1. Decimal System
The radix or base of a number system determines
the total number of different symbols or digits
used by the system.
The decimal system has a base of 10.
In the decimal system, 10 unique numbers or
digits ( 0 through 9) are used: the total number
of symbols is the same as the base, and the symbol
with the largest value is 1 less than the base.

2. Decimal System The decimal system can be summarized as follows:
Ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Base: 10
Weights: 1, 10, 100, 1000, …(powers of base 10)

3. Decimal System
Weighted value in the decimal system

4. Binary System The binary system has a base of 2.
The only allowable digits are 0 and 1
Digital Signal Waveform: with digital circuits it is easy
to distinguish between two voltage levels - +5 V and O V,
which can be related to the binary digits 1 and 0.
Time
Volts +5
High (H) (1)
Low (L) (0)

5. Binary System The binary system can be summarized as follows:
Two digits: 0, 1
Base: 2
Weights: 1, 2, 4, 8, 16, 32, …(powers of base 2)

6. Binary System Since the binary system uses only
Decimal
Binary
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 8
1000
Since the binary system uses only
two digits, each position of a binary
number can go through only two
changes, and then a 1 is carried to
the immediate left position.
To express the number in the
binary system requires many
more digits than in the decimal
system.

7. Converting For Binary To Decimal

8. Converting For Binary To Decimal
Another Method
In the binary number
when you see a 1, multiply
that 1 times the value that
is directly over it. Where
you see a 0 in the box,
just ignore it.
If we add only those numbers which have a binary 1
in the box under them, we come up with
which equals 173.

9. Bits – Bytes - Words Each digit of a binary number is known as a bit.
A group of 8 bits is known as a byte.
A group of bits that occupies one or more storage
locations and is treated as a unit is known as a word.
A 16-bit word is made up of two bytes (Upper and Lower).
The least significant bit (LSB) is the digit that represents
the smallest value.
The most significant bit (MSB) is the digit that represents
the largest value.
16-Bit Word
Bit
Upper Byte
MSB
LSB

10. PLC Processor Memory Size
The size of the programmable controller relates to the
amount of user program that can be stored.
The 1 K word memory
size shown can store
1,024 words, or 16,380
(1,024 x 16) bits of
information using
16-bit words or 32,768
(1,024 x 32) using
32 bit words.

11. Converting For Decimal To Binary

12. Binary Representation Of Data
Even though the binary system has only two digits,
it can be used to represent any quantity that can be
represented in the decimal system. Computer memory
is then a series of binary 1s and 0s.
SLC 500 Modular Chassis Output Status File
A word will be created in the table only if the processor
finds an output module residing in a particular slot.
Each bit represents the “on” or “off” state of one
output point. These points are numbered 0 through15.
One 16-bit output file word is reserved for each slot
in the chassis.
The column on the right lists the output module address.
Made up of single bits grouped into 16-bit words

13. Read chapter 7-1 thru 7-3 p

14. Binary Addition When adding with binary numbers, there are only
four conditions that can occur.

15. Binary Addition When adding larger binary numbers, the resulting
1’s are carried into higher-order columns.

16. Binary Subtraction To subtract from larger binary numbers, subtract
column by column, borrowing from the adjacent
column when necessary. Remember that when
borrowing from the adjacent column, there are
two digits, i. e., 0 borrow 1 gives 10.

17. Binary Subtraction To subtract using the 1’s complement:
Change the subtrahend to 1’s complement
Add the two numbers
Remove the last carry and add it to the number
1’s complement

18. Binary Multiplication
When multiplying binary numbers, there are only
four conditions that can occur.
0 x 0 = 0
0 x 1 = 0
1 x 0 = 0
1 x 1 = 1

19. x 110 Binary Multiplication
To multiply numbers with more than one digit,
form partial products and add them together.
101
x 110
000
101
101
11110

20. Binary Division The process for dividing one binary number by
another is the same for both binary and decimal
numbers.
111 10 11 00

21. 1. The binary number system has a base of 8.
(True/False)
2. The decimal number 7 would be written in
binary as (True/False)
3. To express a number in decimal requires fewer
digits than in the binary system. (True/False)
4. For a base 2 number system, the weight value
associated with the 3rd digit would be 4.
(True/False)

22. 5. What is the decimal value of binary 110010 ?
a c. 13
b d. 50
6. The decimal number 15 would be written in
binary as:
a c. 4C
b d
7. Data can be stored in one 16-bit word as two
separate groups of 8-bit data. (True/False)

23. 8. A group of 8 bits is known as a byte.
(True/False)
9. The MSB is the digit that represents the
smallest value. (True/False)
10. Since the binary system has only two digits,
it is limited as far as representing very large
quantities. (True/False)