Computer numbering formats

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The term computer numbering formats refers to the schemes implemented in digital computer and calculator hardware and software to represent numbers.[citation needed]

Contents

Bits, bytes, nibbles, and unsigned integers

Bits

The concept of a bit can be understood as a value of either 1 or 0, on or off, yes or no, true or false, or encoded by a switch or toggle of some kind. A single bit must represent one of two states:

  one-digit binary value:       decimal value:
  -----------------------       --------------
  0                             0
  1                             1                  two distinct values
  

While a single bit, on its own, is able to represent only two values, a string of two bits together are able to represent twice as many values:

  two-digit binary value:       decimal value:
  -----------------------       --------------
  00                            0
  01                            1
  10                            2
  11                            3                  four distinct values
  

A series of three binary digits can likewise designate twice as many distinct values as the two-bit string.

  three-digit binary value:     decimal value:
  -------------------------     --------------
  000                           0
  001                           1
  010                           2
  011                           3
  100                           4
  101                           5
  110                           6
  111                           7                   eight distinct values
  

As the number of bits within a sequence goes up, the number of possible 0 and 1 combinations increases exponentially. The examples above show that a single bit allows only two value-combinations, while two bits combined can make four separate values; three bits yield eight possibilities, and the amount of possible combinations doubles with each binary digit added:

  bits in series (b):           number of possible values (N):
  -------------------------     ------------------------------
  1                             2
  2                             4
  3                             8
  4                             16
  5                             32
  6                             64
  7                             128
  8                             256
  
  ...                                               2b = N

[edit] Bytes

A byte is a sequence of eight bits or binary digits that can represent one of 256 possible values. Modern computers process information in 8-bit units, or some other multiple thereof (such as 16, 32, or 64 bits) at a time. A group of 8 bits is now widely used as a fundamental unit, and has been given the name of octet. A computer's smallest addressable memory unit (a byte) is typically an octet, so the word byte is now generally understood to mean an octet.

[edit] Nibbles

A unit of four bits, or half an octet, is often called a nibble (or nybble). It can encode 16 different values, such as the numbers 0 to 15. Any arbitrary sequence of bits could be used in principle, but in practice the most common scheme is:

  0000  =  decimal 00           1000  =  decimal 08
  0001  =  decimal 01           1001  =  decimal 09
  0010  =  decimal 02           1010  =  decimal 10
  0011  =  decimal 03           1011  =  decimal 11
  0100  =  decimal 04           1100  =  decimal 12
  0101  =  decimal 05           1101  =  decimal 13
  0110  =  decimal 06           1110  =  decimal 14
  0111  =  decimal 07           1111  =  decimal 15

This order (rather than gray code) is used because it is a positional notation, like the decimal notation that humans are more used to. For example, given the decimal number:

7531

is commonly interpreted as:

(7 × 1000) + (5 × 100) + (3 × 10) + (1 × 1)

or, using powers-of-10 notation:

(7 × 103) + (5 × 102) + (3 × 101) + (1 × 100)

(Note that any number to the zero power is 1.)

Each digit in the number represents a value from 0 to 9 (hence ten different possible values) which is why this is called a decimal or base-10 number. Each digit also has a weight of a power of ten associated with its position.

Similarly, in the binary number encoding scheme mentioned above, the (decimal) value 13 is encoded as:

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