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| {{numeral systems}}
| | mais aswell de ses admirables aux paquets acclament ous d'acheter GHD sur internet pour ��conomiser de l'argent ous un excellent.? Bien fers GHD sont faites aec les plus hauts standards de qualit��, et sont lir��s aec une garantie d'un an, elles disposent ��galement de deux modes qui prot��gent le fer contre les dommages de sorte que ous pouez ��tre assur�� que otre GHD Styler a durer et durer? Herbe derait normalement ��tre de deux �� trois pouces de haut. |
| A '''negative base''' (or negative [[radix]]) may be used to construct a [[non-standard positional numeral system]]. Like other place-value systems, each position holds multiples of the appropriate power of the system's base; but that base is negative—that is to say, the base <math>\scriptstyle b</math> is equal to <math>\scriptstyle -r</math> for some natural number <math>\scriptstyle r</math> (''r ≥ 2'').
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| Negative-base systems can accommodate all the same numbers as standard place-value systems, but both positive and negative numbers are represented without the use of a [[minus sign]] (or, in computer representation, a [[sign bit]]); this advantage is countered by an increased complexity of arithmetic operations. The need to store the information normally contained by a negative sign often results in a negative-base number being one digit longer than its positive-base equivalent.
| | Seulement couper une ��me aec le haut en m��me temps pour emp��cher l'herbe de se mettre en ��tat de choc. Son imp��ratif que la lame de la tondeuse est toujours forte pour arr��ter les dommages de l'herbe. L'herbe haute pousse des racines plus longues, ombrage le sol et emp��che les mauaises herbes infiltration. il est possible de rep��rer une imitation GHD si le redresseur est exceptionnellement forte. Tous les ��ritables d��frisants GHD sont ultra-l��ger car les plaques sont produites �� partir de c��ramique de porcelaine. |
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| The common names for negative-base positional numeral systems are formed by [[prefix (linguistics)|prefixing]] ''nega-'' to the name of the corresponding positive-base system; for example, '''negadecimal''' (base −10) corresponds to [[decimal]] (base 10), '''negaternary''' (base −3) to [[ternary numeral system|ternary]] (base 3), and '''negabinary''' (base −2) to [[binary numeral system|binary]] (base 2).<ref>{{harvnb|Knuth|1998}} and [[#WeissteinNegadecimal|Weisstein]] each refer to the negadecimal system. In the index {{harvnb|Knuth|1998}} refers to the negabinary system, as does [[#WeissteinNegabinary|Weisstein]]. The negaternary system is discussed briefly in {{Citation | last1=Petkovšek | first1=Marko | author1-link=Marko Petkovšek | title=Ambiguous numbers are dense | doi=10.2307/2324393 | mr=1048915 | year=1990 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=97 | issue=5 | pages=408–411}}.</ref>
| | Les eneloppes de biens authentiques GHD sont ��galement cr����s �� partir de l'��tat mince mais assez solide aec les fournitures d'art. C'est pourquoi chaque unit�� de d��friser les cheeux GHD est tr��s l��ger. Quand ous aez obtenu un redresseur lourd aec un logo GHD, y aller et trouer plus de redresseurs de GHD et faire des r��parations plus de GHD. S'il ous pla?t noter que nous serions toujours proposer une r��paration �� ��rifier par une r��paration GHD qualifi��s et technicien de maintenance et les r��parations GHD sont faites �� os propres risques. |
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| ==Example==
| | En cas de doute trouer un technicien de r��paration qualifi�� GHD, beaucoup peut ��tre trou�� en faisant une recherche sur Internet. S'il ous pla?t oir la bo?te des ressources pour plus de d��tails. Vous ��tes susceptibles de regarder �� traers un grand nombre d'?ures de bon go?t par ce composant qui offrent un confort id��al et la d��fense du cuir cheelu, m��me si l'apparence. Yourrrre pu repousser aec rouleau os cheeux fait de ous habiller rapidement pour n'importe quelle affaire sociale urgente parler ��. |
| Consider what is meant by the representation ''12,243'' in the negadecimal system, whose base <math>\scriptstyle b</math> is −10:
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| {| class="wikitable"
| | Mode rapide ��tant l'une des cl��s de cr��dit pour ces stylers ceux-ci sont assez populaire aupr��s des filles qui contiennent un proacties plusieurs mod��les de style routine.Contributing, Il aut mieux que tous les autres fers plats disponibles sur le march��. Bien que toute autre redresseur de cheeux r��guli��re prend eniron minutes pour traailler sur le style que ous oulez, GHD peut le faire dans les minutes. |
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| |align="center"| multiples of <math>\scriptstyle b^4</math> <br> (i.e., 10,000)
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| |align="center"| multiples of <math>\scriptstyle b^3</math> <br> (i.e., −1,000)
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| |align="center"| multiples of <math>\scriptstyle b^2</math> <br> (i.e., 100)
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| |align="center"| multiples of <math>\scriptstyle b^1</math> <br> (i.e., −10)
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| |align="center"| multiples of <math>\scriptstyle b^0</math> <br> (i.e., 1)
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| |-
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| |align="center"| 1
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| |align="center"| 2
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| |align="center"| 2
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| |align="center"| 4
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| |align="center"| 3
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| |-
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| |}
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| Since 10,000 + (−2,000) + 200 + (−40) + 3 = 8,163, the representation ''12,243'' in negadecimal notation is equivalent to ''8,163'' in decimal notation.
| | Ainsi, en prenant presque la moiti�� du temps, Il ya juste quelque chose sur cet ingr��dient secret qui rend les cheeux look superbe et fabuleux. Il n'est pas ��tonnant coiffeurs professionnels partout dans le monde utilisent Cloud redresseurs et sp��culer sur ce que l'ingr��dient pourrait ��tre! Le d��tecteur est un excellent dispositif de s��curit�� comme de nombreux redresseurs ne contiennent pas de dispositif de s��curit��. Le d��tecteur peut dire si oui ou non une personne utilise le dispositif. |
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| ==History==
| | Si une personne a accidentellement laiss�� son lisseur sur, Il semble qu'il n'y ait toujours pas de r��ponse. Mais je pense que les lecteurs, en particulier les utilisateurs de fer plat peut-��tre d��j�� leurs propres r��ponses, et c'est nos objectifs, pour aider les lecteurs �� faire le bon choix! article--ghd-hair-straighteners-cheapGhd Lisseur pour la plupart ifficile cheeux-Info Les cheeux afro des Cara?<br><br> |
| Negative numerical bases were first considered by [[Vittorio Grünwald]] in his work ''Giornale di Matematiche di Battaglini'', published in 1885. Grünwald gave algorithms for performing addition, subtraction, multiplication, division, root extraction, divisibility tests, and radix conversion. Negative bases were later independently rediscovered by [[Aubrey J. Kempner|A. J. Kempner]] in 1936 and [[Zdzisław Pawlak]] and A. Wakulicz in 1959{{Citation needed|date=May 2012}}.
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| Negabinary was implemented in the early [[Poland|Polish]] computer [[BINEG]], built 1957–59, based on ideæ by Z. Pawlak and A. Lazarkiewicz from the [[Mathematical Institute]] in [[Warsaw]].<ref>[http://chc60.fgcu.edu/images/articles/Marczynski.pdf Marczynski, R. W., "The First Seven Years of Polish Computing"], IEEE Annals of the History of Computing, Vol. 2, No 1, January 1980</ref> Implementations since then have been rare.
| | If you have any type of concerns relating to where and how to use [http://tinyurl.com/m63r8fp GHD lisseur Pas Cher], you could contact us at the web-site. |
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| ==Notation and use==
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| Denoting the base as <math>-r</math>, every [[integer]] <math>a</math> can be written uniquely as
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| :<math>a = \sum_{i=0}^{n}d_{i}(-r)^{i}</math>
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| where each digit <math>\scriptstyle d_k</math> is an integer from 0 to <math>\scriptstyle r - 1</math> and the leading digit <math>\scriptstyle d_n</math> is <math>\scriptstyle > 0</math> (unless <math>\scriptstyle n=0</math>). The base <math>\scriptstyle -r</math> expansion of <math>\scriptstyle a</math> is then given by the string <math>\scriptstyle d_n d_{n-1} \ldots d_1 d_0</math>.
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| Negative-base systems may thus be compared to [[signed-digit representation]]s, such as [[balanced ternary]], where the radix is positive but the digits are taken from a partially negative range.
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| Some numbers have the same representation in base <math>\scriptstyle -r</math> as in base <math>r</math>. For example, the numbers from 100 to 109 have the same representations in decimal and negadecimal. Similarly,
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| :<math>17=2^4+2^0=(-2)^4+(-2)^0</math>
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| and is represented by 10001 in binary and 10001 in negabinary.
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| Some numbers with their expansions in a number of positive and corresponding negative bases are:
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| {|class="wikitable" style="margin: 1em auto 1em auto; text-align: right;"
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| ! Decimal !! Negadecimal !! Binary !! Negabinary !! Ternary !! Negaternary
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| | −15 || 25 || −1111 || 110001 || −120 || 1220
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| |-
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| | : || : || : || : || : || :
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| |-
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| | −5 || 15 || −101 || 1111 || −12 || 21
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| |-
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| | −4 || 16 || −100 || 1100 || −11 || 22
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| |-
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| | −3 || 17 || −11 || 1101 || −10 || 10
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| |-
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| | −2 || 18 || −10 || 10 || −2 || 11
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| |-
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| | −1 || 19 || −1 || 11 || −1 || 12
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| |-
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| | 0 || 0 || 0 || 0 || 0 || 0
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| |-
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| | 1 || 1 || 1 || 1 || 1 || 1
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| |-
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| | 2 || 2 || 10 || 110 || 2 || 2
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| |-
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| | 3 || 3 || 11 || 111 || 10 || 120
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| | 4 || 4 || 100 || 100 || 11 || 121
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| |-
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| | 5 || 5 || 101 || 101 || 12 || 122
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| |-
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| | 6 || 6 || 110 || 11010 || 20 || 110
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| | 7 || 7 || 111 || 11011 || 21 || 111
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| |-
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| | 8 || 8 || 1000 || 11000 || 22 || 112
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| |-
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| | 9 || 9 || 1001 || 11001 || 100 || 100
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| |-
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| | 10 || 190 || 1010 || 11110 || 101 || 101
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| |-
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| | 11 || 191 || 1011 || 11111 || 102 || 102
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| |-
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| | 12 || 192 || 1100 || 11100 || 110 || 220
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| |-
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| | 13 || 193 || 1101 || 11101 || 111 || 221
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| |-
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| | 14 || 194 || 1110 || 10010 || 112 || 222
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| |-
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| | 15 || 195 || 1111 || 10011 || 120 || 210
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| |-
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| | 16 || 195 || 10000 || 10000 || 121 || 211
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| |-
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| | 17 || 197 || 10001 || 10001 || 122 || 212
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| |}
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| Note that the base <math>\scriptstyle -r</math> expansions of negative integers have an [[even number]] of digits, while the base <math>\scriptstyle -r</math> expansions of the non-negative integers have an [[odd number]] of digits.
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| ==Calculation==
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| The base <math>\scriptstyle -r</math> expansion of a number can be found by repeated division by <math>\scriptstyle -r</math>, recording the non-negative remainders of <math>\scriptstyle 0, 1,\ldots r-1</math>, and concatenating those remainders, starting with the last. Note that if <math>\scriptstyle a / b = c</math>, remainder <math>d</math>, then <math>\scriptstyle bc + d = a</math>. For example, in negaternary:
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| :<math>\begin{align}
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| 146 & ~/~ -3 = & -48, & ~\mbox{remainder}~ 2 \\
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| -48 & ~/~ -3 = & 16, & ~\mbox{remainder}~ 0 \\
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| 16 & ~/~ -3 = & -5, & ~\mbox{remainder}~ 1 \\
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| -5 & ~/~ -3 = & 2, & ~\mbox{remainder}~ 1 \\
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| 2 & ~/~ -3 = & 0, & ~\mbox{remainder}~ 2 \\
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| \end{align}</math>
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| Therefore, the negaternary expansion of 146 is 21,102.
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| Note that in most [[programming languages]], the result (in integer arithmetic) of dividing a negative number by a negative number is rounded towards 0, usually leaving a negative remainder. In such a case we have <math>\scriptstyle a = (-r)c + d = (-r)c + d - r + r = (-r)(c + 1) + (d + r)</math>. Because <math>\scriptstyle |d| < r</math>, <math>\scriptstyle (d + r)</math> is the positive remainder. Therefore, to get the correct result in such case, computer implementations of the above algorithm should add 1 and <math>r</math> to the quotient and remainder respectively (shown below in the [[Python (programming language)|Python]] programming language):
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| <source lang="python">
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| def negaternary(i):
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| digits = ''
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| if not i:
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| digits = '0'
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| else:
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| while i != 0:
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| i, remainder = divmod(i, -3)
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| if remainder < 0:
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| i, remainder = i + 1, remainder + 3
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| digits = str(remainder)+ digits
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| return digits
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| </source>
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| C# implementation:
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| <source lang="csharp">
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| static string negatenary(int value)
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| {
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| string result = string.Empty;
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| while (value != 0)
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| {
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| int remainder = value % -3;
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| value = value / -3;
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| if (remainder < 0)
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| {
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| remainder += 3;
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| value += 1;
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| }
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| result = remainder.ToString() + result;
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| }
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| return result;
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| }
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| </source>
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| Common Lisp implementation:
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| <source lang="lisp">
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| (defun negaternary (i)
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| (if (zerop i)
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| "0"
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| (let ((digits "")
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| (rem 0))
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| (loop while (not (zerop i)) do
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| (progn
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| (multiple-value-setq (i rem) (truncate i -3))
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| (when (minusp rem)
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| (incf i)
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| (incf rem 3))
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| (setf digits (concatenate 'string (write-to-string rem) digits))))
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| digits)))
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| </source>
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| ==Arithmetic operations==
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| The following describes the arithmetic operations for negabinary; calculations in larger bases are similar.
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| ===Addition===
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| To add two negabinary numbers, start with a carry of 0, and, starting from the [[least significant bit]]s, add the bits of the two numbers plus the carry. The resulting number is then looked up in the following table to get the bit to write down as result, and the next carry:
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| {| class="wikitable"
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| ! Number !! Bit !! Carry !! Comment
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| | −2 || 0 || 1 || −2 occurs only during subtraction.
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| |-
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| | −1 || 1 || 1 ||
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| |-
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| | 0 || 0 || 0 ||
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| |-
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| | 1 || 1 || 0 ||
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| |-
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| | 2 || 0 || −1 ||
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| |-
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| | 3 || 1 || −1 || 3 occurs only during addition.
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| |}
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| The second row of this table, for instance, expresses the fact that '''−1''' = '''1''' + '''1''' × −2; the fifth row says '''2''' = '''0''' + '''−1''' × −2; etc.
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| As an example, to add 1010101 (1 + 4 + 16 + 64 = 85) and 1110100 (4 + 16 − 32 + 64 = 52),
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| carry: 1 −1 0 −1 1 −1 0 0 0
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| first number: 1 0 1 0 1 0 1
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| second number: 1 1 1 0 1 0 0 +
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| --------------------------
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| number: 1 −1 2 0 3 −1 2 0 1
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| bit (result): 1 1 0 0 1 1 0 0 1
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| carry: 0 1 −1 0 −1 1 −1 0 0
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| so the result is 110011001 (1 − 8 + 16 − 128 + 256 = 137).
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| ==== Another Method ====
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| While adding two negabinary numbers, every time a carry is generated an extra carry should be propagated to next bit.
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| Consider same example as above
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| extra carry: 1 1 0 1 0 0 0
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| carry: 1 0 1 1 0 1 0 0 0
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| first number: 1 0 1 0 1 0 1
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| second number: 1 1 1 0 1 0 0 +
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| --------------------------
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| Answer: 1 1 0 0 1 1 0 0 1
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| ===Subtraction===
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| To subtract, multiply each bit of the second number by −1, and add the numbers, using the same table as above.
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| As an example, to compute 1101001 (1 − 8 − 32 + 64 = 25) minus 1110100 (4 + 16 − 32 + 64 = 52),
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| carry: 0 1 −1 1 0 0 0
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| first number: 1 1 0 1 0 0 1
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| second number: −1 −1 −1 0 −1 0 0 +
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| --------------------
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| number: 0 1 −2 2 −1 0 1
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| bit (result): 0 1 0 0 1 0 1
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| carry: 0 0 1 −1 1 0 0
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| so the result is 100101 (1 + 4 −32 = −27).
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| To negate a number, compute 0 minus the number.
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| ===Multiplication and division===
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| Shifting to the left multiplies by −2, shifting to the right divides by −2.
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| To multiply, multiply like normal [[decimal]] or [[binary numeral system|binary]] numbers, but using the negabinary rules for adding the carry, when adding the numbers.
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| first number: 1 1 1 0 1 1 0
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| second number: 1 0 1 1 0 1 1 *
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| -------------------------------------
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| 1 1 1 0 1 1 0
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| 1 1 1 0 1 1 0
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|
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| 1 1 1 0 1 1 0
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| 1 1 1 0 1 1 0
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|
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| 1 1 1 0 1 1 0 +
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| -------------------------------------
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| carry: 0 −1 0 −1 −1 −1 −1 −1 0 −1 0 0
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| number: 1 0 2 1 2 2 2 3 2 0 2 1 0
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| bit (result): 1 0 0 1 0 0 0 1 0 0 0 1 0
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| carry: 0 −1 0 −1 −1 −1 −1 −1 0 −1 0 0
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| For each column, add ''carry'' to ''number'', and divide the sum by −2, to get the new ''carry'', and the resulting bit as the remainder.
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| <!--
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| (Todo: Division by arbitrary numbers?)
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| --><!--
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| "To be written things", like below, better be written. Empty sections are ugly.
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| ===Divisibility tests===
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| To be written.
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| ===Root extraction===
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| To be written.
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| -->
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| ==Fractional numbers==
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| Base <math>\scriptstyle -r</math> representation may of course be carried beyond the [[radix point]], allowing the representation of non-integral numbers.
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| As with positive-base systems, terminating representations correspond to fractions where the denominator is a power of the base; repeating representations correspond to other rationals, and for the same reason.
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| ===Non-unique representations===
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| Unlike positive-base systems, where integers and terminating fractions have non-unique representations (for example, in decimal [[0.999… = 1]]) in negative-base systems the integers have only a single representation. However, there do exist rationals with non-unique representations; for example, in negaternary,
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| : <math>0.(02)\ldots_{(-3)} = \frac{1}{4} = 1.(20)\ldots_{(-3)}</math>.
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| Such non-unique representations can be found by considering the largest and smallest possible representations with integral parts 0 and 1 respectively, and then noting that they are equal. (Indeed, this works with any integral-base system.) The rationals thus non-uniquely expressible are those of form
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| : <math>\frac{ar + 1}{b(r + 1)}</math>.
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| == Imaginary base ==
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| {{main|Complex base systems}}
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| Just as using a negative base allows the representation of negative numbers without an explicit negative sign, using an [[imaginary number|imaginary]] base allows the representation of [[Gaussian integer]]s. [[Donald Knuth]] proposed the [[quater-imaginary base]] (base 2i) in 1955.<ref>D. Knuth. The Art of Computer Programming. Volume 2, 3rd Edition. Addison-Wesley. pp. 205, "Positional Number Systems"</ref>
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| Imaginary-base arithmetic is not much different from negative-base arithmetic, since an imaginary-base number may be considered as the interleave of its real and imaginary parts; using [[INTERCAL]]-72 notation,
| |
| : ''x''<sub>(2i)</sub> + (2''i'')''y''<sub>(2i)</sub> = ''x''<sub>(2i)</sub> ¢ ''y''<sub>(2i)</sub>.
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| == See also ==
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| * [[Quater-imaginary base]]
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| * [[Binary numeral system|Binary]]
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| * [[Balanced ternary]]
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| * [[Numeral system]]s
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| == Notes ==
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| <references/>
| |
| | |
| ==References==
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| * Vittorio Grünwald. ''Giornale di Matematiche di Battaglini'' (1885), 203-221, 367
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| * A. J. Kempner. (1936), 610-617
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| * Z. Pawlak and A. Wakulicz ''Bulletin de l'Academie Polonaise des Scienses'', Classe III, 5 (1957), 233-236; Serie des sciences techniques 7 (1959), 713-721
| |
| * L. Wadel ''IRE Transactions EC-6'' 1957, 123
| |
| * N. M. Blachman, ''Communications of the ACM'' (1961), 257
| |
| * IEEE Transactions 1963, 274-276
| |
| * ''Computer Design'' May 1967, 52-63
| |
| * R. W. Marczynski, ''Annotated History of Computing'', 1980, 37-48
| |
| * {{citation|first=Donald|last=Knuth|title=[[The Art of Computer Programming]], Volume 2|year=1998|edition=3rd|pages=204–205}}.
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| * {{anchor|WeissteinNegabinary}} {{mathworld|title=Negabinary|urlname=Negabinary}}
| |
| * {{anchor|WeissteinNegadecimal}} {{mathworld|title=Negadecimal|urlname=Negadecimal}}
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| [[Category:Non-standard positional numeral systems]]
| |
| [[Category:Computer arithmetic]]
| |
mais aswell de ses admirables aux paquets acclament ous d'acheter GHD sur internet pour ��conomiser de l'argent ous un excellent.? Bien fers GHD sont faites aec les plus hauts standards de qualit��, et sont lir��s aec une garantie d'un an, elles disposent ��galement de deux modes qui prot��gent le fer contre les dommages de sorte que ous pouez ��tre assur�� que otre GHD Styler a durer et durer? Herbe derait normalement ��tre de deux �� trois pouces de haut.
Seulement couper une ��me aec le haut en m��me temps pour emp��cher l'herbe de se mettre en ��tat de choc. Son imp��ratif que la lame de la tondeuse est toujours forte pour arr��ter les dommages de l'herbe. L'herbe haute pousse des racines plus longues, ombrage le sol et emp��che les mauaises herbes infiltration. il est possible de rep��rer une imitation GHD si le redresseur est exceptionnellement forte. Tous les ��ritables d��frisants GHD sont ultra-l��ger car les plaques sont produites �� partir de c��ramique de porcelaine.
Les eneloppes de biens authentiques GHD sont ��galement cr����s �� partir de l'��tat mince mais assez solide aec les fournitures d'art. C'est pourquoi chaque unit�� de d��friser les cheeux GHD est tr��s l��ger. Quand ous aez obtenu un redresseur lourd aec un logo GHD, y aller et trouer plus de redresseurs de GHD et faire des r��parations plus de GHD. S'il ous pla?t noter que nous serions toujours proposer une r��paration �� ��rifier par une r��paration GHD qualifi��s et technicien de maintenance et les r��parations GHD sont faites �� os propres risques.
En cas de doute trouer un technicien de r��paration qualifi�� GHD, beaucoup peut ��tre trou�� en faisant une recherche sur Internet. S'il ous pla?t oir la bo?te des ressources pour plus de d��tails. Vous ��tes susceptibles de regarder �� traers un grand nombre d'?ures de bon go?t par ce composant qui offrent un confort id��al et la d��fense du cuir cheelu, m��me si l'apparence. Yourrrre pu repousser aec rouleau os cheeux fait de ous habiller rapidement pour n'importe quelle affaire sociale urgente parler ��.
Mode rapide ��tant l'une des cl��s de cr��dit pour ces stylers ceux-ci sont assez populaire aupr��s des filles qui contiennent un proacties plusieurs mod��les de style routine.Contributing, Il aut mieux que tous les autres fers plats disponibles sur le march��. Bien que toute autre redresseur de cheeux r��guli��re prend eniron minutes pour traailler sur le style que ous oulez, GHD peut le faire dans les minutes.
Ainsi, en prenant presque la moiti�� du temps, Il ya juste quelque chose sur cet ingr��dient secret qui rend les cheeux look superbe et fabuleux. Il n'est pas ��tonnant coiffeurs professionnels partout dans le monde utilisent Cloud redresseurs et sp��culer sur ce que l'ingr��dient pourrait ��tre! Le d��tecteur est un excellent dispositif de s��curit�� comme de nombreux redresseurs ne contiennent pas de dispositif de s��curit��. Le d��tecteur peut dire si oui ou non une personne utilise le dispositif.
Si une personne a accidentellement laiss�� son lisseur sur, Il semble qu'il n'y ait toujours pas de r��ponse. Mais je pense que les lecteurs, en particulier les utilisateurs de fer plat peut-��tre d��j�� leurs propres r��ponses, et c'est nos objectifs, pour aider les lecteurs �� faire le bon choix! article--ghd-hair-straighteners-cheapGhd Lisseur pour la plupart ifficile cheeux-Info Les cheeux afro des Cara?
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