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48-564: December's name derives from the Latin word decem (meaning ten ) because it was originally the tenth month of the year in the calendar of Romulus c.  750 BC , which began in March. The winter days following December were not included as part of any month. Later, the months of January and February were created out of the monthless period and added to the beginning of the calendar, but December retained its name. In Ancient Rome , as one of

96-422: A plane-vertex alongside other regular polygons alone; the first regular polygon unable to do so is the eleven-sided hendecagon . While the regular decagon cannot tile alongside other regular figures, ten of the eleven regular and semiregular tilings of the plane are Wythoffian (the elongated triangular tiling is the only exception); however, the plane can be covered using overlapping decagons, and

144-409: A finite even number, and odd otherwise. Let R be a commutative ring and let I be an ideal of R whose index is 2. Elements of the coset 0 + I {\displaystyle 0+I} may be called even , while elements of the coset 1 + I {\displaystyle 1+I} may be called odd . As an example, let R = Z (2) be the localization of Z at

192-571: A negative Cartan matrix determinant , of −1. There are precisely ten affine Coxeter groups that admit a formal description of reflections across n {\displaystyle n} dimensions in Euclidean space. These contain infinite facets whose quotient group of their normal abelian subgroups is finite. They include the one-dimensional Coxeter group I ~ 1 {\displaystyle {\tilde {I}}_{1}} [ ∞ ], which represents

240-468: A permutation (as defined in abstract algebra ) is the parity of the number of transpositions into which the permutation can be decomposed. For example (ABC) to (BCA) is even because it can be done by swapping A and B then C and A (two transpositions). It can be shown that no permutation can be decomposed both in an even and in an odd number of transpositions. Hence the above is a suitable definition. In Rubik's Cube , Megaminx , and other twisting puzzles,

288-566: Is divisible by 2: 2   |   x {\displaystyle 2\ |\ x} and an odd number is not: 2 ⧸ |   x {\displaystyle 2\not |\ x} The sets of even and odd numbers can be defined as following: { 2 k : k ∈ Z } {\displaystyle \{2k:k\in \mathbb {Z} \}} { 2 k + 1 : k ∈ Z } {\displaystyle \{2k+1:k\in \mathbb {Z} \}} The set of even numbers

336-535: Is "deca-". The meaning "10" is part of the following terms: Also, the number 10 plays a role in the following: The metric system is based on the number 10, so converting units is done by adding or removing zeros (e.g. 1 centimetre = 10 millimetres, 1 decimetre = 10 centimetres, 1 meter = 100 centimetres, 1 dekametre = 10 meters, 1 kilometre = 1,000 meters). The Ten Commandments in the Hebrew Bible are ethical commandments decreed by God (to Moses ) for

384-614: Is a prime ideal of Z {\displaystyle \mathbb {Z} } and the quotient ring Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } is the field with two elements . Parity can then be defined as the unique ring homomorphism from Z {\displaystyle \mathbb {Z} } to Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } where odd numbers are 1 and even numbers are 0. The consequences of this homomorphism are covered below. The following laws can be verified using

432-410: Is a number that has an odd number of 1's in its binary representation; these numbers play an important role in the strategy for the game Kayles . The parity function maps a number to the number of 1's in its binary representation, modulo 2 , so its value is zero for evil numbers and one for odious numbers. The Thue–Morse sequence , an infinite sequence of 0's and 1's, has a 0 in position i when i

480-484: Is an almost simple group, of order , It functions as a point stabilizer of degree 11 inside the smallest sporadic simple group M 11 {\displaystyle \mathrm {M} _{11}} , a group with an irreducible faithful complex representation in ten dimensions, and an order equal to   7920 = 11 ⋅ 10 ⋅ 9 ⋅ 8 {\displaystyle 7920=11\cdot 10\cdot 9\cdot 8}   that

528-427: Is an integer, it will be even if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad , to be neither fully odd nor fully even. Some of this sentiment survived into the 19th century: Friedrich Wilhelm August Fröbel 's 1826 The Education of Man instructs the teacher to drill students with the claim that 1 is neither even nor odd, to which Fröbel attaches

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576-529: Is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2. All prime numbers are odd, with one exception: the prime number 2. All known perfect numbers are even; it is unknown whether any odd perfect numbers exist. Goldbach's conjecture states that every even integer greater than 2 can be represented as a sum of two prime numbers. Modern computer calculations have shown this conjecture to be true for integers up to at least 4 × 10 , but still no general proof has been found. The parity of

624-513: Is equivalent to the Penrose P2 tiling when it is decomposed into kites and rhombi that are proportioned in golden ratio . The regular decagon is also the Petrie polygon of the regular dodecahedron and icosahedron , and it is the largest face that an Archimedean solid can contain, as with the truncated dodecahedron and icosidodecahedron . There are ten regular star polychora in

672-405: Is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any even number is 0, 2, 4, 6, or 8. The same idea will work using any even base. In particular, a number expressed in the binary numeral system is odd if its last digit is 1; and it is even if its last digit is 0. In an odd base,

720-413: Is evil, and a 1 in that position when i is odious. In information theory , a parity bit appended to a binary number provides the simplest form of error detecting code . If a single bit in the resulting value is changed, then it will no longer have the correct parity: changing a bit in the original number gives it a different parity than the recorded one, and changing the parity bit while not changing

768-400: Is far from obvious. The parity of a function describes how its values change when its arguments are exchanged with their negations. An even function, such as an even power of a variable, gives the same result for any argument as for its negation. An odd function, such as an odd power of a variable, gives for any argument the negation of its result when given the negation of that argument. It

816-399: Is not true for normal integer arithmetic. By construction in the previous section, the structure ({even, odd}, +, ×) is in fact the field with two elements . The division of two whole numbers does not necessarily result in a whole number. For example, 1 divided by 4 equals 1/4, which is neither even nor odd, since the concepts of even and odd apply only to integers. But when the quotient

864-399: Is one more than the one-thousandth prime number, 7919. E 10 {\displaystyle \mathrm {E} _{10}} is an infinite-dimensional Kac–Moody algebra which has the even Lorentzian unimodular lattice II 9,1 of dimension 10 as its root lattice. It is the first E n {\displaystyle \mathrm {E} _{n}} Lie algebra with

912-511: Is possible for a function to be neither odd nor even, and for the case f ( x ) = 0, to be both odd and even. The Taylor series of an even function contains only terms whose exponent is an even number, and the Taylor series of an odd function contains only terms whose exponent is an odd number. In combinatorial game theory , an evil number is a number that has an even number of 1's in its binary representation , and an odious number

960-402: Is the even natural number following 9 and preceding 11 . Ten is the base of the decimal numeral system , the most common system of denoting numbers in both spoken and written language. Ten is the fifth composite number , and the smallest noncototient , which is a number that cannot be expressed as the difference between any integer and the total number of coprimes below it. Ten

1008-414: Is the average sum of the proper divisors of the natural numbers N {\displaystyle \mathbb {N} } if the size of the numbers approaches infinity, and it is the smallest number whose status as a possible friendly number is unknown. Figurate numbers that represent regular ten-sided polygons are called decagonal and centered decagonal numbers. On the other hand, 10

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1056-652: Is the eighth Perrin number , preceded by 5 , 5, and 7 . As important sums, The factorial of ten is equal to the product of the factorials of the first four odd numbers as well: 10 ! = 1 ! ⋅ 3 ! ⋅ 5 ! ⋅ 7 ! {\displaystyle 10!=1!\cdot 3!\cdot 5!\cdot 7!} , and 10 is the only number whose sum and difference of its prime divisors yield prime numbers ( 2 + 5 = 7 {\displaystyle (2+5=7} and 5 − 2 = 3 ) {\displaystyle 5-2=3)} . Ten has an aliquot sum of 8 , and

1104-458: Is the first discrete semiprime ( 2 × 5 ) {\displaystyle (2\times 5)} to be in deficit , as with all subsequent discrete semiprimes. It is the second composite in the aliquot sequence for ten (10, 8, 7 , 1 , 0 ) that is rooted in the prime 7 - aliquot tree . It is a largely composite number , as it has 4 divisors and no smaller number has more than 4 divisors. According to conjecture, ten

1152-505: Is the first non-trivial centered triangular number and tetrahedral number . 10 is also the first member in the coordination sequence for body-centered tetragonal lattices . 10 is the fourth telephone number , and the number of Young tableaux with four cells. It is also the number of n {\displaystyle n} - queens problem solutions for n = 5 {\displaystyle n=5} . There are precisely ten small Pisot numbers that do not exceed

1200-556: Is the highest rank for paracompact hyperbolic solutions , with a representation in nine dimensions. There also exist hyperbolic Lorentzian cocompact groups where removing any permutation of two nodes in its Coxeter–Dynkin diagram leaves a finite or Euclidean graph. The tenth dimension is the highest dimensional representation for such solutions, which share a root symmetry in eleven dimensions. These are of particular interest in M-theory of string theory . The SI prefix for 10

1248-909: The Geminids (December 13–14), the Monocerotids (December 7 to December 20, peaking on December 9. This shower can also start in November), the Phoenicids (November 29 to December 9, with a peak occurring around 5/6 December), the Quadrantids (typically a January shower but can also start in December), the Sigma Hydrids (December 4–15), and the Ursids (December 17-to December 25/26, peaking around December 22). The zodiac signs for

1296-595: The apeirogonal tiling , as well as the five affine Coxeter groups G ~ 2 {\displaystyle {\tilde {G}}_{2}} , F ~ 4 {\displaystyle {\tilde {F}}_{4}} , E ~ 6 {\displaystyle {\tilde {E}}_{6}} , E ~ 7 {\displaystyle {\tilde {E}}_{7}} , and E ~ 8 {\displaystyle {\tilde {E}}_{8}} that are associated with

1344-422: The face-centered cubic lattice and its higher-dimensional generalizations (the D n lattices ) consist of all of the integer points whose coordinates have an even sum. This feature also manifests itself in chess , where the parity of a square is indicated by its color: bishops are constrained to moving between squares of the same parity, whereas knights alternate parity between moves. This form of parity

1392-418: The fourth dimension , all of which have orthographic projections in the H 3 {\displaystyle \mathrm {H} _{3}} Coxeter plane that contain various decagrammic symmetries, which include compound forms of the regular decagram. M 10 {\displaystyle \mathrm {M} _{10}} is a multiply transitive permutation group on ten points. It

1440-448: The golden ratio . As a constructible polygon with a compass and straight-edge, the regular decagon has an internal angle of 12 2 = 144 {\displaystyle 12^{2}=144} degrees and a central angle of 6 2 = 36 {\displaystyle 6^{2}=36} degrees. All regular n {\displaystyle n} -sided polygons with up to ten sides are able to tile

1488-400: The prime ideal (2). Then an element of R is even or odd if and only if its numerator is so in Z . The even numbers form an ideal in the ring of integers, but the odd numbers do not—this is clear from the fact that the identity element for addition, zero, is an element of the even numbers only. An integer is even if it is congruent to 0 modulo this ideal, in other words if it

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1536-695: The summer solstice in the Southern Hemisphere , the day with the most daylight hours (excluding polar regions in both cases, which consistently have none or 24 hours , respectively, near the solstice). December in the Northern Hemisphere is the seasonal equivalent to June in the Southern Hemisphere and vice versa. In the Northern hemisphere, the beginning of the astronomical winter is traditionally 21 December or

1584-426: The date in question unless otherwise noted.) Tuesday immediately following fourth Thursday of November First Friday First Sunday Second Monday December 15, unless the date falls on a Sunday, then December 16 Winter Solstice December 22, unless that date is a Sunday, in which case the 23rd December 26, unless that day is a Sunday, in which case the 27th 10 (number) 10 ( ten )

1632-727: The date of the solstice. Meteor showers occurring in December are the Andromedids (September 25 – December 6, peaking around November 9), the Canis-Minorids (December 4 – December 15, peaking around December 10–11), the Coma Berenicids (December 12 to December 23, peaking around December 16), the Delta Cancrids (December 14 to February 14, the main shower from January 1 to January 24, peaking on January 17),

1680-693: The five exceptional Lie algebras . They also include the four general affine Coxeter groups A ~ n {\displaystyle {\tilde {A}}_{n}} , B ~ n {\displaystyle {\tilde {B}}_{n}} , C ~ n {\displaystyle {\tilde {C}}_{n}} , and D ~ n {\displaystyle {\tilde {D}}_{n}} that are associated with simplex , cubic and demihypercubic honeycombs, or tessellations . Regarding Coxeter groups in hyperbolic space , there are infinitely many such groups; however, ten

1728-406: The four Agonalia , this day in honour of Sol Indiges was held on December 11, as was Septimontium . Dies natalis (birthday) was held at the temple of Tellus on December 13, Consualia was held on December 15, Saturnalia was held December 17–23, Opiconsivia was held on December 19, Divalia was held on December 21, Larentalia was held on December 23, and the dies natalis of Sol Invictus

1776-405: The month of December are Sagittarius (until December 21) and Capricorn (December 22 onward). December's birth flower is the narcissus . Its birthstones are turquoise , zircon and tanzanite . This list does not necessarily imply either official status or general observance. (All Baháʼí, Islamic, and Jewish observances begin at the sundown prior to the date listed, and end at sundown of

1824-506: The mouthpiece, the harmonics produced are odd multiples of the fundamental frequency . (With cylindrical pipes open at both ends, used for example in some organ stops such as the open diapason , the harmonics are even multiples of the same frequency for the given bore length, but this has the effect of the fundamental frequency being doubled and all multiples of this fundamental frequency being produced.) See harmonic series (music) . In some countries, house numberings are chosen so that

1872-432: The moves of the puzzle allow only even permutations of the puzzle pieces, so parity is important in understanding the configuration space of these puzzles. The Feit–Thompson theorem states that a finite group is always solvable if its order is an odd number. This is an example of odd numbers playing a role in an advanced mathematical theorem where the method of application of the simple hypothesis of "odd order"

1920-410: The number is even according to the sum of its digits—it is even if and only if the sum of its digits is even. An even number is an integer of the form x = 2 k {\displaystyle x=2k} where k is an integer; an odd number is an integer of the form x = 2 k + 1. {\displaystyle x=2k+1.} An equivalent definition is that an even number

1968-400: The number it was derived from again produces an incorrect result. In this way, all single-bit transmission errors may be reliably detected. Some more sophisticated error detecting codes are also based on the use of multiple parity bits for subsets of the bits of the original encoded value. In wind instruments with a cylindrical bore and in effect closed at one end, such as the clarinet at

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2016-442: The people of Israel to follow. Even and odd numbers In mathematics , parity is the property of an integer of whether it is even or odd . An integer is even if it is divisible by 2, and odd if it is not. For example, −4, 0, and 82 are even numbers, while −3, 5, 7, and 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See

2064-401: The philosophical afterthought, It is well to direct the pupil's attention here at once to a great far-reaching law of nature and of thought. It is this, that between two relatively different things or ideas there stands always a third, in a sort of balance, seeming to unite the two. Thus, there is here between odd and even numbers one number (one) which is neither of the two. Similarly, in form,

2112-473: The properties of divisibility . They are a special case of rules in modular arithmetic , and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, and multiplication is distributive over addition. However, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which

2160-429: The right angle stands between the acute and obtuse angles; and in language, the semi-vowels or aspirants between the mutes and vowels. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this and other important laws. Integer coordinates of points in Euclidean spaces of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance,

2208-427: The section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system

2256-418: Was famously used to solve the mutilated chessboard problem : if two opposite corner squares are removed from a chessboard, then the remaining board cannot be covered by dominoes, because each domino covers one square of each parity and there are two more squares of one parity than of the other. The parity of an ordinal number may be defined to be even if the number is a limit ordinal, or a limit ordinal plus

2304-472: Was held on December 25. These dates do not correspond to the modern Gregorian calendar. The Anglo-Saxons referred to December–January as Ġēolamonaþ (modern English: " Yule month"). The French Republican Calendar contained December within the months of Frimaire and Nivôse . December contains the winter solstice in the Northern Hemisphere , the day with the fewest daylight hours, and

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