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2.5 CONVERSION BETWEEN NUMBER SYSTEMS 37

Table 2.3 The BCH and BCO number systems

Binary BCH BCO Decimal Binary BCH BCO Decimal
0000 0 0 0 1010 A 12 10
0001 1 1 1 1011 B 13 11
0010 2 2 2 1100 C 14 12
0011 3 3 3 1101 D 15 13
0100 4 4 4 1110 E 16 14
0101 5 5 5 1111 F 17 15
0110 6 6 6 10000 10 20 16
0111 7 7 7 11011 IB 33 27
1000 8 10 8 110001 31 61 49
1001 9 11 9 1001110 4E 116 78

When the hexadecimal and octal number systems are used to represent bit patterns in
binary, they are called binary coded hexadecimal (BCH) and binary coded octal (BCO),
respectively. These two number systems are examples of binary-derived radices. Table 2.3
lists several selected examples showing the relationships between BCH, BCO, binary and
decimal.
What emerges on close inspection of Table 2.3 is that each hexadecimal digit corresponds
to four binary digits, and that each octal digit corresponds to three binary digits. The
following example illustrate the relationships between these number systems:

5 B F . D 8
10110111111.11011 2=0101 1011 1111.1101 1000

= 5BF.D8 16
2 677.6 6

= 010 110 111 111 . 110 110
= 2677.66 8
= 1471.84375i 0.

To separate the binary digits into groups of four (for BCH) or groups of three (for BCO),
counting must begin from the radix point and continue outward in both directions. Then,
where needed, zeros are added to the leading and trailing ends of the binary representation
to complete the MSDs and LSDs for the BCH and BCO forms.

2.5 CONVERSION BETWEEN NUMBER SYSTEMS

It is not the intent of this section to cover all methods for radix (base) conversion. Rather,
the plan is to provide general approaches, separately applicable to the integer and fraction
portions, followed by specific examples.