Spring 2015
Week 9 Module 48
Digital Circuits and
Systems
Number Representation
Shankar Balachandran*
Associate Professor, CSE Department
Indian Institute of Technology Madras
*Currently a Visiting Professor at IIT Bombay
Positional Number System – a review


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Value of a number is determined by a weighted sum of its digits
Weighting is implicit and is determined for each digit by the position
of the digit in the number
Let the number have n digits to the left of the radix point and m digits
to the right of the radix point. Di is the ith digit, R is the radix or base
of the number,V is the value of the number.
n 1
V   Di R i
n digits
m digits
i  m
Example : 12.3410  4  10 2  3  10 1  2  100  1  101
Other common radices – binary (2), octal (8), hexadecimal/hex (16):
0-9,a,b,c,d,e,f.
Number Representation
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Representation of Signed Numbers

How to represent positive and negative integers using 1’s and 0’s
of the binary notation ?

In mathematics, there are infinitely many positive and negative
integers. However, in a practical hardware system only a fixed
number of integers can be represented.

In most modern computer systems, numbers are represented in
32 bits. If an arithmetic operation results in a number outside the
range, an overflow occurs.

There are 4 popular schemes for representing signed numbers.
1. Sign Magnitude
2. Ones Complement
3. Twos Complement
4. Excess-B or Biased Representation
Number Representation
3
Sign Magnitude Representation

Use MSB as a sign bit as follows,


MSB = 0 for positive integers
MSB = 1 for negative integers
S
Magnitude (n-1 bits)
n bits

Other bits encode magnitude of integer.
 2 n 1  X  2 n 1
Range of representation with n bits is:
The range is symmetric around 0.
There are two representations for 0, i.e., 0000 and 1000.

Examples:



Assume a 4-bit representation,
5
-5
3
-7
=
=
=
=
Convert the following sign magnitude
numbers to a 6-bit representation.
0101
1101
0011
1111
0101 = 000101
1010 = 100010
Number Representation
4
Ones Complement Representation

Positive integers are represented “as is”. Negative integers are
represented by performing bit-wise complement of the integer. E.g.,


5 = 0101
-5 = 1010

All positive integers have MSB=0; negative integers have MSB=1
Range of representation with n bits is:
 2 n 1  X  2 n 1
The range is symmetric around 0.
There are two representations for 0, i.e., 0000 and 1111.

Examples:



Assume a 5-bit representation,
01110 = 14
10111 = -8
15 = 01111
16 = not with 5 bits
-16 = not with 5 bits
Convert the following signed
numbers to a 8-bit representation.
Number Representation
01011 = 00001011
10111 = 11110111
5
Twos Complement Representation

Positive integers are represented “as is”. Negative integers are
formed by subtracting magnitude of negative integer from 0;
borrowing from imaginary position to the left of MSB.
(Shortcut: bit-wise complement of the integer and add 1).

Example:
5 = 0101
-5 = 0000 – 0101 = 1011

All positive integers have MSB=0; negative integers have MSB=1
Range of representation with n bits is:
 2 n 1  X  2 n 1
The range is asymmetric around 0.
There is only one representation for 0, i.e., 0000.

Examples:



Assume a 5-bit representation,
01110 = 14
10111 = -9
15 = 01111
16 = not with 5 bits
-16 = 10000
Convert the following signed
numbers to a 8-bit representation.
Number Representation
01011 = 00001011
10111 = 11110111
6
Excess-B (or Biased) Representation
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Integer representations are biased by B.
A signed integer X is represented by the binary number X+B
Range of representation with n bits is:  B  X  2 n  B
Usually, B=2n-1  -2n-1 ≤ X < 2n-1
There is only one representation for 0, i.e., binary representation for
bias B.
Examples:
Assume a 5-bit representation and B = 24,
10001
01100
00000
0
15
= 17 – 24 = 1
= 12 – 24 = -4
= 0 – 24 = -16
= 0 + 24 = 10000
= 15 + 24 = 11111
Number Representation
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Important Points to Remember
Given a bit vector B (bn-1 to b0), it can be
“interpreted” in many ways
 The interpretation gives the value to the
vector
 Example: bit vector 1111

 Interpreted
as unsigned it is 15
 Interpreted as signed it is -7
 Interpreted as 2’s complement it is -1

Mixing interpretations can be disastrous
Number Representation
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Important Points to Remember
Arithmetic operations are basic operations
for computers
 Number Representation should take

 Numbers
in the same interpretation
 And produce results that are consistent in the
same interpretation

Circuits are usually designed so that the
interpretations are consistent.
Number Representation
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Number Circle
1111
1110
0000 0001
0010
1101
0011
1100
0100
0101
1011
0110
1010
1001 1000 0111
Number Representation
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Interpretation of 4-bit Binary Numbers
Number Representation
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End of Week 9: Module 48
Thank You
Number Representation
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