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Matematikçiler
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Matematik Seçkileri
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Sayılar Teorisi
=> Algebraic Curves-Mordell Curve
=> Algebraic Curves-Ochoa Curve
=> Algebraic Integer
=> Algebraic Number
=> Algebraic Number Theory
=> Chebotarev Density Theorem
=> Class Field
=> Cyclotomic Field
=> Dedekind Ring
=> Fractional Ideal
=> Global Field
=> Local Field
=> Number Field Signature
=> Picard Group
=> Pisot Number
=> Weyl Sum
=> Casting Out Nines
=> A-Sequence
=> Anomalous Cancellation
=> Archimedes' Axiom
=> B2-Sequence
=> Calcus
=> Calkin-Wilf Tree
=> Egyptian Fraction
=> Egyptian Number
=> Erdős-Straus Conjecture
=> Erdős-Turán Conjecture
=> Eye of Horus Fraction
=> Farey Sequence
=> Ford Circle
=> Irreducible Fraction
=> Mediant
=> Minkowski's Question Mark Function
=> Pandigital Fraction
=> Reverse Polish Notation
=> Division by Zero
=> Infinite Product
=> Karatsuba Multiplication
=> Lattice Method
=> Pippenger Product
=> Reciprocal
=> Russian Multiplication
=> Solidus
=> Steffi Problem
=> Synthetic Division
=> Binary
=> Euler's Totient Rule
=> Goodstein Sequence
=> Hereditary Representation
=> Least Significant Bit
=> Midy's Theorem
=> Moser-de Bruijn Sequence
=> Negabinary
=> Negadecimal
=> Nialpdrome
=> Nonregular Number
=> Normal Number
=> One-Seventh Ellipse
=> Quaternary
=> Radix
=> Regular Number
=> Repeating Decimal
=> Saunders Graphic
=> Ternary
=> Unique Prime
=> Vigesimal
Ziyaretçi defteri
 

Binary

The base 2 method of counting in which only the digits 0 and 1 are used. In this base, the number 1011 equals 1·2^0+1·2^1+0·2^2+1·2^3=11. This base is used in computers, since all numbers can be simply represented as a string of electrically pulsed ons and offs. In computer parlance, one binary digit is called a bit, two digits are called a crumb, four digits are called a nibble, and eight digits are called a byte.

An integer n may be represented in binary in Mathematica using the command BaseForm[n, 2], and the first d digits of a real number x may be obtained in binary using RealDigits[x, 2, d]. Finally, a list of binary digits l can be converted to a decimal rational number or integer using FromDigits[l, 2].

Binary

The illustration above shows the binary numbers from 0 to 63 represented graphically (Wolfram 2002, p. 117), and the following table gives the binary equivalents of the first few decimal numbers.

1 1 11 1011 21 10101
2 10 12 1100 22 10110
3 11 13 1101 23 10111
4 100 14 1110 24 11000
5 101 15 1111 25 11001
6 110 16 10000 26 11010
7 111 17 10001 27 11011
8 1000 18 10010 28 11100
9 1001 19 10011 29 11101
10 1010 20 10100 30 11110

A negative number -n is most commonly represented in binary using the complement of the positive number n-1, so -11=00001011_2 would be written as the complement of 10=00001010_2, or 11110101. This allows addition to be carried out with the usual carrying and the leftmost digit discarded, so 17-11=6 gives

 00010001   17
            11110101__  -11__
            00000110   6.

The number of times k that a given binary number b_n...b_2b_1b_0 is divisible by 2 is given by the position of the first b_k=1 counting from the right. For example, 12=1100 is divisible by 2 twice, and 13=1101 is divisible by 2 zero times. The number of times that 1, 2, ... are divisible by 2 are 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, ... (Sloane's A007814), which is the binary carry sequence.

Real numbers can also be represented using binary notation by interpreting digits past the "decimal" point as negative powers of two, so the binary digits ...b_2b_1b_0.b_(-1)b_(-2)... would represent the number

 ...+b_2·2^2+b_1·2^1+b_0·2^0+b_(-1)·2^(-1)+b_(-2)·2^(-2)+....

Therefore, 1/2 would be represented as 0.1_2, 1/4 as 0.01_2, 3/4 as 0.11_2, and so on. The sequence of binary digits for the integers n=0, 1, ... concatenated together and interpreted as a binary constant give the binary Champernowne constant C=0.11011100..._2 (Sloane's A030190).

Unfortunately, the storage of binary numbers in computers is not entirely standardized. Because computers store information in 8-bit bytes (where a bit is a single binary digit), depending on the "word size" of the machine, numbers requiring more than 8 bits must be stored in multiple bytes. The usual FORTRAN77 integer size is 4 bytes long. However, a number represented as (byte1 byte2 byte3 byte4) in a VAX would be read and interpreted as (byte4 byte3 byte2 byte1) on a Sun. The situation is even worse for floating-point (real) numbers, which are represented in binary as a mantissa and characteristic, and worse still for long (8-byte) reals!

Binary multiplication of single bit numbers (0 or 1) is equivalent to the AND operation, as can be seen in the following multiplication table.

× 0 1
0 0 0
1 0 1
BinarySums

Consider the cumulative digit sum of all binary numbers up to 1, 2, ..., n. The first few terms are then 1, 2, 4, 5, 7, 9, 12, 13, 15, 17, 20, 22, ... (Sloane's A000788). This sequence in monotonic increasing (left figure), but if the main asymptotic term is removed, a sequence of humped curves (right figure; Trott 2004, p. 218) tending towards the Blancmange function is obtained.


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