Page 66 - Programmable Logic Controllers, Fifth Edition

3.1 Decimal System
+5 High (H) (1)
Knowledge of different number systems and digital codes Volts
is quite useful when working with PLCs or with almost Low (L) (0)
any type of digital computer. This is true because a basic 0 Time
requirement of these devices is to represent, store, and op-
erate on numbers. In general, PLCs work on binary num- Figure 3-2 Digital signal waveform.
bers in one form or another; these are used to represent
various codes or quantities. changes, and then a 1 is carried to the immediate left po-
The decimal system, which is most common to us, has sition. Table 3-1 shows a comparison among four com-
a base of 10. The radix or base of a number system de- mon number systems: decimal (base 10), octal (base 8),
termines the total number of different symbols or digits hexadecimal (base 16), and binary (base 2). Note that all
used by that system. For instance, in the decimal system, numbering systems start at zero.
10 unique numbers or digits—i.e., the digits 0 through 9— The decimal equivalent of a binary number can be de-
are used: the total number of symbols is the same as the base, termined in a manner similar to that used for a decimal
and the symbol with the largest value is 1 less than the base. number. This time the weighted values of the positions
The value of a decimal number depends on the dig- are 1, 2, 4, 8, 16, 32, 64, and so on. The weighted value,
its that make up the number and the place value of each instead of being 10 raised to the power of the position, is
digit. A place (weight) value is assigned to each position 2 raised to the power of the position. Figure 3-3 illustrates
that a digit would hold from right to left. In the decimal how the binary number 10101101 is converted to its deci-
system the first position, starting from the rightmost po- mal equivalent: 173.
sition, is 0; the second is 1; the third is 2; and so on up Each digit of a binary number is known as a bit. In a
to the last position. The weighted value of each position PLC the processor-memory element consists of hundreds
can be expressed as the base (10 in this case) raised to or thousands of locations. These locations, or registers,
the power of the position. For the decimal system then,
the position weights are 1, 10, 100, 1000, and so on.
Figure 3-1 illustrates how the value of a decimal number Table 3-1 Number System Comparisons
can be calculated by multiplying each digit by the weight
of its position and summing the results. Decimal Octal Hexadecimal Binary
0 0 0 0
3.2 Binary System 1 1 1 1

The binary system uses the number 2 as the base. The 2 2 2 10
only allowable digits are 0 and 1. With digital circuits it is 3 3 3 11
easy to distinguish between two voltage levels (i.e., +5 V 4 4 4 100
and 0 V), which can be related to the binary digits 1 and 0 5 5 5 101
(Figure 3-2). Therefore, the binary system can be applied 6 6 6 110
quite easily to PLCs and computer systems. 7 7 7 111
Since the binary system uses only two digits, each 8 10 8 1000
position of a binary number can go through only two
9 11 9 1001
10 12 A 1010
Decimal 11 13 B 1011
number 12 14 C 1100
3 2 1 0
13 15 D 1101
1 9 6 2
10 14 16 E 1110
0
2 × 10 = 2 × 1 = 2 15 17 F 1111
1
6 × 10 = 6 × 10 = 60 16 20 10 10000
2
9 × 10 = 9 × 100 = 900 17 21 11 10001
3
1 × 10 = 1 × 1000 = 1000
18 22 12 10010
1962 10
(Sum of products) 19 23 13 10011
20 24 14 10100
Figure 3-1 Weighted value in the decimal system.

Number Systems and Codes Chapter 3 47

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