Number Systems by mhmmdmousa255

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									               Number Systems
The 0s and 1s present in the logic
circuits discussed in the course can be
used to represent real data inside logic
circuitry in, for example,
microprocessors. To do this a binary
format has to be adopted.
The usual practise is to use so-called
pure binary coding whereby each binary
digit (either 0 or 1) carries a certain
weight according to its position in the
binary number. So, for example

1101 = 1x + 1x + 0x + 1x + 0x + 0x
00     25   24   23   22   21   20
     =    +    +    +    +    +
       32 16 0        4    0    0
      =
          52

The same approach applies to non-
integral numbers so, for example

110.1 = 1x2 + 1x + 0x + 1x + 0x + 1x2-
        2
01            21   20   2-1 2-2 3
        =       +       +       + 0. +     + 0.1
            4       2       0          0
                                  5          25
        = 6.6
          25

These examples illustrate binary to
decimal conversion. To convert a
fractional decimal number to binary then
the procedure to follow is
    first divide the number at the decimal
     point and treat the two parts
     separately.
    For the integer part then repeatedly
     divide it by 2 and store the remainder
     until nothing is left.
    The remainders when reverse-
     ordered gives the first part of the
     binary number. The reverse-ordering
     comes about since the first division
     by 2 gives the least significant bit
     (lsb) and so on until the last division
     which gives the most significant bit
     (msb).
    For the fractional part repeatedly
     multiply by 2 and record the carries
   i.e. when the resulting number is
   greater than 1. Repeat this process
   until the desired precision is
   achieved.
An full example of this technique is
given in the Solved Problems.
A useful way of expressing long pure
binary coded numbers is by the use of
hexadecimal numbers i.e. base 16. This
is because each group of four bits
(called a nibble since 2 nibbles make a
byte!) can be converted into one
hexadecimal number. The mapping
between binary, decimal and
hexadecimal (hex.) numbers is shown
below.
Decima Binar He Decima Binar He
l      y     x l       y     x
     0 0000 0         8 1000 8
     1 0001 1         9 1001 9
     2 0010 2        10 1010 A
     3 0011 3        11 1011 B
     4 0100 4        12 1100 C
       5 0101      5       13 1101       D
       6 0110      6       14 1110       E
       7 0111      7       15 1111       F
To convert a binary number into its
hexadecimal equivalent first ensure that
the binary number has a number of
digits that is a multiple of 4, if not add
zeros to the left hand side of the
number until it does. Then split the
number into nibbles and convert each
nibble into its hexadecimal counterpart.
An example of binary to hexadecimal
conversion can be found in the Solved
Problems.

								
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