Топ-100 ★ Signed-digit representation - non-standard positional num
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Signed-digit representation
                                     

★ Signed-digit representation

In mathematical notation for numbers, signed-digit representation is a positional system with signed digits.

Signed-digit representation may be used to accomplish fast addition of integers because it can eliminate the chain of dependent bears. In binary notation, a special case of signed-digit representation, non-adjacent forms, which can offer the advantage of speed with minimal space overhead.

                                     

1. Notation and use. (Обозначения и применения)

Denoting the positional numeral notation as B {\the style property display value b}, each integer a {\the style property display the value} can be written uniquely as

a = ∑ i = 0 n d i b i {\displaystyle a=\sum _{i=0}^{n}d_{i}b^{i}}

where each digit d K {\d_ properties of the style display value{to}} is an integer such that α ≤ D ≤ β {\the style property display the value of -\alpha \leq d_{K}\beta \leq }. given that α ≤ 0 {\the style property display the value of -\alpha \leq 0} is the minimum number and β ≥ 0 {\the style property display the value of \beta \geq 0} is the maximum of digits, where the digits d n > 0 {\the style property display the value of d_{N} & gt; 0}, if n = 0 {\the style property display the value of N=0}, where base B = α β 1 {\the style property display value b=\alpha \Beta 1}. Negative-integers are usually represented in a bar over a number, i.e. α = α {\the style property display the value\alpha ={bar \{\alpha }}}.

                                     

1.1. Notation and use. Balanced form. (Сбалансированная форма)

Balanced presentation of presentation, where α = β {\the style property display the value \alpha =\beta }, or, equivalently, where there are an equal number of negative and positive numbers in the set of digits to represent. Can only odd base a balanced view of the form, and when α = β {\the style property display the value \alpha =\beta }, b = α β 1 {\the style property display value b=\alpha \Beta 1} is always an odd number.

For example, consider the ternary number system with only three digits. In standard ternary, the digits { 0, 1, 2 } {\the style property display set to \lbrace 0.1.2\rbrace }, but, in balanced ternary, balanced ternary uses digits { 1, 0, 1 } {\the style property display set to \lbrace {\bar {1}},0.1\rbrace }. This Convention is adopted in finite fields of odd Prime order Q, and {\the style property display the value of g}:

G F q = { 0, 1, 1 ¯ = − 1. β = q − 1 2, β ¯ = 1 − q 2 | q = 0 }. {\displaystyle GFq=\lbrace 0.1,{\bar {1}}=-1.\beta ={\frac {q-1}{2}},\ {\bar {\beta }}={\frac {1-q}{2}}\ |\ q=0\rbrace.}

In the balanced form, truncation and rounding be one and the same operation.

                                     

1.2. Notation and use. Signed-digit decimal. (Подпись-значное десятичное)

Signed numbers are usually associated with the decimal number system. The following table shows the signed-digit representation of integers 10 ≤ N ≤ 10 {\the style property display the value -10\leq N\leq 10} for the base B = 10 = α β 1 {\the style property display the value of B=10=\alpha \Beta 1}.

                                     

1.3. Notation and use. Table. (Таблица)

The following table displays the set of numbers for each signed-digit representations α ≤ 5 {\the style property display set to \alpha \leq 5} and β ≤ 5 {\the style property display the value of \beta \leq 5}.

Regarded as a matrix, the matrix is cotaskmemrealloc matrix. The view down diagonally balanced form, signed-digit representations, and submissions on both sides of the diagonal is isomorphic to its counterpart on the other side of the diagonal through negation, where positive numbers on one side correspond to negative numbers and Vice versa. When α = 0 {\the style property display the value \alpha =0}, the representation can only represent non-negative integers, unsigned, and if β = 0 {\the style property display the value of \beta =0}, the representation can only represent non-positive unsigned numbers. When both α = 0 {\the style property display the value \alpha =0} and β = 0 {\beta \properties display style =0}, only zero can be represented, yielding a trivial ring.



                                     

1.4. Notation and use. Non-uniqueness of rationals. (Неединственность рационалы)

As in the case of the unsigned positional number system, when extended to represent rational numbers, uniqueness is lost. For example, consider the signed-digit decimal number with α = 5 {\the style property display the value \alpha =5}

0.4444 … 10 = 4 9 = 1. 5 ¯ 5 ¯ 5 ¯ 5 ¯ … 10 {\displaystyle {0.4444\ldots}_{10}={\frac {4}{9}}={1.{\bar {5}}{\bar {5}}{\bar {5}}{\bar {5}}\ldots}_{10}}

Such examples can be provided subject to maximum and minimum possible representations with integral parts 0 and 1, respectively, and then noting that they are equal.

                                     

1.5. Notation and use. Additional properties. (Дополнительные свойства)

For every integer b {\the style property display value b}, the base - b {\the style property display value b} of integers, when presented with positive and negative numbers, is the direct limit of the groups z / b n z {\the style property display set to \mathbb {Z} /b^{n}\mathbb {z} } with an index set of natural numbers with the usual order and the homomorphisms Z / b n z → z / B N 1 Z {\the style property display set to \mathbb {z} /b^{n}\mathbb {z} \rightarrow \mathbb {Z} /b^{n 1}\mathbb {z} }, the induced module B {\the style property display value b} operations. If there are either no negative or positive numbers in a database, then when you try direct limits prevent the group but only a monoid, consisting only of nonnegative or nonpositive numbers, respectively, the additive inverse property cannot be represented by a finite number of nonzero digits without the negation operator, and as the basis of the complements require an infinite number of nonzero digits. When there is no negative or positive numbers, only the digits to the left number-zero, yielding a trivial direct system of groups and direct restriction that is still the trivial group, not a group of integers. Hence, by definition of direct limit groups that is mandatory digit representation must contain both positive and negative numbers for the system to be isomorphic to the integers.

Widely used in the theory of numbers are b {\the style property display value b} -adic integers is the inverse limit of the groups z / b n z {\the style property display set to \mathbb {Z} /b^{n}\mathbb {z} } with an index set of natural numbers with the usual order, and the homomorphisms Z / b n z → z / B N 1 Z {\the style property display set to \mathbb {Z} /b^{n}\mathbb {z} \rightarrow \mathbb {Z} /b^{n 1}\mathbb {z} }, the induced module B {\the style property display value b} operations.

For every integer b {\the style property display value B}, then the fractional part of the base - b {\the style property display value b} as a decimal is called a Prufer b {\the style property display value B} -group, and is a direct limit of the groups z / b n z {\the style property display set to \mathbb {Z} /b^{n}\mathbb {z} } with an index set of natural numbers with the usual order and the homomorphisms Z / b n z → z / B N 1 Z {\the style property display set to \mathbb {Z} /b^{n}\mathbb {z} \rightarrow \mathbb {Z} /b^{n 1}\mathbb {z}} induced by multiplication by B {\the style property display the value of b}. Direct sum of Prufer b {\the style property display value B} -groups and Basic - b {\the style property display value b} of integers is a group of base - b {\the style property display value b} decimal fractions himself, otherwise known as B {\the style property display value b} -adic rational.

Inverse limit of the groups z / b n z {\the style property display set to \mathbb {Z} /b^{n}\mathbb {z} }, with index set of natural numbers with the usual order, and the homomorphisms Z / b n z → z / B N 1 Z {\the style property display set to \mathbb {Z} /b^{n}\mathbb {z} \rightarrow \mathbb {Z} /b^{n 1}\mathbb {z} }induced by multiplication by B {\the style property display value b}, the base - b {\the style property display value b} representation in a circle the group and the direct sum of a base - b {\the style property display value b} is a circle group and the base - b {\the style property display value b} of integers with basis b {\the style property display value b} representation of real numbers. This provides a formalization of the standard construction of real numbers as taught in high school WED. 0.999.#Infinite_decimal_representation for signed-digit representations, and is similar to the construction of the reals by Cauchy sequences.

For a Prime number p {\the style property display the value of n}, a base P {\the style property display the value of n} of integers can be considered as the Prufer p {\the style property display the value of R} group shall have R {\the style property display the value of P} -adic norm, not the Euclidean norm. A similar phenomenon occurs with P {\the style property display the value of p} -adic integers, which are the base - p {\the style property display the value of R} group circle endowed p {\the style property display the value of P} -adic norm, not the Euclidean norm. R {\the style property display the value of P} -adic rational is endowed with a p {\the style property display the value of P} -adic norms is isomorphic to R {\the style property display the value of P} -adic rational itself, endowed with the Euclidean norm.



                                     

2. History. (История)

Problems in the calculation of stimulated early writers Coulson Cauchy 1726, and 1840 to use a signed-digit representation. The next step is to replace the negated numbers with the new proposed selling screening 1887 and 1928.

In 1928, Florian screening noted the recurring theme of signed numbers, beginning with the 1726 Coulson and Cauchy 1840. In his book a history of mathematical notations, cinema under the title "negative numbers". For completeness, Coulson uses examples and describes the addition of PP 163.4, multiplication, and division 165.6 170.1 PP using a table of multiples of the divisor. He explains the convenience of the approximation, the truncation in multiplication. Coulson has also developed a counting table device, which is calculated using signed numbers.

Edward sale was made by inverting the digits 1, 2, 3, 4, and 5 to indicate a negative sign. He also suggested, Soviet, Jess, jerd, reff, and niff as all the names out loud. Most of the other early sources used in a bar over the number indicates a negative sign for him. Another German use of signed numbers has been described in 1902 in the encyclopedia of Kleins.

                                     

3. Other systems. (Другие системы)

There are other signed-digit bases such that the base B ≠ α β 1 {\the style property display the value of b\neq \alpha \Beta 1}. Striking examples of this is the booth encoding which α = 1 {\the style property display the value \alpha =1} β = 1 {\the style property display the value of \beta =1}, with values { 1, 0, 1 } {\the style property display set to \lbrace {\bar {1}},0.1\rbrace }, but which uses base B = 2 < 3 = α β 1 {\the style property display value B=2

                                     
  • canonical - signed - digit CSD is a special manner for encoding a value in a signed - digit representation which itself is non - unique representation and allows
  • numerals. The concept of signed - digit representation has also been taken up in computer design. Despite the essential role of digits in describing numbers
  • be used efficiently for signed integers. Another approach is to give each digit a sign yielding the signed - digit representation For instance, in 1726
  • A redundant binary representation RBR is a numeral system that uses more bits than needed to represent a single binary digit so that most numbers have
  • The non - adjacent form NAF of a number is a unique signed - digit representation Like the name suggests, non - zero values cannot be adjacent. For example:
  • the negative integers by use of a signed - digit representation to represent each integer. The concept of a decimal digit sum is closely related to, but not
  • any branch. A form of redundant binary representation called balanced ternary or signed - digit representation is sometimes used in low - level software
  • but then every integer may not have a unique representation For example, Fibonacci coding uses the digits 0 and 1, weighted according to the Fibonacci
  • they provide a human - friendly representation of binary - coded values. Each hexadecimal digit represents four binary digits also known as a nibble, which
                                     
  • A signed overpunch is a code used to store the sign of a number by changing the last digit It is used in character data on IBM mainframes by languages
  • usual decimal representation of whole numbers gives every nonzero whole number a unique representation as a finite sequence of digits beginning with
  • 20, 000 digits if they do exist. The multiplicative digital root can be extended to the negative integers by use of a signed - digit representation to represent
  • particular integer is exactly the integer s Zeckendorf representation with the order of its digits reversed and an additional 1 appended to the end. For
  • represent digits but it was not really a mixed - radix system of bases 10 and 6, since the ten sub - base was used merely to facilitate the representation of the
  • values. In a signed - digit representation each digit of a number may have a positive or negative sign The ideas of signed area and signed volume are sometimes
  • complement of a number given in decimal representation is formed by replacing each digit with nine minus that digit To subtract a decimal number y the
  • order of magnitude and unit of measurement of the first digit in the numerical representation In this case 十元 which stands for ten yuan When put
  • significant digits and decimal places of a number are digits that carry meaning contributing to its measurement resolution. This includes all digits except:
  • mixed - radix representation the indices of the output values are expanded in a corresponding mixed - radix representation with the order of the bases and digits reversed
  • allow representation of arbitrarily large numbers using multiple words of memory. Computers represent data in sets of binary digits The representation is


                                     
  • appending digits to the most significant side of the number, following a procedure dependent on the particular signed number representation used. For
  • base - φ representation can be uniquely standardized in this manner. If we get to a point where all digits are 0 or 1 except for the first digit being
  • used in digit grouping, so the latter is also treated in this article. Any such symbol can be called a decimal mark, decimal marker or decimal sign But
  • In the table below the digit of value 1 is written as the single character T. Some numbers have the same representation in base r as in base r. For
  • base 8 Some programming languages also permit digit group separators. The internal representation of this datum is the way the value is stored in the
  • base - 2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit. Because of its straightforward implementation in
  • 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
  • trials. Items may include words, numbers, or letters. The task is known as digit span when numbers are used. Memory span is a common measure of short - term
  • but as place value of digits By converting a number less than n to factorial representation one obtains a sequence of n digits that can be converted
  • contribution of a digit to the value of a number is the product of the value of the digit by a factor determined by the position of the digit In early numeral

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