The binary system uses only two digits, 0 and 1, with each column in the binary number representing a successive power of two. The hexadecimal system uses the digits 0-9 and letters A-F, with each column representing a power of 16, providing a more compact notation than binary. Binary Coded Decimal (BCD) represents each decimal digit with a 4-bit binary group to allow decimal numbers to be displayed digitally.

1. Computers – Numbering Systems
The Binary System
This is a numbering system to the base of two, meaning that it only has two digits 0 and 1.
The binary system is the base numbering system used in computer and digital logic control
systems.
Binary numbers are written as a sequence of zero’s and one’s, where: 0101; 1000; 1001 are
examples of four-bit binary numbers.
Weighting
Each column in the binary number represents a denary number as shown below
and doubles with each column.
This is called the ‘weighting’ of the numbering system.
Each 1 or 0 multiplies a successive power of 2.
8 4 2 1
LSBMSB
23
22
21
20
Bin 1 1 0 1 = (1x8) + (1x4) + (0x2) + (1x1)= 13d

2. Computers – Numbering Systems
The Hexadecimal System
Numbers in the binary system tend to get very long, the hex system is more compact and
less prone to error.
Each column represents a successive power of sixteen (base 16).
Hex symbols 0 – 9 for decimal 0 – 9 and A to F for decimal 10 to 15.
256 16 1
162
161
160
Hex 2 3 4 = (2x256) + (3x16) + (4x1) = 564d
Decimal Binary Hex
101
(10)
100
(1)
24
(16)
23
(8)
22
(4)
21
(2)
20
(1)
161
(16)
160
(1)
0 7 0 0 1 1 1 0 7
1 0 0 1 0 1 0 0 A
1 2 0 1 1 0 0 0 C
1 5 0 1 1 1 1 0 F
2 7 1 1 0 1 1 1 8

3. Computers – Numbering Systems
The Hexadecimal System
To write a binary number in hexadecimal group the number into four-bit groups beginning
with the LSB and write the hex equivalent of each group.
Example
Decimal 245 = 11110101
Group into four bits, = 1111 0101
Hex = F 5
Decimal 245 = F5 (Hex)
The Octal System
A base-8 system from an early era, used during the development of the first computer
systems and is not commonly used these days.
Although it does not require the extra symbols (A-F “hex”) it is extremely awkward when
applied to byte-organised words of today’s computer systems.

4. BINARY CODED DECIMAL (BCD)
By converting (encoding) each decimal digit into a four-bit binary group we obtain a
system called Binary Coded Decimal (BCD).
This system is ideal for use when we wish to display a decimal digit from its binary
equivalent eg digital displays.
Each four-bit group is binary weighted and represents a single digit.
Computers – Numbering Systems
Example
Decimal 245 = 2 4 5
BCD = 0010 0100 0101

5. Computers – Numbering Systems
2. State the decimal equivalents represented by the binary numbers shown below,
a) 01100101, b) 10110011, c) 10111101.
Assessment
1. Write down the binary equivalents for denary numbers 0 to 12, 28, 32 and 64.
3. State the hexadecimal equivalents for the denary numbers shown below,
a) 8, b) 12, c) 18, d) 28.
4. Convert the binary numbers in question 2 to their hexadecimal equivalents.
5. Convert the binary numbers in question 2 to their BCD equivalents.

6. ASCII CODE
This is the American Standard Code for Information Interchange.
It is a method of coding alphabetic, numeric and punctuation characters into
groups of 7-bits.
Computers – Coding Systems
Char Hex Dec Char Hex Dec
A 41H 65 a 61H 97
B 42H 66 b 62H 98
C 43H 67 c 63H 99
0 30H 48 5CH 92
1 31H 49 ] 5DH 93
2 32H 50 < 3CH 60

7. GRAY CODE
The Gray code is used for mechanical shaft-angle encoders, the feature of this
code is that only one bit changes in going from one state to the next.
This method prevents errors, since there is no way of guaranteeing that all bits will
change simultaneously at the boundary between two encoded states.
Computers – Coding Systems
Position Gray Code Position Gray Code
0 0000 8 1100
1 0001 9 1101
2 0011 10 1111
3 0010 11 1110
4 0110 12 1010
5 0111 13 1011
6 0101 14 1001
7 0100 15 1000
D0
D1
D2
D3
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 position
LED’s

8. 2’s COMPLEMENT
The “2’s complement” is the method most widely used for integer computation in
electronic systems.
To obtain a negative number (2’s complement) first complement each bit of the
positive number then add 1.
Computers – Arithmetic Operations
Check this by adding the original positive number to the 2’s complement
representation.
The result should be zero, since the 2’s complement changes the sign of the
original positive number.
Example
Decimal 5 = 0101
1’s complement = 1010
+1 (2’s complement) = 1011

9. Using the 2’s Complement
Subtract B from A, take the 2’s complement of B (create a negative of B) then add
this to A.
Computers – Arithmetic Operations
Example
5 - 2 = 5 + (-2)
2 = 0010
1’s complement = 110
+1 (2’s complement) = 1110
5 = 0101
Add = 0011 = 3

10. Computers – Data Systems
BITS to GIGABYTES
The bit is short for binary digit and has two values (0 or 1)
When four binary bits are grouped together they form a nybble,
‘0110’ and ‘1110’.
When two nibbles, (eight binary bits) are grouped together they form a
byte,‘10110101’ and ‘00110111’.
A word is the term used to represent the unit of data in a particular system.
In an eight bit system the word contains eight bits.
Personal computers today use 32 and 64 bit words while pic’s and micro-
controller’s and Z80cpu systems use four and eight bit words.

11. Computers – Data Systems
BITS to GIGABYTES – Common Units of Measure
Kilobyte (kB) – Describes the data storage capacity of small systems e.g. memory
capacity for micro-controllers and size of data files.
1kB = 1024 bytes (210
bytes).
Megabyte (MB) – Describes the data storage capacity of memory devices, image
files (bitmaps).
1MB = 1024 kB (220
bytes).
Gigabyte (GB) – Describes the data storage capacity of hard drives, CD/DVD data
storage devices. The gigabyte is the largest unit of capacity in use at present.
1GB = 1024 MB (230
bytes).