082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

1.3 Number bases 17
Table1.4isusedtoexpresseachgroupoffourasitshexequivalent.Forexample,0111 =
7 16
, and continuing inthis waywe obtain
1AE7
Thus 110101110 0111 2
= 1AE7 16
.
1.3.4 Binarycodeddecimal
We have seen in Section 1.3.2 that decimal numbers can be expressed in an equivalent
binary form where the position of each binary digit, moving from the right to the left,
represents an increasing power of 2. There is an alternative way of expressing numbers
using the binary digits 1 and 0 that is often used in electronic engineering because for
some applications it is more straightforward to build the necessary hardware. This systemiscalledbinary
coded decimal(b.c.d.).
First of all, recall how the decimal digits 0,1,2,...,9 are expressed in their usual
binary form. Note that the largest decimal digit 9 is 1001 in binary, and so we need
at most four digits to store the binary representations of 0,1,...,9. Expressing each
decimal digitas a four-digitbinary number we obtain Table 1.5.
Table1.5
Decimal digits and their four-digit
binaryrepresentations.
0 0000 5 0101
1 0001 6 0110
2 0010 7 0111
3 0011 8 1000
4 0100 9 1001
A four-digit binary number is referred to as a nibble. To express a multi-digit decimal
number, such as 347, in b.c.d. each decimal digit in turn is converted into its binary
representation as shown. Note thatanibble isused foreach decimal digit.
3 4 7
↓ ↓ ↓
0011 0100 0111
Recall from Section 1.3.2 that a byte is a group of 8 bits (binary digits). Computers
usually store numbers in 8-bit bytes so there are two common ways of encoding b.c.d.
The first is to use a whole byte for each nibble, with the first 4 bits always set to 0. So,
forexample, 347 10
can be storedas
00000011 00000100 00000111
Alternatively, eachbytecanbeusedtostoretwonibbles,inwhichcase347 10
wouldbe
storedas
00000011 01000111
Ruleshavebeendevelopedforperformingcalculationsinb.c.d.butthesearebeyondthe
scope of thisbook.