Systems Number Representation Shankar Balachandran* Associate - - PowerPoint PPT Presentation

Shankar Balachandran* Associate Professor, CSE Department Indian Institute of Technology Madras

*Currently a Visiting Professor at IIT Bombay

Digital Circuits and Systems

Spring 2015 Week 9 Module 48

Number Representation

Number Representation 2

Positional Number System – a review

 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

• f 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

• f the number,V is the value of the number.

Other common radices – binary (2), octal (8), hexadecimal/hex (16): 0-9,a,b,c,d,e,f.



  



1 n m i i iR

D V

1 1 2 10

10 1 10 2 10 3 10 4 34 12       

  

. : Example

n digits m digits

Number Representation 3

Representation of Signed Numbers



How to represent positive and negative integers using 1’s and 0’s

• f 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 4

Sign Magnitude Representation

 Use MSB as a sign bit as follows,

 MSB = 0 for positive integers  MSB = 1 for negative integers

 Other bits encode magnitude of integer.  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:

1 1

2 2

 

  

n n

X

Assume a 4-bit representation, 5 = 0101

• 5 = 1101

3 = = 1111 0011

• 7

Convert the following sign magnitude numbers to a 6-bit representation. 0101 = 1010 = 000101 100010 S Magnitude (n-1 bits) n bits

Number Representation 5

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:  The range is symmetric around 0.  There are two representations for 0, i.e., 0000 and 1111.



Examples:

1 1

2 2

 

  

n n

X

Assume a 5-bit representation, 01110 = 10111 = 15 = 16 =

• 16 =

Convert the following signed numbers to a 8-bit representation. 01011 = 10111 = 00001011 11110111 14

• 8

01111 not with 5 bits not with 5 bits

Number Representation 6

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:  The range is asymmetric around 0.  There is only one representation for 0, i.e., 0000.



Examples:

1 1

2 2

 

  

n n

X

Assume a 5-bit representation, 01110 = 10111 = 15 = 16 =

• 16 =

Convert the following signed numbers to a 8-bit representation. 01011 = 10111 = 00001011 11110111 14

• 9

01111 not with 5 bits 10000

Number Representation 7

Excess-B (or Biased) Representation

 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:  Usually, B=2n-1  -2n-1 ≤ X < 2n-1  There is only one representation for 0, i.e., binary representation for

bias B.



Examples: B X B

n 

   2 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 8

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 9

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 Circle

Number Representation 10

0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1111 1011 1100 1101 1110

Interpretation of 4-bit Binary Numbers

Number Representation 11

End of Week 9: Module 48

Thank You

Number Representation 12