime 34. Only 2 other square primes are abundant 12 and 18,20 can be written as the sum of three Fibonacci numbers uniquely, i. e.20 =13 +5 +2. The product of the number of divisors and the number of divisors of 20 is exactly 20. 20 is the number of required to optimally solve a Rubiks Cube in the worst case. 20 is the number with more than one digit that can be written from base 2 to base 20 using only the digits 0 to 9. The third magic number in physics, the IAU shower number for Coma Berenicids. The number of amino acids that are encoded by the standard genetic code. In some countries, the number 20 is used as an index in measuring visual acuity, 20/20 indicates normal vision at 20 feet, although it is commonly used to mean perfect vision. When someone is able to see only after an event how things turned out, the Baltimore Orioles and Cincinnati Reds, both for Hall of Famer Frank Robinson. The Kansas City Royals, for Frank White, the Los Angeles Dodgers, for Hall of Famer Don Sutton. The Philadelphia Phillies, for Hall of Famer Mike Schmidt, the Pittsburgh Pirates, for Hall of Famer Pie Traynor. The St. Louis Cardinals, for Hall of Famer Lou Brock, the San Francisco Giants, for Hall of Famer Monte Irvin, who played for the team when it was the New York Giants

20 (number)
–
An

icosahedron has 20

faces
36.
21 (number)
–
21 is the natural number following 20 and preceding 22. In a 24-hour clock, the twenty-first hour is in conventional language called nine or nine oclock,21 is, the fifth discrete semiprime and the second in the family. With 22 it forms the second discrete semiprime pair, a Blum integer, since it is a semiprime with both its prime factors being Gaussian primes. A composite number, its divisors being 1,3 and 7. The sum of the first six numbers, making it a triangular number. The sum of the sum of the divisors of the first 5 positive integers, the smallest non-trivial example of a Fibonacci number whose digits are Fibonacci numbers and whose digit sum is also a Fibonacci number. The smallest natural number that is not close to a power of 2, 2n,21 has an aliquot sum of 11 though it is the second composite number found in the 11-aliquot tree with the abundant square prime 18 being the first such member. Twenty-one is the first number to be the sum of three numbers 18,51,91. 21 appears in the Padovan sequence, preceded by the terms 9,12,16, in several countries 21 is the age of majority. In most US states,21 is the drinking age, however, in Puerto Rico and U. S. Virgin Island, the drinking age is 18. In Hawaii and New York,21 is the age that one person may purchase cigarettes. In some countries it is the voting age, in the United States,21 is the age at which one can purchase multiple tickets to an R-rated film without providing Identifications. It is also the age to one under the age of 17 as their parent or adult guardian for an R-rated movie. In some states,21 is the age, persons may gamble or enter casinos. In 2011, Adele named her second studio album 21, because of her age at the time, the Milwaukee Braves, for Hall of Famer Warren Spahn, the number continues to be honored by the team in its current home of Atlanta. The Pittsburgh Pirates, for Hall of Famer Roberto Clemente, following his death in a crash while attempting to deliver humanitarian aid to victims of an earthquake in Nicaragua. In the NBA, The Atlanta Hawks, for Hall of Famer Dominique Wilkins, the Boston Celtics, for Hall of Famer Bill Sharman. The Detroit Pistons, for Hall of Famer Dave Bing, the Sacramento Kings, for Vlade Divac

21 (number)
–
Number 21 on the road bicycle of

Ellen van Dijk at the

Ronde van Drenthe.

21 (number)
–
Building called "21" in

Zlín,

Czech Republic.

21 (number)
–
Detail of the building entrance

37.
24 (number)
–
24 is the natural number following 23 and preceding 25. The SI prefix for 1024 is yotta, and for 10−24 yocto and these numbers are the largest and smallest number to receive an SI prefix to date. In a 24-hour clock, the hour is in conventional language called twelve or twelve oclock. 24 is the factorial of 4 and a number, being the first number of the form 23q. It follows that 24 is the number of ways to order 4 distinct items and it is the smallest number with exactly eight divisors,1,2,3,4,6,8,12, and 24. It is a composite number, having more divisors than any smaller number. 24 is a number, since adding up all the proper divisors of 24 except 4 and 8 gives 24. Subtracting 1 from any of its divisors yields a number,24 is the largest number with this property. 24 has a sum of 36 and the aliquot sequence. It is therefore the lowest abundant number whose aliquot sum is itself abundant, the aliquot sum of only one number,529 =232, is 24. There are 10 solutions to the equation φ =24, namely 35,39,45,52,56,70,72,78,84 and 90 and this is more than any integer below 24, making 24 a highly totient number. 24 is the sum of the prime twins 11 and 13, the product of any four consecutive numbers is divisible by 24. This is because among any four consecutive numbers there must be two numbers, one of which is a multiple of four, and there must be a multiple of three. The tesseract has 24 two-dimensional faces,24 is the only nontrivial solution to the cannonball problem, that is,12 +22 +32 + … +242 is a perfect square. In 24 dimensions there are 24 even positive definite unimodular lattices, the Leech lattice is closely related to the equally nice length-24 binary Golay code and the Steiner system S and the Mathieu group M24. The modular discriminant Δ is proportional to the 24th power of the Dedekind eta function η, Δ = 12η24, the Barnes-Wall lattice contains 24 lattices. 24 is the number whose divisors — namely 1,2,3,4,6,8,12,24 — are exactly those numbers n for which every invertible element of the commutative ring Z/nZ is a square root of 1. It follows that the multiplicative group × = is isomorphic to the additive group 3 and this fact plays a role in monstrous moonshine

24 (number)
–
Astronomical clock in Prague

38.
25 (number)
–
25 is the natural number following 24 and preceding 26. It is a number, being 52 =5 ×5. It is one of two numbers whose square and higher powers of the number also ends in the same last two digits, e. g.252 =625, the other is 76. It is the smallest square that is also a sum of two squares,25 =32 +42, hence it often appears in illustrations of the Pythagorean theorem. 25 is the sum of the odd natural numbers 1,3,5,7 and 9. 25 is an octagonal number, a centered square number. 25 percent is equal to 1/4,25 has an aliquot sum of 6 and number 6 is the first number to have an aliquot sequence that does not culminate in 0 through a prime. 25 is the sum of three integers,95,119, and 143. Twenty-five is the second member of the 6-aliquot tree. It is the smallest base 10 Friedman number as it can be expressed by its own digits,52 and it is also a Cullen number. 25 is the smallest pseudoprime satisfying the congruence 7n =7 mod n.25 is the smallest aspiring number — a composite non-sociable number whose aliquot sequence does not terminate. Within base 10 one can readily test for divisibility by 25 by seeing if the last two digits of the number match 00,25,50 or 75. 25 and 49 are the perfect squares in the following list,13,25,37,49,511,613,715,817,919,1021,1123,1225,1327,1429. The formula in this list can be described as 10nZ + where n depends on the number of digits in Z, in base 30,25 is a 1-automorphic number, and in base 10 a 2-automorphic number. The percent DNA overlap of a half-sibling, grandparent, grandchild, aunt, uncle, niece, nephew, identical twin cousin, in Ezekiels vision of a new temple, The number twenty-five is of cardinal importance in Ezekiels Temple Vision. In The Book of Revelations New International Version, Surrounding the throne were twenty-four other thrones and they were dressed in white and had crowns of gold on their heads. In Islam, there are 25 prophets mentioned in the Quran, the size of the full roster on a Major League Baseball team for most of the season, except for regular-season games on or after September 1, when teams expand their roster to 40 players. The size of the roster on a Nippon Professional Baseball team for a particular game

25 (number)
–
25 is a square

39.
30 (number)
–
30 is the natural number following 29 and preceding 31. Thirty is the sum of the first four squares, which makes it a square pyramidal number and it is a primorial and is the smallest Giuga number. 30 is the smallest sphenic number, and the smallest of the form 2 ×3 × r,30 has an aliquot sum of 42, the second sphenic number and all sphenic numbers of this form have an aliquot sum 12 greater than themselves. The aliquot sequence of 30 is 16 members long, it comprises Thirty has but one number for which it is the aliquot sum, adding up some subsets of its divisors gives 30, hence 30 is a semiperfect number. 30 is the largest number such that all smaller than itself. A polygon with thirty sides is called a triacontagon, the icosahedron and the dodecahedron are Platonic solids with 30 edges. The icosidodecahedron is an Archimedean solid with 30 vertices, and the Tutte–Coxeter graph is a graph with 30 vertices. The atomic number of zinc is 30 Messier object M30, a magnitude 8, the duration of Saros series 30 was 1496.5 years, and it contained 84 solar eclipses. Further, the Saros number of the lunar eclipse series began on June 19,1803 BC. The duration of Saros series 30 was 1316.2 years, Thirty is, Used to indicate the end of a newspaper story, a copy editors typographical notation. S. Judas Iscariot betrayed Jesus for 30 pieces of silver, one of the rallying-cries of the 1960s student/youth protest movement was the slogan, Dont trust anyone over thirty. In Franz Kafkas novel The Trial Joseph wakes up on the morning of his birthday to find himself under arrest for an unspecified crime. After making many attempts to find the nature of the crime or the name of his accuser. The number of uprights that formed the Sarsen Circle at Stonehenge, western Christianitys most prolific 20th century essayist, F. W. Also in that essay Boreham writes It was said of Keats, in tennis, the number 30 represents the second point gained in a game. Under NCAA rules for basketball, the offensive team has 30 seconds to attempt a shot. As of 2012, three of the four major leagues in the United States and Canada have 30 teams each. The California Angels baseball team retired the number in honor of its most notable wearer, Nolan Ryan, the San Francisco Giants extended the same honor to Orlando Cepeda

30 (number)
–
For other uses, see

The Thirty.

40.
31 (number)
–
31 is the natural number following 30 and preceding 32. As a Mersenne prime,31 is related to the perfect number 496,31 is also the 4th lucky prime and the 11th supersingular prime. 31 is a triangular number, the lowest prime centered pentagonal number. For the Steiner tree problem,31 is the number of possible Steiner topologies for Steiner trees with 4 terminals, at 31, the Mertens function sets a new low of −4, a value which is not subceded until 110. No integer added up to its base 10 digits results in 31,31 is a repdigit in base 5, and base 2. The numbers 31,331,3331,33331,333331,3333331, for a time it was thought that every number of the form 3w1 would be prime. Here,31 divides every fifteenth number in 3w1, the atomic number of gallium Messier object M31, a magnitude 4.5 galaxy in the constellation Andromeda. It is also known as the Andromeda Galaxy, and is visible to the naked eye in a modestly dark sky. The New General Catalogue object NGC31, a galaxy in the constellation Phoenix The Saros number of the solar eclipse series which began on -1805 January 31. The duration of Saros series 31 was 1316.2 years, the Saros number of the lunar eclipse series which began on -1774 May 30 and ended on -476 July 17. The duration of Saros series 31 was 1298.1 years, the jersey number 31 has been retired by several North American sports teams in honor of past playing greats, In Major League Baseball, The San Diego Padres, for Dave Winfield. The Chicago Cubs, for Ferguson Jenkins and Greg Maddux, the Atlanta Braves, also for Maddux. The New York Mets, for Mike Piazza, in the NBA, The Boston Celtics, for Cedric Maxwell. The Indiana Pacers, for Reggie Miller, in the NHL, The Edmonton Oilers, for Grant Fuhr. The New York Islanders, for Billy Smith, in the NFL, The Atlanta Falcons, for William Andrews. The New Orleans Saints, for Jim Taylor, NASCAR driver Jeff Burton drives #31, a car which was subject to a controversy when one of the sponsors changed its name after merging with another company. In ice hockey goaltenders often wear the number 31, in football the number 31 has been retired by Queens Park Rangers F. C.31 from the Prime Pages

31 (number)
–
31 is a

centered pentagonal number
41.
34 (number)
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34 is the natural number following 33 and preceding 35. 34 is the ninth distinct semiprime and has four divisors including one and its neighbors,33 and 35, also are distinct semiprimes, having four divisors each, and 34 is the smallest number to be surrounded by numbers with the same number of divisors as it has. It is also in the first cluster of three distinct semiprimes, being within 33,34,35, the next cluster of semiprimes is 85,86,87. It is the ninth Fibonacci number and a companion Pell number, since it is an odd-indexed Fibonacci number,34 is a Markov number, appearing in solutions with other Fibonacci numbers, such as, etc. This number is the constant of a 4 by 4 normal magic square. It has the sum,20, in the following descending sequence 34,20,22,14,10,8,7,1. There is no solution to the equation φ =34, making 34 a nontotient, nor is there a solution to the equation x − φ =34, making 34 a noncototient. The atomic number of selenium One of the numbers in physics. Messier object M34, a magnitude 6, the duration of Saros series 34 was 1532.5 years, and it contained 86 solar eclipses. The Saros number of the lunar eclipse series began on 1633 BC May. The duration of Saros series 34 was 1298.1 years, the Minnesota Twins, for Hall of Famer Kirby Puckett. The Oakland Athletics and Milwaukee Brewers, both for Hall of Famer Rollie Fingers, the Boston Red Sox have announced they will retire the number for David Ortiz in 2017. Additionally, the Los Angeles Dodgers have not issued the number since the departure of Fernando Valenzuela following the 1990 season, under current team policy, Valenzuelas number is not eligible for retirement because he is not in the Hall of Fame. In the NBA, The Houston Rockets, for Hall of Famer Hakeem Olajuwon, the Los Angeles Lakers retired the number for Hall of Famer Shaquille ONeal on April 2,2013. In the NFL, The Chicago Bears, for Hall of Famer Walter Sweetness Payton, the Houston Oilers, for Hall of Famer Earl Campbell. The franchise continues to honor the number in its current incarnation as the Tennessee Titans, in the NCAA, The Auburn University Tigers, for Hall of Famer Bo Jackson. In The Count of Monte Cristo, Number 34 is how Edmond Dantès is referred to during his imprisonment in the Château dIf.34 from the Prime Pages

34 (number)
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← 33
42.
35 (number)
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35 is the natural number following 34 and preceding 36. 35 is the sum of the first five numbers, making it a tetrahedral number. 35 is the number of ways that three things can be selected from a set of seven unique things also known as the combination of seven things taken three at a time,35 is a centered cube number, a pentagonal number and a pentatope number. 35 is a highly cototient number, since there are solutions to the equation x − φ =35 than there are for any other integers below it except 1. There are 35 free hexominoes, the polyominoes made from six squares, since the greatest prime factor of 352 +1 =1226 is 613, which is obviously more than 35 twice,35 is a Størmer number. 35 is a semiprime, the tenth, and the first with 5 as the lowest non-unitary factor. The aliquot sum of 35 is 13 this being the composite number with such an aliquot sum. 35 is the last member of the first triple cluster of semiprimes 33,34,35, the second such triple discrete semiprime cluster is 85,86,87. 35 is the highest number one can count to on ones fingers using base 6, the Chicago White Sox, for 2014 Hall of Fame inductee Frank Thomas. The San Diego Padres, for Randy Jones, in the NBA, The Boston Celtics, for Reggie Lewis. The Indiana Pacers, for Roger Brown, the Utah Jazz, for Darrell Griffith. The Golden State Warriors, for Kevin Durant In the NHL, The Chicago Blackhawks, in MotoGP,35 is the rider number of British rider, Cal Crutchlow. 35 mm film is the film gauge most commonly used for both analog photography and motion pictures The minimum age of candidates for election to the United States or Irish Presidency. 35 is used as a slang term throughout North America to denote failure, hardship, or self-defeat

35 (number)
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The 35 free hexominoes

43.
37 (number)
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37 is the natural number following 36 and preceding 38. Thirty-seven is the 12th prime number, a prime with 73. It is a hexagonal number and a star number. Every positive integer is the sum of at most 37 fifth powers,37 appears in the Padovan sequence, preceded by the terms 16,21, and 28. Since the greatest prime factor of 372 +1 =1370 is 137, the atomic number of rubidium The normal human body temperature in degrees Celsius Messier object M37, a magnitude 6. The duration of Saros series 37 was 1298.1 years, the Saros number of the lunar eclipse series which began on -1492 April 3 and ended on -194 May 22. The duration of Saros series 37 was 1298.1 years, kepler-37b is the smallest known planet. The New York Yankees, also for Stengel and this honor made him the first manager to have had his number retired by two different teams. In the NFL, The Detroit Lions, for Doak Walker, the San Francisco 49ers, for Jimmy Johnson. Thirty-seven is, The number of plays William Shakespeare is thought to have written, today the +37 prefix is shared by Lithuania, Latvia, Estonia, Moldova, Armenia, Belarus, Andorra, Monaco, San Marino and Vatican City. A television channel reserved for radio astronomy in the United States The number people are most likely to state when asked to give a number between 0 and 100. The inspiration for the album 37 Everywhere by Punchline List of highways numbered 37 Number Thirty-Seven, Pennsylvania, unincorporated community in Cambria County, Pennsylvania I37

37 (number)
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House number in

Baarle (in its Belgian part)

44.
38 (number)
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38 is the natural number following 37 and preceding 39. 38 is the 11th distinct semiprime and the 7th in the family and it is the initial member of the third distinct semiprime pair. 38 has a sum of 22 which is itself a distinct semiprime In fact 38 is the first number to be at the head of a chain of four distinct semiprimes in its 8-member aliquot sequence. 38 is the 8th member of the 7-aliquot tree, −1 yields 523022617466601111760007224100074291199999999, which is the 16th factorial prime. There is no answer to the equation φ =38, making 38 a nontotient,38 is the sum of the squares of the first three primes. 37 and 38 are the first pair of positive integers not divisible by any of their digits. 38 is the largest even number which cannot be written as the sum of two odd composite numbers, there are only two normal magic hexagons, order 1 and order 3. The sum of row of an order 3 magic hexagon is 38. The duration of Saros series 38 was 1298.1 years, the lunar eclipse series which began on -1408 April 16 and ended on -111 June 3. The duration of Saros series 38 was 1298.1 years, the New General Catalogue object NGC38, a spiral galaxy in the constellation Pisces Thirty-eight is also, The 38th parallel north is the pre-Korean War boundary between North Korea and South Korea

38 (number)
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Most people will see the number 38, but people with red-green color blindness might see

88 instead.

45.
39 (number)
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39 is the natural number following 38 and preceding 40. Thirty-nine is the sum of consecutive primes and also is the product of the first, among small semiprimes only three other integers share this attribute. 39 also is the sum of the first three powers of 3, given 39, the Mertens function returns 0. 39 is the smallest natural number which has three partitions into three parts which all give the product when multiplied. 39 is the 12th distinct semiprime and the 4th in the family and it is the last member of the third distinct biprime pair. 39 has a sum of 17 which is itself a prime. 39 is the 4th member of the 17-aliquot tree and it is a perfect totient number. The thirteenth Perrin number is 39, which comes after 17,22,29, since the greatest prime factor of 392 +1 =1522 is 761, which is obviously more than 39 twice,39 is a Størmer number. The F26A graph is a graph with 39 edges. The atomic number of yttrium Astronomy Messier object Open Cluster M39, the duration of Saros series 39 was 1298.1 years, and it contained 73 lunar eclipses. The retired jersey number of baseball player Roy Campanella The book series The 39 Clues revolves around 39 clues hidden around the world. Glorious 39 is a 2009 drama film set at the beginning of World War II In the CBS reality show Survivor, the number of episodes done during its one season in 1955-1956 of The Honeymooners television series is commonly referred to as the Classic 39. I-39 is the 39th shortest of the two digit Interstates. The bowling lane normally consists of 39 wooden boards

39 (number)
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The

F26A graph has 39 edges, all equivalent.

46.
40 (number)
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Despite being related to the word four, the modern spelling of 40 is forty. The archaic form fourty is now considered a misspelling, the modern spelling possibly reflects a pronunciation change due to the horse–hoarse merger. Forty is a number, an octagonal number, and as the sum of the first four pentagonal numbers. Adding up some subsets of its divisors gives 40, hence 40 is a semiperfect number, given 40, the Mertens function returns 0. 40 is the smallest number n with exactly 9 solutions to the equation φ = n, Forty is the number of n-queens problem solutions for n =7. Since 402 +1 =1601 is prime,40 is a Størmer number,40 is a repdigit in base 3 and a Harshad number in base 10. Negative forty is the temperature at which the Fahrenheit and Celsius scales correspond. It is referred to as either minus forty or forty below, the planet Venus forms a pentagram in the night sky every eight years with it returning to its original point every 40 years with a 40-day regression. The duration of Saros series 40 was 1280.1 years, lunar eclipse series which began on -1387 February 12 and ended on -71 April 12. The duration of Saros series 40 was 1316.2 years, the number 40 is used in Jewish, Christian, Islamic, and other Middle Eastern traditions to represent a large, approximate number, similar to umpteen. In the Hebrew Bible, forty is often used for periods, forty days or forty years. Rain fell for forty days and forty nights during the Flood, spies explored the land of Israel for forty days. The Hebrew people lived in the Sinai desert for forty years and this period of years represents the time it takes for a new generation to arise. Moses life is divided into three 40-year segments, separated by his growing to adulthood, fleeing from Egypt, and his return to lead his people out, several Jewish leaders and kings are said to have ruled for forty years, that is, a generation. Examples include Eli, Saul, David, and Solomon, goliath challenged the Israelites twice a day for forty days before David defeated him. He went up on the day of Tammuz to beg forgiveness for the peoples sin. He went up on the first day of Elul and came down on the day of Tishrei. A mikvah consists of 40 seah of water 40 lashes is one of the punishments meted out by the Sanhedrin, One of the prerequisites for a man to study Kabbalah is that he is forty years old

40 (number)
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The number on the logo for the American-Japanese hard rock band Crush 40.

47.
42 (number)
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42 is the natural number that succeeds 41 and precedes 43. Forty-two is a number and an abundant number, its prime factorization 2 ·3 ·7 makes it the second sphenic number. As with all numbers of this form, the aliquot sum is abundant by 12. 42 is also the second number to be bracketed by twin primes,30 is also a pronic number. 42 has a 14-member aliquot sequence 42,54,66,78,90,144,259,45,33,15,9,4,3,1,0 and is part of the aliquot sequence commencing with the first sphenic number 30. Further,42 is the 10th member of the 3-aliquot tree, additional properties of the number 42 include, It is the third primary pseudoperfect number. It is an alternating sign matrix number, that is, the number of 4-by-4 alternating sign matrices and it is the number of partitions of 10—the number of ways of expressing 10 as a sum of positive integers. It is the third pentadecagonal number and it is a meandric number and an open meandric number. It is conjectured to be the factor in the leading order term of the sixth moment of the Riemann zeta function. In particular, Conrey & Ghosh have conjectured that 1 T ∫0 T | ζ |6 d t ∼429, ∏ p 4 log 9 T. where the infinite product is over all prime numbers, p.42 is a Størmer number. Whether there are other remains a open question. 42 is a number, as σ2 = σ = 6n. 42 is the number of the original Smith number, Both the sum of its digits. The dimension of the Borel subalgebra in the exceptional Lie algebra e6 is 42,42 is a perfect score on the USA Math Olympiad and International Mathematical Olympiad. 42 is the maximum of core points awarded in International Baccalaureate Diploma Programme,42 is the sum of the first 6 positive even numbers. 42 is the number of molybdenum. 42 is the mass of one of the naturally occurring stable isotopes of calcium. The angle rounded to whole degrees for which a rainbow appears, the first half of the journey consists of free-fall acceleration, while the second half consists of an exactly equal deceleration

42 (number)
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Jackie Robinson in his now-retired number 42 jersey.

42 (number)
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The 3 × 3 × 3

magic cube with rows summing to 42.

48.
45 (number)
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45 is the natural number following 44 and followed by 46. Forty-five is a number, and in particular the sum of all the decimal digits. It is the smallest triangle number which can be written as the sum of two squares and it is also a hexagonal and 16-gonal number. 45 is the positive integer with a prime factorization of the form p2q. 33 is the sum of 45 and the aliquot sequence of 45 is. Since the greatest prime factor of 452 +1 =2026 is 1013, in base 10, it is a Kaprekar number and a Harshad number. The duration of Saros series 45 was 1280.1 years, the Saros number of the lunar eclipse series which began on −1369 August 19 and ended on 182 March. The duration of Saros series 45 was 1550.6 years, a card game, Forty-five.45, a 2006 motion picture. In the United States,45 is often a reference to one of two specific.45 caliber cartridges— the.45 Colt or the.45 ACP, in years of marriage, the sapphire wedding anniversary. Forty Five a Big Finish 2008 audio play made for the forty fifth anniversary of the British science fiction television show Doctor Who, issue 45 of The North Briton was thought to be seditious but its publisher, John Wilkes, was celebrated as a champion of liberty. The number 45 was used as a symbol of support for him, banquets were held with a theme of 45 while many items were produced showing the number or featuring it in some way. For example, a wig was produced with 45 curls

45 (number)
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45

rpm gramophone record
49.
46 (number)
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46 is the natural number following 45 and preceding 47. Forty-six is a Wedderburn-Etherington number, a number and a centered triangular number. It is the sum of the totient function for the first twelve integers,46 is the largest even integer that can not be expressed as a sum of two abundant numbers. 46 is the third semiprime with an aliquot sum. The aliquot sequence of 46 is, since it is possible to find sequences of 46 consecutive integers such that each inner member shares a factor with either the first or the last member,46 is an Erdős–Woods number. The approximate molar mass of ethanol Messier object M46, a magnitude 6.5 open cluster in the constellation Puppis, the New General Catalogue object NGC46, a star in the constellation Pisces. The Saros number of the solar eclipse series began on April 1,1371 BC. The duration of Saros series 46 was 1280.1 years, the Saros number of the lunar eclipse series which began on July 19,1358 BC and ended on October 8,12. The duration of Saros series 46 was 1370.5 years, the number of mountains in the 46 peaks of the Adirondack mountain range. People who have climbed all of them are called forty-sixers, there is also an unofficial 47th peak, the name of a defensive scheme used in American football, see 46 defense. The total of books in the Old Testament, Catholic version, the number corresponding to the word ADAM where A=1, D=4, M=40. Forty-six is also, The code for international direct dial phone calls to Sweden, the number of samurai, out of 47, who carried out the attack in the historical Ako vendetta, sometimes referred to as the 46 Ronins to discount the one samurai forced to turn back. In the title of the movie Code 46, starring Tim Robbins, several routes numbered 46 exist throughout the world. Because 46 in Japanese can be pronounced as yon roku, and yoroshiku（よろしく） means my best regards in Japanese,46 is the number of the City Chevrolet and Superflo cars driven by Cole Trickle in the movie Days of Thunder. The number of the French department Lot,46 is the number that unlocks the Destiny spaceship on the popular Sci-Fi TV show Stargate Universe. Dr. Rush discovers that the number 46 relates to the amount of human chromosomes, the number depicted in the first flag of Oklahoma, signifying the fact that Oklahoma was the 46th state to join the United States

46 (number)
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Flag of Oklahoma (1911–1925)

50.
52 (number)
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52 is the natural number following 51 and preceding 53. Fifty-two is the 6th Bell number and a decagonal number and it is an untouchable number, since it is never the sum of proper divisors of any number, and it is a noncototient since it is never the answer to the equation x − φ. The atomic number of tellurium Messier object M52, a magnitude 8.0 open cluster in the constellation Cassiopeia, the New General Catalogue object NGC52, a spiral galaxy in the constellation Pegasus. The Saros number of the solar eclipse series began on -1378 August 14. The duration of Saros series 52 was 1532.5 years, the Saros number of the lunar eclipse series which began on -1026 May 27 and ended on 204 June. The duration of Saros series 52 was 1280.1 years, U. S. Route 52 that runs from South Carolina to North Dakota Fifty-two is, The approximate number of weeks in a year. 52 weeks is 364 days, while the year is 365.24 days long. According to ISO8601, most years have 52 weeks while some have 53, the New 52 is a 2011 revamp and relaunch by DC Comics of its entire line of ongoing monthly superhero books. 52 is the car number of retired NASCAR driver Jimmy Means 52 American hostages were held in the Iran hostage crisis 52 BC, AD52,1952,2052, etc

52 (number)
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The

piano has 52 white keys