Number system

Digital Logics Number System
Prof. K. Adisesha
Presidency College
1
Numbering System
Many number systems are in use in digital technology. The most common are the decimal, binary, octal, and
hexadecimal systems. The decimal system is clearly the most familiar to us because it is a tool that we use
every day. Examining some of its characteristics will help us to better understand the other systems. In the
next few pages we shall introduce four numerical representation systems that are used in the digital system.
There are other systems, which we will look at briefly.
 Decimal
 Binary
 Octal
 Hexadecimal
Decimal System
The decimal system is composed of 10 numerals or symbols. These 10 symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8,
9. Using these symbols as digits of a number, we can express any quantity. The decimal system is also
called the base-10 system because it has 10 digits.
103
102
101
100
10-1
10-2
10-3
=1000 =100 =10 =1 . =0.1 =0.01 =0.001
Most Significant Digit Decimal point
Least Significant
Digit
Even though the decimal system has only 10 symbols, any number of any magnitude can be expressed by
using our system of positional weighting.
Decimal Examples
 3.1410
 5210
 102410
 6400010
Binary System
In the binary system, there are only two symbols or possible digit values, 0 and 1. This base-2 system can be
used to represent any quantity that can be represented in decimal or other base system.
23
22
21
20
2-1
2-2
2-3
=8 =4 =2 =1 . =0.5 =0.25 =0.125
Most Significant Digit Binary point Least Significant Digit
Binary Counting
The Binary counting sequence is shown in the table:
23
22
21
20
Decimal
0 0 0 0 0
0 0 0 1 1
0 0 1 0 2
0 0 1 1 3
0 1 0 0 4
0 1 0 1 5
0 1 1 0 6
0 1 1 1 7
1 0 0 0 8

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1 0 0 1 9
1 0 1 0 10
1 0 1 1 11
1 1 0 0 12
1 1 0 1 13
1 1 1 0 14
1 1 1 1 15
Representing Binary Quantities
In digital systems the information that is being processed is usually presented in binary form. Binary
quantities can be represented by any device that has only two operating states or possible conditions. E.g.. a
switch is only open or closed. We arbitrarily (as we define them) let an open switch represent binary 0 and a
closed switch represent binary 1. Thus we can represent any binary number by using series of switches.
Typical Voltage Assignment
Binary 1: Any voltage between 2V to 5V
Binary 0: Any voltage between 0V to 0.8V
Not used: Voltage between 0.8V to 2V in 5 Volt CMOS and TTL Logic, this may cause error in a digital
circuit. Today's digital circuits works at 1.8 volts, so this statement may not hold true for all logic circuits.
We can see another significant difference between digital and analog systems. In digital systems, the exact
voltage value is not important; eg, a voltage of 3.6V means the same as a voltage of 4.3V. In analog
systems, the exact voltage value is important.
The binary number system is the most important one in digital systems, but several others are also
important. The decimal system is important because it is universally used to represent quantities outside a
digital system. This means that there will be situations where decimal values have to be converted to binary
values before they are entered into the digital system.
In additional to binary and decimal, two other number systems find wide-spread applications in digital
systems. The octal (base-8) and hexadecimal (base-16) number systems are both used for the same
purpose- to provide an efficient means for representing large binary system.
Octal System
The octal number system has a base of eight, meaning that it has eight possible digits: 0,1,2,3,4,5,6,7.
83
82
81
80
8-1
8-2
8-3
=512 =64 =8 =1 . =1/8 =1/64 =1/512
Most Significant Digit Octal point Least Significant Digit
Octal to Decimal Conversion
 2378 = 2 x (82
) + 3 x (81
) + 7 x (80
) = 15910
 24.68 = 2 x (81
) + 4 x (80
) + 6 x (8-1
) = 20.7510
 11.18 = 1 x (81
) + 1 x (80
) + 1 x (8-1
) = 9.12510
 12.38 = 1 x (81
) + 2 x (80
) + 3 x (8-1
) = 10.37510
Hexadecimal System

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Presidency College
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The hexadecimal system uses base 16. Thus, it has 16 possible digit symbols. It uses the digits 0 through 9
plus the letters A, B, C, D, E, and F as the 16 digit symbols.
163
162
161
160
16-1
16-2
16-3
=4096 =256 =16 =1 . =1/16 =1/256 =1/4096
Most Significant Digit
Hexa Decimal
point
Least Significant
Digit
Hexadecimal to Decimal Conversion
 24.616 = 2 x (161
) + 4 x (160
) + 6 x (16-1
) = 36.37510
 11.116 = 1 x (161
) + 1 x (160
) + 1 x (16-1
) = 17.062510
 12.316 = 1 x (161
) + 2 x (160
) + 3 x (16-1
) = 18.187510
Binary Codes
Binary codes are codes which are represented in binary system with modification from the original ones.
Below we will be seeing the following:
 Weighted Binary Systems
 Non Weighted Codes
Weighted Binary Systems
Weighted binary codes are those which obey the positional weighting principles, each position of the
number represents a specific weight. The binary counting sequence is an example.
Decimal 8421 2421 5211 Excess-3
0 0000 0000 0000 0011
1 0001 0001 0001 0100
2 0010 0010 0011 0101
3 0011 0011 0101 0110
4 0100 0100 0111 0111
5 0101 1011 1000 1000
6 0110 1100 1010 1001
7 0111 1101 1100 1010
8 1000 1110 1110 1011
9 1001 1111 1111 1100
8421 Code/BCD Code
The BCD (Binary Coded Decimal) is a straight assignment of the binary equivalent. It is possible to assign
weights to the binary bits according to their positions. The weights in the BCD code are 8,4,2,1.
Example: The bit assignment 1001, can be seen by its weights to represent the decimal 9 because:
1x8+0x4+0x2+1x1 = 9
2421 Code
This is a weighted code, its weights are 2, 4, 2 and 1. A decimal number is represented in 4-bit form and
the total four bits weight is 2 + 4 + 2 + 1 = 9. Hence the 2421 code represents the decimal numbers from 0

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to 9.
5211 Code
This is a weighted code, its weights are 5, 2, 1 and 1. A decimal number is represented in 4-bit form and
the total four bits weight is 5 + 2 + 1 + 1 = 9. Hence the 5211 code represents the decimal numbers from 0
to 9.
Reflective Code
A code is said to be reflective when code for 9 is complement for the code for 0, and so is for 8 and 1
codes, 7 and 2, 6 and 3, 5 and 4. Codes 2421, 5211, and excess-3 are reflective, whereas the 8421 code
is not.
Sequential Codes
A code is said to be sequential when two subsequent codes, seen as numbers in binary representation,
differ by one. This greatly aids mathematical manipulation of data. The 8421 and Excess-3 codes are
sequential, whereas the 2421 and 5211 codes are not.
Non Weighted Codes
Non weighted codes are codes that are not positionally weighted. That is, each position within the binary
number is not assigned a fixed value.
Excess-3 Code
Excess-3 is a non weighted code used to express decimal numbers. The code derives its name from the
fact that each binary code is the corresponding 8421 code plus 0011(3).
Example: 1000 of 8421 = 1011 in Excess-3
Gray Code
The gray code belongs to a class of codes called minimum change codes, in which only one bit in the code
changes when moving from one code to the next. The Gray code is non-weighted code, as the position of
bit does not contain any weight. The gray code is a reflective digital code which has the special property
that any two subsequent numbers codes differ by only one bit. This is also called a unit-distance code. In
digital Gray code has got a special place.
Decimal Number Binary Code Gray Code
0 0000 0000
1 0001 0001
2 0010 0011
3 0011 0010
4 0100 0110
5 0101 0111
6 0110 0101
7 0111 0100
8 1000 1100
9 1001 1101
10 1010 1111
11 1011 1110
12 1100 1010
13 1101 1011
14 1110 1001
15 1111 1000
Binary to Gray Conversion
 Gray Code MSB is binary code MSB.

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 Gray Code MSB-1 is the XOR of binary code MSB and MSB-1.
 MSB-2 bit of gray code is XOR of MSB-1 and MSB-2 bit of binary code.
 MSB-N bit of gray code is XOR of MSB-N-1 and MSB-N bit of binary code.
Error Detecting and Correction Codes
For reliable transmission and storage of digital data, error detection and correction is required. Below are a
few examples of codes which permit error detection and error correction after detection.
Error Detecting Codes
When data is transmitted from one point to another, like in wireless transmission, or it is just stored, like in
hard disks and memories, there are chances that data may get corrupted. To detect these data errors, we
use special codes, which are error detection codes.
Parity
In parity codes, every data byte, or nibble (according to how user wants to use it) is checked if they have
even number of ones or even number of zeros. Based on this information an additional bit is appended to
the original data. Thus if we consider 8-bit data, adding the parity bit will make it 9 bit long.
At the receiver side, once again parity is calculated and matched with the received parity (bit 9), and if they
match, data is ok, otherwise data is corrupt.
There are two types of parity:
 Even parity: Checks if there is an even number of ones; if so, parity bit is zero. When the number
of ones is odd then parity bit is set to 1.
 Odd Parity: Checks if there is an odd number of ones; if so, parity bit is zero. When number of
ones is even then parity bit is set to 1.
Check Sums
The parity method is calculated over byte, word or double word. But when errors need to be checked over
128 bytes or more (basically blocks of data), then calculating parity is not the right way. So we have
checksum, which allows to check for errors on block of data. There are many variations of checksum.
 Adding all bytes
 CRC
 Fletcher's checksum
 Adler-32
The simplest form of checksum, which simply adds up the asserted bits in the data, cannot detect a
number of types of errors. In particular, such a checksum is not changed by:
 Reordering of the bytes in the message
 Inserting or deleting zero-valued bytes
 Multiple errors which sum to zero
Example of Checksum : Given 4 bytes of data (can be done with any number of bytes): 25h, 62h, 3Fh, 52h
 Adding all bytes together gives 118h.
 Drop the Carry Nibble to give you 18h.
 Get the two's complement of the 18h to get E8h. This is the checksum byte.
To Test the Checksum byte simply add it to the original group of bytes. This should give you 200h.
Drop the carry nibble again giving 00h. Since it is 00h this means the checksum means the bytes were
probably not changed.
Error-Correcting Codes
Error-correcting codes not only detect errors, but also correct them. This is used normally in Satellite
communication, where turn-around delay is very high as is the probability of data getting corrupt.

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ECC (Error correcting codes) are used also in memories, networking, Hard disk, CDROM, DVD etc.
Normally in networking chips (ASIC), we have 2 Error detection bits and 1 Error correction bit.
Hamming Code
Hamming code adds a minimum number of bits to the data transmitted in a noisy channel, to be able to
correct every possible one-bit error. It can detect (not correct) two-bits errors and cannot distinguish
between 1-bit and 2-bits inconsistencies. It can't - in general - detect 3(or more)-bits errors.
The idea is that the failed bit position in an n-bit string (which we'll call X) can be represented in binary with
log2(n) bits, hence we'll try to get it adding just log2(n) bits.
First, we set m = n + log2(n) to the encoded string length and we number each bit position starting from 1
through m. Then we place these additional bits at power-of-two positions, that is 1, 2, 4, 8..., while
remaining ones (3, 5, 6, 7...) hold the bit string in the original order.
Now we set each added bit to the parity of a group of bits. We group bits this way: we form a group for
every parity bit, where the following relation holds:
position(bit) AND position(parity) = position(parity)
(Note that: AND is the bit-wise boolean AND; parity bits are included in the groups; each bit can belong to
one or more groups.)
So bit 1 groups bits 1, 3, 5, 7... while bit 2 groups bits 2, 3, 6, 7, 10... , bit 4 groups bits 4, 5, 6, 7, 12, 13...
and so on.
Thus, by definition, X (the failed bit position defined above) is the sum of the incorrect parity bits positions
(0 for no errors).
To understand why it is so, let's call Xn the nth
bit of X in binary representation. Now consider that each
parity bit is tied to a bit of X: parity1 -> X1, parity2 -> X2, parity4 -> X3, parity8 -> X4 and so on - for
programmers: they are the respective AND masks -. By construction, the failed bit makes fail only the parity
bits which correspond to the 1s in X, so each bit of X is 1 if the corresponding parity is wrong and 0 if it is
correct.
Note that the longer the string, the higher the throughput n/m and the lower the probability that no more
than one bit fails. So the string to be sent should be broken into blocks whose length depends on the
transmission channel quality (the cleaner the channel, the bigger the block). Also, unless it's guaranteed
that at most one bit per block fails, a checksum or some other form of data integrity check should be
added.
Alphanumeric Codes
The binary codes that can be used to represent all the letters of the alphabet, numbers and mathematical
symbols, punctuation marks, are known as alphanumeric codes or character codes. These codes enable
us to interface the input-output devices like the keyboard, printers, video displays with the computer.
ASCII Code
ASCII stands for American Standard Code for Information Interchange. It has become a world standard
alphanumeric code for microcomputers and computers. It is a 7-bit code representing 27
= 128 different
characters. These characters represent 26 upper case letters (A to Z), 26 lowercase letters (a to z), 10
numbers (0 to 9), 33 special characters and symbols and 33 control characters.
The 7-bit code is divided into two portions, The leftmost 3 bits portion is called zone bits and the 4-bit
portion on the right is called numeric bits.
An 8-bit version of ASCII code is known as USACC-II 8 or ASCII-8. The 8-bit version can represent a
maximum of 256 characters.
EBCDIC Code
EBCDIC stands for Extended Binary Coded Decimal Interchange. It is mainly used with large computer
systems like mainframes. EBCDIC is an 8-bit code and thus accomodates up to 256 characters. An
EBCDIC code is divided into two portions: 4 zone bits (on the left) and 4 numeric bits (on the right).
Floating Point Numbers

Prof. K. Adisesha
Presidency College
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A real number or floating point number is a number which has both an integer and a fractional part.
Examples for real real decimal numbers are 123.45, 0.1234, -0.12345, etc. Examples for real binary
numbers are 1100.1100, 0.1001, -1.001, etc. In general, floating point numbers are expressed in
exponential notation.
For example the decimal number
 30000.0 can be written as 3 x 104
.
 312.45 can be written as 3.1245 x 102
.
Similarly, the binary number 1010.001 can be written as 1.010001 x 103
.
The general form of a number N can be expressed as
N = ± m x b±e
.
Where m is mantissa, b is the base of number system and e is the exponent. A floating point number is
represented by two parts. The number first part, called mantissa, is a signed fixed point number and the
second part, called exponent, specifies the decimal or binary position.
Binary Representation of Floating Point Numbers
A floating point binary number is also represented as in the case of decimal numbers. It means that
mantissa and exponent are expressed using signed magnitude notation in which one bit is reserved for
sign bit.
Consider a 16-bit word used to store the floating point numbers; assume that 9 bits are reserved for
mantissa and 7 bits for exponent and also assume that the mantissa part is represented in fraction system.
This implies the assumed binary point is at the mantissa sign bit immediate right.
Example
A binary number 1101.01 is represented as
Mantissa = 110101 = (1101.01)2 = 0.110101 X 24
Exponent = (4)10
Expanding mantissa to 8 bits we get 11010100
Binary representation of exponent (4)10 = 000100
The required representation is

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