Introduction to Industrial Automation

A.3   Binary Arithmetic System

The binary arithmetic system uses the two digits 0 and 1, so its base is 2. Any number can be 

expressed as a combination of digits 0 and 1. In order for an electronic circuit to process numbers, 

which is called a “digital circuit”, it is necessary to join the numbers with corresponding electric 

voltage signals. Therefore, if a digital device, such as the PLC, were to process decimal numbers, 

then ten different voltages would be required, one voltage level for each digit. Consequently, each 

decimal number will be expressed by a combination of these ten voltage levels. The implementa-

tion of a digital circuit for processing decimal numbers, due to many electric elements needed for 

creating the ten different voltages, will lead to a very complex circuitry. For this reason, to avoid 

complex electronic circuits, the binary arithmetic system is the most suitable for developing digital 

circuits for processing numbers, since only two voltage levels are required. Assuming that 5 V and 

0 V represent the digits 1 and 0 correspondingly, a four-digit binary number (such as 1011) will 

be represented electronically by the combination of voltages 5 V, 0 V, 5 V, and 5 V, which can be 

recognized by a processor. It is widely known that a combination of eight binary digits (bits) is 

called a “byte” and a combination of 16 binary digits is called a “word”.

The position weighting that is assigned to each digit of the binary number, due to the base 2, 

will be doubled for each step to the left of the number, with the first right position to correspond 

to the value 1 (2

0

). As in the case of the decimal arithmetic system, all the position values of the 

binary arithmetic system can be expressed as powers of two, as shown in Figure A.2. The digit of 

Decimal number

Position value

Position value as

power of 10

Multiplication of

decimal digit and

position value

Decimal number

6 0 5 3

6 2

.

1000

10

0

10

1

1/1

0

1/100

10

3

10

2

10

1

10

0

10

–1

10

–2

6000

6053.62

50

3

0.7

0.02

0