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It is defined by indexing the elements of the sequence by the numbers from to , representing each of these numbers by its binary representation (padded to have length exactly ), and mapping each item to the item whose representation has the same bits in the reversed order.
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" and "1" ().
Both instances and solutions are represented by binary strings, namely elements of {0, 1} *. [ a ] For example, natural numbers are usually represented as binary strings using binary encoding . This is important since the complexity is expressed as a function of the length of the input representation.
In computing, signed number representations are required to encode negative numbers in binary number systems. In mathematics, negative numbers in any base are represented by prefixing them with a minus sign ("−"). However, in RAM or CPU registers, numbers are represented only as sequences of bits, without extra symbols.
The binary representation of a number is an expression for as a sum of distinct powers of two, = + + + where each bit in this expression is either 0 or 1. It is commonly written in binary notation as just the sequence of these bits, ⋯ b 3 b 2 b 1 b 0 {\displaystyle \cdots b_{3}b_{2}b_{1}b_{0}} .
The modern binary number system, the basis for binary code, is an invention by Gottfried Leibniz in 1689 and appears in his article Explication de l'Arithmétique Binaire (English: Explanation of the Binary Arithmetic) which uses only the characters 1 and 0, and some remarks on its usefulness. Leibniz's system uses 0 and 1, like the modern ...
The length of the grid's last line is given by the remainder. The key is written above the grid, and the ciphertext is written down the columns of the grid in the order given by the letters of the key. The plaintext appears on the rows. A partial decipherment of the above ciphertext, after writing in the first column: 6 3 2 4 1 5 . . . . E ...
A bitwise AND is a binary operation that takes two equal-length binary representations and performs the logical AND operation on each pair of the corresponding bits. Thus, if both bits in the compared position are 1, the bit in the resulting binary representation is 1 (1 × 1 = 1); otherwise, the result is 0 (1 × 0 = 0 and 0 × 0 = 0).