# Number Systems by mhmmdmousa255

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• pg 1
```									               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
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
   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
between binary, decimal and
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