Data representation
1. Unsigned Magnitude Representation
2. Signed Magnitude
3. 1’s Complement
4. 2’s Complement
1. Unsigned Number Representation
In this representation all the bits represent only the magnitude of the number without consideration to the sign of the number.
Example:
D7 
D6 
D5 
D4 
D3 
D2 
D1 
D0 
Value 

Minimum Value 
0 
0 
0 
0 
0 
0 
0 
0 
0 
Maximum Value 
1 
1 
1 
1 
1 
1 
1 
1 
2^{N}1 =255 
The range of the numbers that can be represented in this representation is 0 to 2^{N}1
For 1 4bit number the range will be = 0 to 2^{4}1 = 0 to 15
For an 8bit number the range will be = 0 to 2^{8}1 = 0 to 255
· Note: We cannot represent signed number in this representation.
Signed and Magnitude number Representation:
We have only +ve or the –ve numbers. Two states can be represented by only a single bit. For convenience and also as standard we take the MSB as ‘0’ to mean +ve and ‘1’ to mean –ve.
In this representation we can represent the signed numbers i.e. both +ve and –ve numbers can be written. Example
D7
(Sign Bit 
D6 
D5 
D4 
D3 
D2 
D1 
D0 
Value 

Min +ve Value 
0 
0 
0 
0 
0 
0 
0 
0 
0 
Max +ve Value 
0 
1 
1 
1 
1 
1 
1 
1 
+2^{N1 }– 1 = + 127 
Max –ve Value 
1 
0 
0 
0 
0 
0 
0 
0 
0 
Min –Ve value 
1 
1 
1 
1 
1 
1 
1 
1 
– 2^{N1 }– 1 =127 
The range of numbers in signed and magnitude representation is
+/ 0 to +/ 2^{N1 }– 1
For 8bit number N=8, therefore range of numbers will be: +/ 0 to +/ 127
1’s and 2’s Complement Representation:
These representations are well suited for use in computer systems. They provide very powerful representation of signed numbers and their arithmetic operations. It is worth noting that both addition and subtraction are performed using addition only.
1’s Complement Representation:
This representation uses an additional bit generally the MSB, as the sign bit. All the other bits represent the magnitude of the number. Positive numbers are represented by appending a ‘0’ at the MSB position, negative numbers are represented by taking the bitwise complement of the number.
There are two ways of representing the –ve numbers
Bitwise complement 
By subtraction Method 
Example:
+25 in 8bit = 00011001 To write 25, complement each bit of +25 So, 25 in 1’s complement = 11100110 
Example:
+25 in 8bit = 00011001 To write 25, Subtract it from all 1’s 11111111 00011001 —————————– 25 =11100110 So, 25 in 1’s complement = 11100110 
Example
Positive Decimal numbers

1’s Complement representation of + numbers 5bits (MSB sign bit and rest 4 bits to represent 0 to 9 
Negative Decimal numbers

1’s Complement Representation of –ve numbers 5bits (MSB sign bit and rest 4 bits to represent 0 to 9 
0 
00000 
0 
11111 
1 
00001 
1 
11110 
2 
00010 
2 
11101 
3 
00011 
3 
11100 
4 
00100 
4 
11011 
5 
00101 
5 
11010 
6 
00110 
6 
11001 
7 
00111 
7 
11000 
8 
01000 
8 
10111 
9 
01001 
9 
10110 
Range of Numbers in 1’s Complement
In General the range of representable numbers is: +/ 0 to +/ 2^{N1} 1
· There are two zeros +0 and 0 in this representation. This being disadvantage.
2’s Complement Representation:
This representation also uses an additional bit, generally the MSB, as the sign bit. All the other bits represent the magnitude of the number. Positive numbers are represented by appending a ‘0’ at the MSB position, negative numbers are represented by taking the bitwise complement of the number and adding 1 to it.
There are two ways of representing the –ve numbers
2’s Complement Representation of –ve Numbers  
Bitwise complement and adding ‘1’  By subtraction Method and adding ‘1’ 
Example:
+25 in 8bit = 00011001 To write 25, complement each bit of +25 1’s 1’s complement of 25 = 11100110 Add ‘1’ +1 —————————————————– So, 25 in 2’s Complement =1110111 
Example:
+25 in 8bit = 00011001 To write 25, Subtract it from all 1’s 11111111 00011001 —————————– 25 =11100110 , 1’s complement Add ‘1’ +1 ——————————— = 11100111
So, 25 in 2’s Complement =1110111 
Example
Positive Decimal numbers

1’s Complement representation of + numbers 5bits (MSB sign bit and rest 4 bits to represent 0 to 9 
Negative Decimal numbers

1’s Complement Representation of –ve numbers 5bits (MSB sign bit and rest 4 bits to represent 0 to 9 
0 
00000 
0 
00000 
1 
00001 
1 
11111 
2 
00010 
2 
11110 
3 
00011 
3 
11101 
4 
00100 
4 
11100 
5 
00101 
5 
11011 
6 
00110 
6 
11010 
7 
00111 
7 
11001 
8 
01000 
8 
11000 
9 
01001 
9 
10111 
Range of Numbers in 2’s Complement
In General the range of representable numbers is: 0 to + 2^{N1} 1 and 1 to 2^{N1}
Example: For an 8bit number the range will be
Positive Numbers 0 to 2^{81} 1 = 0 to 127
Negative Numbers 1 to 2^{N1} = 1 to 128
Representation of Real Numbers:
Real numbers can be represented using the following two representations.
· Fixed point Notation
· Floating point notation
Fixed Point Notation:
In the fixed point notation the decimal point is fixed so that there are fixed number of digits after the decimal point.
It has the following parts
Integer Part
Decimal Point
Fraction Part
The general format of in this notation is:
IIIII…….II . FFFF (Decimal point fixed at four places from right
Where I is Integer part and F means the fraction part
Example
Represent +45.525 in the fixed point notation.
Solution: Assuming the decimal point to fixed at four decimal places.
Range in Signed Fixed Point Notation
(2^{N1} 1) to (2^{N1} 1) for Nbits
Example:
Represent 50.675 in fixed point with 1 sign bit, 16 bit integer and 15 bit fraction part.
Solution:
Convert to decimal:
50 = 0000000000110010
.675 = 101011001100110
So 50.675 = 1 0000000000110010 . 101011001100110
Floating Point Notation.
This notation is the scientific notation. It does not reserve specific number of bits to the integer or the fraction part of the number. Instead the decimal point is floating.
It has the following Parts.
Sign Bit
Exponent
Mantissa
The number is represented in the following format.
(1)^{s} (1+M)x2^{Ebias}.
Where S is sign bit of Mantissa
E is exponent value
Example
Represent 53.5 in floating point notation assuming 8 bit exponent, 1 sign bit, 23 bit mantissa
53 in binary = 110101
.5 = .10000
53.5 = 110101.1
= 1.101011 x 2^{5}.
IEEE Floating Point Number Representation
Half Precession ( 16 bits== 1 Sign bit, 5 bit exponent, 10 bit mantissa)
Single Precession (32 bits== 1 Sign bit, 8 bit exponent, 23 bit mantissa)
Double Precession (64 bits== 1 Sign bit, 11 bit exponent, 52 bit mantissa)
Quadruple Precession (128 bits== 1 Sign bit, 15 bit exponent, 112 bit mantissa)
Summary of Range of Numbers:
Representation System 
General Case 
Example for 8bit Number 
Unsigned Magnitude representation 
0 to 2^{N} 1 
0 to 255 
Signed and Magnitude Representation 
+/ 0 to +/ 2^{N1} 1 
+0 to +127
0 to 127 
1’s Complement Representation 
+/ 0 to +/ 2^{N1} 1 
+0 to +127
0 to 127 
2’s Complement Representation 
0 to + 2^{N1} 1
1 to – 2^{N1} 
0 to + 127
1 to 128 