Binary
The Base 2 method of counting in which only the digits 0 and 1 are used. In thisBase, the number 1011 equals 
. ThisBase is used in computers, since all numbers can be simply represented as a string of electricallypulsed ons and offs. A Negative 
is most commonly represented as the complement of the Positive number 
, so
would be written as the complement of 
, or 11110101. This allows addition to be carriedout with the usual carrying and the left-most digit discarded, so 
gives
 | |
 | |
 |
The number of times 
a given binary number 
is divisible by 2 is given by the position of the first
counting from the right. For example, 
is divisible by 2 twice, and 
is divisible by 2 0 times.
Unfortunately, the storage of binary numbers in computers is not entirely standardized. Because computers storeinformation in 8-bit bytes (where a bit is a single binary digit), depending on the ``word size'' of the machine, numbersrequiring 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 byte2byte1) on a Sun. The situation is even worse for floating point (real) numbers, which are represented in binary as aMantissa 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 thefollowing Multiplication Table.
 | 0 | 1 |
| 0 | 0 | 0 |
| 1 | 0 | 1 |
References
Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 6-9, 1991. Pappas, T. ``Computers, Counting, & Electricity.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 24-25, 1989. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Error, Accuracy, and Stability'' and ``Diagnosing Machine Parameters.'' §1.2 and §20.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 18-21, 276, and 881-886, 1992.
Weisstein, E. W. ``Bases.'' Mathematica notebook Bases.m.
- Cyclic Redundancy Check
- Cyclid
- Cyclide
- Cyclidic Coordinates
- Cycloid
- Cycloid Evolute
- Cycloid Involute
- Cycloid Radial Curve
- Cyclomatic Number
- Cyclotomic Equation
- Cyclotomic Factorization
- Cyclotomic Field
- Cyclotomic Integer
- Cyclotomic Invariant
- Cyclotomic Number
- Cyclotomic Polynomial
- Cylinder
- Cylinder Cutting
- Cylinder-Cylinder Intersection
- Cylinder Function
- Cylinder-Sphere Intersection
- Cylindrical Coordinates
- Cylindrical Equal-Area Projection
- Cylindrical Equidistant Projection
- Cylindrical Function