Introduction to Industrial Automation

Table A.1  The Hexadecimal Digits and Their Equivalents 

in Both Binary and Decimal Systems

Hexadecimal 

Number

Binary Number

Decimal 

Number

0

0

0

0

0

0

1

0

0

0

1

1

2

0

0

1

0

2

3

0

0

1

1

3

4

0

1

0

0

4

5

0

1

0

1

5

6

0

1

1

0

6

7

0

1

1

1

7

8

1

0

0

0

8

9

1

0

0

1

9

A

1

0

1

0

10

B

1

0

1

1

11

C

1

1

0

0

12

D

1

1

0

1

13

E

1

1

1

0

14

F

1

1

1

1

15

---

416

 

◾

  Appendix A

divisions of the decimal number by 16, and by denoting the remainders of divisions as the digits 

of the hexadecimal number, as in the following example for decimal 4670:

4670/16

= 291 with remainder 14 (=E)

291/16

= 18 with remainder 3

18/16

= 1 with remainder 2

1/16

= 0 with remainder 1, therefore hexadecimal number is 123E

16

The conversion of a hexadecimal number to a decimal number is performed by a procedure sim-

ilar to this, which was applied in previous arithmetic systems, such as the one shown in Figure A.9. 

Finally, the conversion of a hexadecimal number to a binary number is done with a simple replace-

ment of each hexadecimal digit with its equivalent binary form of 4-bits, according to Table A.1.

A.6.1   Parity  Checking

Arithmetic digital data, independently of the utilized arithmetic system and the way that they 

are represented, do not stay static at some memory location of a digital device, but instead they 

are transported from one point to another. Digital data are transferred continuously, especially in 

PLCs, mainly due to the real-time control that PLCs perform (such as from a PLC to a peripheral 

device, from a PLC to another PLC via a communication network, etc.). It is very likely for a single 

bit of binary data to change value from 1 to 0, due to electromagnetic noise or a transitional phe-

nomenon, or any other imponderable reason. For the proper detection of a possible error during 

the transmission or storage of binary data, the parity checking technique is applied which consists 

of adding an extra bit, called a parity bit, in the transferred data word.

Even Parity. The parity bit is added to the transferred data is such way that the total number of 

units (1) are even. For example, if the byte 01010100 is transmitted, then the parity bit 1 has 

to be added and becomes 010101001.

Odd Parity. The parity bit is added to the transferred data in such a way that the total number 

of units (1) are odd. For example, if the byte 01010100 is transmitted, then the parity bit 0 

has to be added and becomes 010101000.

Hexadecimal number

Position value

Decimal number

Position value as

power of 16

Multiplication of

hexadecimal digit and

position value

1 2 3 E

16

10

4096

256

16

1

4096

512

48

14

16

3

16

2

16

1

16

0

+

4670