Representation of Integers

Computer and Network
Technology (CNT)
Chapter: Fundamentals
Lesson: Data Representation in Computers
Representation of Integers
Lecturer: Susantha Herath PGD in IT (MBCS), PGD in Marketing (Uni. Of Kln)
Lecture 02 bcsonlinelectures.com

Few basic things to know
 Most Significant Bit (MSB) also called as Most Significant Digit (MSD). This is
indeed the first digit (from left) in a binary number. More accurately the digit
with the highest radix (base) value.
 Least Significant Bit (LSB) or Least Significant Digit (LSD). This is the digit of a
binary number with the least radix value, probably the last digit.
 In binary number one digit is called as a bit.
 Computer memory is a place that store data and it has a fixed length of bits.
Generally 8bit, 16bit, 32bit and 64bit in length.
1001100100111
212 (MSD) 20 (LSD)

Data Representation
 Computer uses a fixed number of bits to represent a single piece of data. Generally
computers use 8bit, 16bit, 32bit and 64bit in size of binary numbers to represent
different type and size of data.
 N-number of bit memory (location) can store 2n of different types of values.
 For an example 8bit memory can store 256 of different data values.
 3bit memory can store 8 distinct values (patterns)
000, 001, 010, 011, 100, 101, 110, or 111
 When we interpret these values, we need to know exactly what represent with
these values.
 If we know that these represent decimal numbers, then we know it is from 0-7 then
it is a different value.
 So this interpretation of a binary pattern called data representation or encoding.

Integer Representation
 Integers are whole numbers or fixed point numbers. The radix point is fixed after
the LSB.
Example:
377410 is a fixed point number (integer)
2949.9410 is a floating point number (real number)
54533. 4573.55
LSB Radix Point Fixed
Radix Point Moved from LSB

 In a computer integers and floating point numbers are treated differently and
processed separately. Floating point numbers are processed in a separate
processor called floating point processor.
 Computers use fixed number of bits to represent integers. Commonly use bit
lengths are 8bit, 16bit, 32bit and 64bit.
 There are two types of integers
 Unsigned integers: can represent zero and positive numbers only
 Signed integers: can represent zero, positive and negative numbers.
 There are three schemas (ways) to represent signed integers.
 Sign Magnitude representation
 1’s complement representation
 2’s complement representation

 We learned about two different types of Integers;
 Unsigned Integers – that can represent only zero and positive numbers.
The value of an unsigned integer interpreted as the “magnitude of
underlying binary pattern or the normal way we convert binary to decimal.
Example: 8bit binary number 0100 0010B (B = Binary)
0100 0010B = 2x21 + 2x26 = 4+64 = 68D (D = Decimal)
8bit = 0000 0000
16bit = 0000 0000 0000 0000
32bit = 0000 0000 0000 0000 0000 0000 0000 0000
64bit = 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000
All above numbers represent zero in different bit systems. The bit length of each system is different.

n-bit Unsigned Integers
 An n-bit pattern can represent 2n distinct integers. This range from 0 to (2n)-1
n-bit Minimum 2n – 1 Maximum
8bit 0 28 – 1 255
16bit 0 216 – 1 65,535
32bit 0 232 – 1 4,294,967,295
64bit 0 264 – 1 18,446,744,073,709,551,615
When you are programming you need to select the correct and appropriate
bit size to store integers. If you want to store small numbers less than 255,
you can select 8bit number, selecting 32bit length to store small numbers is a
waste of resources.

Signed Integers
 Signed integers can represent zero, positive and negative numbers.
 The MSB (Most Significant Bit) – Normally the first bit from left, represent the
sign of the integer. Zero (0) to represent positive (+) sign and One (1) to
represent negative (-) sign.
 The magnitude of the integer represent differently in different schemas.
 There are three schemas to represent signed integers.
1. Sign Magnitude Representation
2. 1’s Complement Representation
3. 2’s Complement Representation
Example of Signed Integers:
+2344 is a positive integer -23434 is a negative integer

n-bit Sign Magnitude Representation
 Most Significant Bit (MSB) is the sign bit. 0 representing positive and 1
representing negative signs.
 The remaining (n-1) bits represent the magnitude / absolute value of the
integer interpreted according to its binary pattern.
Examples:
n=8 (8bit number) and binary number is 01000001
Sign bit (MSB) is 0  this is a positive number
Absolute value 1000001  65
01000001B = +65D
B = Binary, D = Decimal

Examples:
Sign bit (MSB) is 0  this is a positive number
00000000B = +0D
Sign bit (MSB) is 1  this is a negative number
10000000B = -0D

Problems (Drawbacks / Limitations) of Sign Magnitude Representation:
 There are two
representation for the
number zero
 Even the number zero
need to have a sign, since
it is not necessary nor
accurate and may leads to
confusions.
 Positive and negative
numbers need to process
separately.

n-bit 1’s Complement Representation
In this representation, the MSB is the sign bit 0 represent positive and 1
represent negative numbers. The remaining n-1 bits represent the magnitude
(value) of the integer.
 In positive integers, the value of the integer is the binary pattern (value) of
the n-1 bit magnitude. Same as Sign Magnitude Representation.
 For negative integers, the absolute value is the n-1 binary pattern value of
the inverse magnitude (complement).
Example: 1 000 0001B (sign bit = 1, negative integer)
Complement of 000 0001 is 111 1110 = 126D
1 000 0001B = -126D

In the 1’s Complement
Representation, there are two
different representation for zero;
0000 0000B and 1111 1111B
hence +0 and -0
Since the processing method of
positive and negative integers
are different, computer need to
process the positive and
negative numbers separately.

In 2’s Complement Representation also the MSB is the sign bit 0 represent
positive and 1 represent negative integers.
For positive integers, the value of the integer equal to the magnitude of n-1
binary pattern
For negative integers, the absolute value of the integer equal to the magnitude
of the complement of n-1 binary pattern plus one.
Example: 1 000 0001B (MSB = 1 is a negative number)
Complement of 000 0001 = 111 1110
Let’s add 1 to the complement = 111 1111 = 127
1 000 0001B = -127D

Advantages of 2’s Complement
Representation
 There is only one
representation for the number 0
 In binary addition and
subtraction, both positive and
negative numbers can treated
together (process together)
 Subtraction can use the addition
logic
 Computers use 2’s Complement
Representation for processing
integers.

Addition, Subtraction, Overflow and Underflow
Addition of positive integers
Example: n=8, 65D + 5D = 70D
65D  0100 0001B
5D  0000 0101B
65D + 5D  0100 0110B OK 70D within the range of -128 to +127
Subtraction is the addition of positive and negative integers
Example: n=8, 65D – 5D = 60D  65D + (-5D)
65D  0100 0001B
-5D  1111 1011B
65D + (-5D)  0011 1100B  60D OK within the range of -128 to +127

Addition, Subtraction, Overflow and Underflow
Overflow and underflow happens when the result is not within the acceptable range
Example of Overflow: n=8, 127D + D = 129D (not within -128 and +127, exceed
maximum value)
127D  0111 1111B
2D  0000 0010B
127D + 2D  1000 0001B  -127D wrong answer
Example of underflow: n=8, -125D – 5D = -130D (not within -128 and +127, below
the minimum value)
-125D  1000 0011B
-5D  1111 1011B
-125D + (-5D)  0111 1110B  +126D wrong answer

Range of n-bit 2’s complement signed integers
An n-bit 2’s complement integer can represent integers from -2(n-1) to +2(n-1)-1. This
can represent all the integers within the range without missing any intergers.
8bit from -128 to +127
16bit from -32,768 to +32,767
32bit from -2,147,483,648 to + 2,147,483,647
64bit from -9,223,372,036,854,775,888 to + 9,223,372,036,854,775,887

decoding 2’s complement numbers
• Check the sign bit first. If sign bit = 0, the number is positive and the absolute
value if the binary value of remaining n-1 integers.
• If the sign bit = 1, the integer is negative. To get the absolute value;
• Invert remaining n-1 bits and add 1 or
• Scan from left until the first occurrence of 1. Flip all values from that point to
the left. Binary value of this pattern gives the absolute value of the negative
integer.
Example:
N=8 bit pattern = 1 100 0100 B
Negative number
Scan from right, until the first occurrence of 1, flip all values from that point to left
011 1100 B = 60 D
Represented negative value is -60 D

Big Endian vs. Little Endian
Modern computers store one byte of memory in single memory location / address.
1 byte = 8 bits. Therefore in 32bit integer stores in 4 memory locations.
Term “Endian” means the order of storing bytes in memory. In Big Endian standard
– the most significant byte stores first in the lowest memory location. In Little
Endian standard, the least significant byte (LSB) stores in the lowest memory
location.

Floating Point Number
Representation
A floating point number (also called as a real number) is a number with a decimal
point. A floating point number can represent a very large positive or negative
number as well as a very small negative or positive number. A real number has a
range of -1.23x1088 to 1.25x1088 including zero.
Scientifically a floating point number represent as a number with a fraction and
exponent of a certain radix. F x rE

Representation
Same floating point number can represent in different ways. Consider 5.456D
We can represent this number in following ways:
54.532 x 100
5.4532 x 101
0.54532 x 102 and so on….
Since there are many ways to represent the same floating point number, we need
to define a standard, so every floating point number should be in this standard. To
make it standard, we can normalized the fraction part. In normalized form, there is
only single none zero digit represent before the radix point. In above example
5.4532 x 101 is the normalized form of 54.532 number.

Representation
In computers, there is an issue of “Loss of Precision” in storing floating point
numbers. An n-bit system can only stores 2n distinct finite numbers. Even within a
small range 0.00 – 0.01, there are infinite number of decimal numbers. But the
number of bits in a computer system is limited, only a the approximate number
used, resulting in loss of accuracy.
Processing of a real number (floating number arithmetic) is very much less efficient
and slower than processing of integers. In computers, real numbers uses fraction
(F) and exponent (E) with radix (r) of 2. Both F and E can be negative or positive.
In modern computing uses IEEE 754 standard to represent floating point numbers.
There are two representation schemes
32 bit single precision and
64 bit double precision

IEEE-754 32-bit Single-Precision
Floating-Point Numbers
In 32-bit single precision floating point number representation, decimal number
represent in following structure:
• The most significant bit is the sign bit. Zero for positive numbers and One for
negative numbers.
• Next following 8bit represent the exponent (E)
• Remaining 23bit represent the fraction (F)
You may wonder why radix (r) is not represent here. Because radix is always 2 for
binary numbers (in computers).

IEEE-754 64-bit Double-Precision
Floating-Point Numbers
64-bit double precision floating point number representation is similar to 32-bit
single precision floating point number representation. Only difference is assignment
of bit range for exponent and fraction.
• The most significant bit is the sign bit. Zero for positive numbers and One for
negative numbers.
• Next following 11-bit represent the exponent (E)
• Remaining 52-bit represent the fraction (F)

Character Encoding
In computer memory, characters are represented (encode) using character
schemes called as “character set”, “charset”, “character map” or “code page”.
There are several standard schemes such as ASCII, EBCDIC, UTF, etc.
7-bit ASCII Code (aka US-ASCII, ISO/IEC 646, ITU-T T.50)
• ASCII (American Standard Code for Information Interchange) is one of the earlier
character coding schemes.
• ASCII is originally a 7-bit code. It has been extended to 8-bit to better utilize the 8-bit
computer memory organization. (The 8th-bit was originally used for parity check in
the early computers). This is called ASCII-8 Code.
EBCDIC Code
This is stands for “Extended Binary Code Decimal Interchange Code”. This is a 8-bit
code scheme without parity. With this up to 256 characters can be coded.

Character Encoding
Unicode (aka ISO/IEC 10646 Universal Character Set)
• Before Unicode, no single character encoding scheme could represent characters in
all languages.
• Unicode aims to provide a standard character encoding scheme, which is universal,
efficient, uniform and unambiguous. Unicode standard is maintained by a non-profit
organization called the Unicode Consortium (@ www.unicode.org). Unicode is an
ISO/IEC standard 10646.
• Unicode is backward compatible with the 7-bit US-ASCII and 8-bit Latin-1 (ISO-8859-
1). That is, the first 128 characters are the same as US-ASCII; and the first 256
characters are the same as Latin-1.
• Unicode originally uses 16 bits , which can represent up to 65,536 characters.

Summary
Inside the computer memory – integer 1, floating point 1.0, character 1 and string 1
are totally different. To write good and high performance programs, you need to
know these differences and use correct data type to store data in memory and later
on in databases as well.
• In 8-bit signed integer, integer number 1 is represented as 00000001B.
• In 8-bit unsigned integer, integer number 1 is represented as 00000001B.
• In 16-bit signed integer, integer number 1 is represented as 00000000 00000001B.
• In 32-bit signed integer, integer number 1 is represented
as 00000000 00000000 00000000 00000001B.
• In 32-bit floating-point representation, number 1.0 is represented as 0 01111111
0000000 00000000 00000000B, i.e., S=0, E=127, F=0.
• In 64-bit floating-point representation, number 1.0 is represented as 0 01111111111 0000
00000000 00000000 00000000 00000000 00000000 00000000B,
i.e., S=0, E=1023, F=0.

End of Lesson
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Representation of Integers

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