The Base 2 method of counting in which only the digits 0 and 1 are used. In this Base, the number 1011 equals . This Base is used in computers, since all numbers can be simply represented as a string of electrically pulsed 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 carried out 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 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 |

**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.

© 1996-9

1999-05-26