Introduction to Industrial Automation

239

Figure A.3  Conversion of a decimal number to binary.

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412

 

◾

  Appendix A

The magnitude of a negative number represented with regard to one’s complement is the one’s 

complement of the corresponding positive number magnitude. Some examples of negative num-

bers represented in one’s complement, are shown in Figure A.5.

The two’s complement of a binary number is the sum of one’s complement plus 1, as expressed 

by Equation A.1,

 

2

1

1

’

’

s complement

s complement

=

+

 (A.1)

Binary

number

Decimal

number

Sign

Magnitude

0 1 1 1

+7

0 1 1 0

+6

0 1 0 1

+5

0 1 0 0

+4

0 0 1 1

+3

0 0 1 0

+2

0 0 0 1

+1

0 0 0 0

+0

1 0 0 1

–1

1 0 1 0

–2

1 0 1 1

–3

1 1 0 0

–4

1 1 0 1

–5

1 1 1 0

–6

1 1 1 1

–7

Figure A.4  Signed binary numbers.

Binary

number

Decimal

number

1’s complement

Sign

Magnitude

Magnitude

1

1

0 0

1

0

1 1

1

1

1 0

1

0

1 0

1

1

0 1

1

0

0 1

1

1

0 0

1

0

0 0

1

0

0 1

1

1

0 1

1

0

1 0

1

1

1 0

1

0

1 1

1

1

1 1

–1

–2

–3

–4

–5

–6

–7

Sign

Figure A.5  Representation of negative numbers using the complement of a binary number 

(one’s complement).

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Appendix A 

◾

 

413

PLCs use the two’s complement to perform subtraction of two binary numbers, which means 

that the subtraction is replaced by an additive operation, as clarified by Equation A.2,

 

(7)

10

 − (5)

10

 = (7)

10

 + (−5)

10

 = (0111)

2

 + (1011)

2

 = (0010)

2

 = (2)

10

2’s complement

 (A.2)

The subtraction operation using the two’s complement facilitates its implementation in a PLC, 

since the same digital circuit will perform both addition and subtraction.

A.4   Binary-Coded Decimal Arithmetic System (BCD)

In a binary-coded decimal arithmetic system, each digit of a decimal number is coded via a four-

digit binary number. This definition means that each digit of a decimal number is simply substi-

tuted by the corresponding binary number, as in the following example,

45

10

0100

2

0101

2

In the past, computer systems had specific units for the execution of the arithmetic operations 

between BCD numbers in order to overcome memory storage problems. In some PLCs today, the 

BCD arithmetic system is even supported since it is compatible with the operation of some output 

devices as an indication display device, or input devices as an arithmetic switch. Figure A.6 shows 

an example of converting a BCD number to an equivalent decimal number.