偶数

関: even、even-numbered

WordNet

used as an intensive especially to indicate something unexpected; "even an idiot knows that"; "declined even to consider the idea"; "I dont have even a dollar!"
become even or more even; "even out the surface" (同)even out
make even or more even (同)even out
equal in degree or extent or amount; or equally matched or balanced; "even amounts of butter and sugar"; "on even terms"; "it was a fifty-fifty (or even) split"; "had a fifty-fifty (or even) chance"; "an even fight" (同)fifty-fifty
occurring at fixed intervals; "a regular beat"; "the even rhythm of his breathing" (同)regular
symmetrically arranged; "even features"; "regular features"; "a regular polygon" (同)regular
to a greater degree or extent; used with comparisons; "looked sick and felt even worse"; "an even (or still) more interesting problem"; "still another problem must be solved"; "a yet sadder tale" (同)yet, still
being level or straight or regular and without variation as e.g. in shape or texture; or being in the same plane or at the same height as something else (i.e. even with); "an even application of varnish"; "an even floor"; "the road was not very even"; "the picture is even with the window"
divisible by two
in spite of; notwithstanding; "even when he is sick, he works"; "even with his head start she caught up with him"
to the full extent; "loyal even unto death"
give numbers to; "You should number the pages of the thesis"
place a limit on the number of (同)keep_down
a concept of quantity involving zero and units; "every number has a unique position in the sequence"
the property possessed by a sum or total or indefinite quantity of units or individuals; "he had a number of chores to do"; "the number of parameters is small"; "the figure was about a thousand" (同)figure
a clothing measurement; "a number 13 shoe"
an item of merchandise offered for sale; "she preferred the black nylon number"; "this sweater is an all-wool number"
a numeral or string of numerals that is used for identification; "she refused to give them her Social Security number" (同)identification number
a select company of people; "I hope to become one of their number before I die"
the grammatical category for the forms of nouns and pronouns and verbs that are used depending on the number of entities involved (singular or dual or plural); "in English the subject and the verb must agree in number"
enumerate; "We must number the names of the great mathematicians" (同)list
so frightened as to be unable to move; stunned or paralyzed with terror; petrified; "too numb with fear to move"
make numb or insensitive; "The shock numbed her senses" (同)benumb, blunt, dull
the latter part of the day (the period of decreasing daylight from late afternoon until nightfall); "he enjoyed the evening light across the lake" (同)eve, even, eventide
a later concluding time period; "it was the evening of the Roman Empire"
the early part of night (from dinner until bedtime) spent in a special way; "an evening at the opera"
the day before; "he always arrives on the eve of her departure"
the period immediately before something; "on the eve of the French Revolution"

PrepTutorEJDIC

『平らな』,平たんな / 『同じ高さの』,同一平面の,平行の / (運動・動作・品質などが)『均一の』,規則的な,一様の,むらのない / (数・量などが)『同じの』,等しい,釣り合のとれた互角の / 『偶数の』 / 公平な,公正な(fair) / 平静な,穏やかな,落ち着いた(calm) / 貸借にない,清算済みの / 端数のない,ちょうどの,きっかりの / 《意味を強めて,不審や意外の念を含み》…『でさえも』,までも / 《比較級の前に用いて》『なおいっそう』,さらに(still,yet) / 《形容詞の前または後の用いて》(…と)すら言える;《古》《時を表す副詞の前に用いて》ちょうど / …‘を'平らにする,ならす / …‘を'釣り合わせる,平均させる《+『up』+『名』,+『名』+『up』》 / …‘の'変動をなくする,‘を'安定させる《+『out』+『名』,+『名』+『out』》
夕べ,晩
〈U〉〈C〉(数えて得られる)『数,数量』 / 〈C〉(概念としての)『数,数字』 / 〈C〉『番号』 / 〈C〉(演奏会や演劇の)番組,出し物;曲目 / 〈C〉(雑誌の)号 / 〈U〉(文法で)数(すう) / 《複数形で》数の上の優勢 / 《複数形で》算数 / 〈C〉《単数形で》《話》(商品としての)洋服の1点;商品,売り物 / 〈C〉《単随形で》《俗》女の子 / …‘を'数える / (…の中に,…として)…‘を'含める,加える《+『among』(『with, as』)+『名』》 / …‘に'番号をつける / …‘の'数となる / 《しばしば受動態で》…‘の'数を制限する / 総計(…に)なる《+『in』+『名』〈数〉》
麻痺(まひ)した,感覚を失った,しびれた / …‘の'感覚を失わせる,‘を'しびれさせる
『夕方』,夕刻,晩(日没から就寝時までの間) / 《文》末期,晩年 / 夜会,夕べ / 夕方の,夕暮れの
(祝祭日の)『前夜』,前日 / 《the~》(重要な事の)直前《+『of』+『名』》

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2016/04/14 00:42:28」(JST)

wiki en

[Wiki en表示]

Parity is a mathematical term that describes the property of an integer's inclusion in one of two categories: even or odd. An integer is even if it is 'evenly divisible' by two (the old-fashioned term "evenly divisible" is now almost always shortened to "divisible") and odd if it is not even.^[1] For example, 6 is even because there is no remainder when dividing it by 2. By contrast, 3, 5, 7, 21 leave a remainder of 1 when divided by 2. Examples of even numbers include −4, 0, 8, and 1738. In particular, zero is an even number.^[2] Some examples of odd numbers are −5, 3, 9, and 73. Parity does not apply to non-integer numbers.

A formal definition of an even number is that it is an integer of the form n = 2k, where k is an integer;^[3] it can then be shown that an odd number is an integer of the form n = 2k + 1. This classification applies only to integers, i.e., non-integers like 1/2, 4.201, or infinity are neither even nor odd.

The sets of even and odd numbers can be defined as following:^[4]

Even
Odd

A number (i.e., integer) expressed in the decimal numeral system 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. 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 even if its last digit is 0. In an odd base, 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.^[5]

1 Arithmetic on even and odd numbers
- 1.1 Addition and subtraction
- 1.2 Multiplication
- 1.3 Division
2 History
3 Higher mathematics
- 3.1 Higher dimensions and more general classes of numbers
- 3.2 Number theory
- 3.3 Group theory
- 3.4 Analysis
- 3.5 Combinatorial game theory
4 Additional applications
5 References

Arithmetic on even and odd numbers

The following laws can be verified using 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 is not true for normal integer arithmetic.

Addition and subtraction

even ± even = even;^[1]
even ± odd = odd;^[1]
odd ± odd = even;^[1]

Rules analogous to these for divisibility by 9 are used in the method of casting out nines.

Multiplication

even × even = even;^[1]
even × odd = even;^[1]
odd × odd = odd.^[1]

The structure ({even, odd}, +, ×) is in fact a field with just two elements.

Division

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 even and odd apply only to integers. But when the quotient is an integer, it will be even if and only if the dividend has more factors of two than the divisor.^[6]

History

The ancient Greeks considered 1, the monad, to be neither fully odd nor fully even.^[7] 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 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, 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.^[8]

Higher mathematics

Higher dimensions and more general classes of numbers

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

The two white bishops are confined to squares of opposite parity; the black knight can only jump to squares of alternating parity.

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, the face-centered cubic lattice and its higher-dimensional generalizations, the D_n lattices, consist of all of the integer points whose sum of coordinates is even.^[9] This feature manifests itself in chess, where the parity of a square is indicated by its color: bishops are constrained to squares of the same parity; knights alternate parity between moves.^[10] This form of parity 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.^[11]

The parity of an ordinal number may be defined to be even if the number is a limit ordinal, or a limit ordinal plus a finite even number, and odd otherwise.^[12]

Let R be a commutative ring and let I be an ideal of R whose index is 2. Elements of the coset maybe called even, while elements of the coset maybe called odd. As an example, let R=Z₍₂₎ be the localization of Z at the prime ideal (2). Then an element of R is even or odd if and only if its numerator is so in Z.

Number theory

The even numbers form an ideal in the ring of integers,^[13] 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 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.^[14] All known perfect numbers are even; it is unknown whether any odd perfect numbers exist.^[15]

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 × 10¹⁸, but still no general proof has been found.^[16]

Group theory

Rubik's Revenge in solved state

The parity of a permutation (as defined in abstract algebra) is the parity of the number of transpositions into which the permutation can be decomposed.^[17] 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, 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.^[18]

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" is far from obvious.^[19]

Analysis

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 is possible for a function to be neither odd nor even, and for the case f(x) = 0, to be both odd and even.^[20] 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.^[21]

Combinatorial game theory

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 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.^[22] 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 is evil, and a 1 in that position when i is odious.^[23]

Additional applications

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 the number it was derived from again produces an incorrect result. In this way, all single-bit transmission errors may be reliably detected.^[24] 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.^[25]

In wind instruments with a cylindrical bore and in effect closed at one end, such as the clarinet at 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).^[26]

In some countries, house numberings are chosen so that the houses on one side of a street have even numbers and the houses on the other side have odd numbers.^[27] Similarly, among United States numbered highways, even numbers primarily indicate east-west highways while odd numbers primarily indicate north-south highways.^[28] Among airline flight numbers, even numbers typically identify eastbound or northbound flights, and odd numbers typically identify westbound or southbound flights.^[29]

References

^ ^a ^b ^c ^d ^e ^f ^g A.V.Vijaya & Dora Rodriguez, Figuring Out Mathematics, Pearson Education India, pp. 20–21, ISBN 9788131703571 .
^ Bóna, Miklós (2011), A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory, World Scientific, p. 178, ISBN 9789814335232 .
^ Bassarear, Tom (2010), Mathematics for Elementary School Teachers, Cengage Learning, p. 198, ISBN 9780840054630 .
^ Sidebotham, Thomas H. (2003), The A to Z of Mathematics: A Basic Guide, John Wiley & Sons, p. 181, ISBN 9780471461630 .
^ Owen, Ruth L. (1992), "Divisibility in bases" (PDF), The Pentagon: A Mathematics Magazine for Students 51 (2): 17–20 .
^ Pólya, George; Tarjan, Robert E.; Woods, Donald R. (2009), Notes on Introductory Combinatorics, Springer, pp. 21–22, ISBN 9780817649524 .
^ Tankha (2006), Ancient Greek Philosophy: Thales to Gorgias, Pearson Education India, p. 136, ISBN 9788177589399 .
^ Froebel, Friedrich; Translator Josephine Jarvis (1885). The Education of Man. New York: A Lovell & Company. p. 240.
^ Conway, J. H.; Sloane, N. J. A. (1999), Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 290 (3rd ed.), New York: Springer-Verlag, p. 10, ISBN 0-387-98585-9, MR 1662447 .
^ Pandolfini, Bruce (1995), Chess Thinking: The Visual Dictionary of Chess Moves, Rules, Strategies and Concepts, Simon and Schuster, pp. 273–274, ISBN 9780671795023 .
^ Mendelsohn, N. S. (2004), "Tiling with dominoes", The College Mathematics Journal 35 (2): 115–120, doi:10.2307/4146865, JSTOR 4146865 .
^ Bruckner, Andrew M.; Bruckner, Judith B.; Thomson, Brian S. (1997), Real Analysis, p. 37, ISBN 0-13-458886-X .
^ Stillwell, John (2003), Elements of Number Theory, Springer, p. 199, ISBN 9780387955872 .
^ Lial, Margaret L.; Salzman, Stanley A.; Hestwood, Diana (2005), Basic College Mathematics (7th ed.), Addison Wesley, p. 128, ISBN 9780321257802 .
^ Dudley, Underwood (1992), "Perfect numbers", Mathematical Cranks, MAA Spectrum, Cambridge University Press, pp. 242–244, ISBN 9780883855072 .
^ Oliveira e Silva, Tomás; Herzog, Siegfried; Pardi, Silvio (2013), "Empirical verification of the even Goldbach conjecture, and computation of prime gaps, up to 4·10¹⁸" (PDF), Mathematics of Computation, doi:10.1090/s0025-5718-2013-02787-1 . In press.
^ Cameron, Peter J. (1999), Permutation Groups, London Mathematical Society Student Texts 45, Cambridge University Press, pp. 26–27, ISBN 9780521653787 .
^ Joyner, David (2008), "13.1.2 Parity conditions", Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys, JHU Press, pp. 252–253, ISBN 9780801897269 .
^ Bender, Helmut; Glauberman, George (1994), Local analysis for the odd order theorem, London Mathematical Society Lecture Note Series 188, Cambridge: Cambridge University Press, ISBN 0-521-45716-5, MR 1311244 ; Peterfalvi, Thomas (2000), Character theory for the odd order theorem, London Mathematical Society Lecture Note Series 272, Cambridge: Cambridge University Press, ISBN 0-521-64660-X, MR 1747393 .
^ Gustafson, Roy David; Hughes, Jeffrey D. (2012), College Algebra (11th ed.), Cengage Learning, p. 315, ISBN 9781111990909 .
^ Jain, R. K.; Iyengar, S. R. K. (2007), Advanced Engineering Mathematics, Alpha Science Int'l Ltd., p. 853, ISBN 9781842651858 .
^ Guy, Richard K. (1996), "Impartial games", Games of no chance (Berkeley, CA, 1994), Math. Sci. Res. Inst. Publ. 29, Cambridge: Cambridge Univ. Press, pp. 61–78, MR 1427957 . See in particular p. 68.
^ Bernhardt, Chris (2009), "Evil twins alternate with odious twins", Mathematics Magazine 82 (1): 57–62, doi:10.4169/193009809x469084, JSTOR 27643161 .
^ Moser, Stefan M.; Chen, Po-Ning (2012), A Student's Guide to Coding and Information Theory, Cambridge University Press, pp. 19–20, ISBN 9781107015838 .
^ Berrou, Claude (2011), Codes and turbo codes, Springer, p. 4, ISBN 9782817800394 .
^ Randall, Robert H. (2005), An Introduction to Acoustics, Dover, p. 181, ISBN 9780486442518 .
^ Cromley, Ellen K.; McLafferty, Sara L. (2011), GIS and Public Health (2nd ed.), Guilford Press, p. 100, ISBN 9781462500628 .
^ Swift, Earl (2011), The Big Roads: The Untold Story of the Engineers, Visionaries, and Trailblazers Who Created the American Superhighways, Houghton Mifflin Harcourt, p. 95, ISBN 9780547549132 .
^ Lauer, Chris (2010), Southwest Airlines, Corporations that changed the world, ABC-CLIO, p. 90, ISBN 9780313378638 .

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Japanese Journal

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