1. Data Representation and Computer Arithmetic
Computer Systems
Data Representation and Computer Arithmetic

2. Data Representation Binary codes used to represent numbers
Conversion between number systems
Binary  decimal
Hexadecimal  decimal
Between arbitrary number systems
Representation of positive/negative numbers
Floating point representation

3. Bit, Byte and Word
Bit is the smallest quantity that can be handled by computer: 1 or 0.
Byte is a group of 8 bits.
Word is the basic unit of data that can be operated by computer: e.g., 16 bits.
Bit
Byte
Word
Some architectures have 8, 32, or 64-bit words

4. Content of Word Bit Pattern May represent many things
Actual meaning of a particular bit pattern is given by the programmer
Computer itself cannot determine the meaning of the word
Classic question: Can you tell the meaning of a word picked randomly from the memory?

5. Instruction: (op-code)
A single word defines an action that is to be performed by CPU
Numeric Quantity
Character:
A to Z, a to z, 0 to 9, *, -, +, etc.
ASCII Code:
- Representation of a character by 7 bits.
- The ASCII for W is , or 5716

6. ASCII Code

7. Pixel (Picture Element)
The smallest unit to construct a picture
1bit/pixel for black-and-white pictures, >1 bits/pixel for gray scale or color pictures, e.g. 24bits/pixel.
A letter is represented by a group of pixels on computer screen.
Pixel (1)

8. Positional Notation
Weight is associated with the location within a number.
The 9 in 95 has weight 10
A number N in base b is represented by

9. Example:

10. Number Bases Decimal (b=10): {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
Binary (b=2): {0, 1}
Octal (b=8): {0, 1, 2, 3, 4, 5, 6, 7}
Hexadecimal (b=16):
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}

11. How many binary bits are needed to represent an n-digits decimal?
The value of an m-bit binary is up to
The value of an n-digit decimal is up to so OR

12. Questions
How many bits are needed to represent an n-digits octal? How many bits are needed to represent an n-digits hexadecimal?

13. Conversion of Integer - Decimal to Binary
Divide a decimal successively by 2, record the remainder, until the result of the division is 0.
67/2 = 33 Rem.=1
33/2 = 16 Rem.=1
16/2 = 8 Rem.=0
8/2 = 4 Rem.=0
4/2 = 2 Rem.=0
2/2 = 1 Rem.=0
1/2 = 0 Rem.=1

14. Conversion of Integer - Decimal to Hexadecimal
Divide a decimal successively by 16, record the remainder, until the result is 0
432/16 = 27 Rem.=0
27/16 = 1 Rem.=B
1/ = 0 Rem.=1

15. Conversion of integer - Binary to Decimal
Conversion of integer - Hexadecimal to Decimal

16. Conversion of integer - Hexadecimal Binary 5 7
1011 B
0010
0111
Conversion of integer: - Octal Binary 2 7 1
010
111
101

17. Conversion of Fraction - Decimal to Binary
The fraction is repeatedly multiplied by 2, the integer part is stripped and recorded. =
Question: Convert 0.625? 0.110?
* 2 =
* 2 =
* 2 =
0.510 * 2 =
0.010 * 2 = end

18. Conversion of Fraction - Binary Hexadecimal=
= 0.AC816
0.7BC16 = =
Convert
Convert 0.6D5?

19. Binary Coded Decimal (BCD)
A decimal digit is coded into 4 bits  1 byte can store 2 digits
Example:
1942 is encoded as
(2 bytes)
Disadvantages
Complex arithmetic
Inefficient use of storage
Advantage:
Can represent “real” number
Decimal BCD

20. Binary Addition Binary Subtraction ? 0 0 1 1 0 1 1 1 + 0 1 0 1 0 1 1 0
Carry
Binary Subtraction ?
use 2’s complementary arithmetic (later … )

21. Signed Integer Representations
Sign-and-Magnitude
1’s Complement
2’s Complement

22. Sign-and-Magnitude Representation
Use the most significant bit (MSB) to indicate the sign of the number.
The MSB is 0 for positive, 1 for negative.
8 bits represent to 12710
Examples: for +4410
for -4410
Problem: = +0, = -0

23. 1’s Complement Just flip the bits!
The MSB is 0 for positive, 1 for negative.
8 bits represent to 12710
Examples: for +4410
for -4410
Problem: = +0, = -0

24. 2’s Complement The 2’s complement for N is
Example: Using 5 bits (n = 5), if N = 7, then 12 -7 5 1)
2’s complement = 1’s complement + 1
1’s complement
2’s complement = -7

25. 2’s Complement (con.)
To form the two’s complement of a number, simply invert the bits (1 to0, 0 to 1), and add 1  1’s complement + 1
1310 =
-1310 = =
2’s comp.
-34?

26. Properties of 2’s Complement
One unique 0
MSB = 0, positive number
MSB = 1, negative number
The range is
For 5 bits, the range is (10000) to (01111)
The 2’s complement of the complement of X is X itself (Prove: )

27. 2’s Complementary Arithmetic
Subtraction is performed using addition
A - B = A + (-B)
X = 910 = , Y = 610 =
-X = -910 = , -Y = -610 = X -Y
310 1)
- X
- Y
-1510 1)

28. Binary Adder/Subtractor

29. Arithmetic Overflow The overflow happens when
positive + positive  negative
negative + negative  positive
If are MSBs, then the overflow bit v is expressed as 12
+13 25
-12
-13
-25 1)

30. Arithmetic Overflow In practice, Case I: A and B are +
an-1 = __ bn-1 = __
cn = __ cn-1 = __
V = _____
Case II: A and B are –
Most significant stage of full adder

31. Fixed-point Arithmetic
 
 
The fraction point is added
>
What if we want to represent ?
We need at least m > 3.3*n = 3.3*20 = 66 bits !

32. Floating Point Numbers
Scientific Notation
a is mantissa, r is radix, e is exponent
is for binary

33. IEEE Floating Point Format
S: Sign bit, 1 bit
E: Exponent, 8 bits
B: Bias, 8 bits, 12710=
F: mantissa

34. IEEE Floating Point - Example
Normalization, e.g.,
Use sign and magnitude representation for signed mantissa.
Use biased exponent, e.g. in excess 127
If we have 8 bits for exponent, to represent -5, we add ( ) to -5, the result is 12210=

35. IEEE Floating Point - Example
Represent in 32 bits ? =
1) Normalization:
2) Negative mantissa, so S = 1
3) Exponent:
E = = =
4) Mantissa F = …000 …0
1.510 in IEEE format?

36. IEEE Floating Point Format – Special Cases
Zero: exponent and fraction all 0s
Denormalized: exponent all 0s,
fraction non-zero
for single precision:
Infinity: exponents all 1s, fraction all 0s
sign bit determines +infinity or -infinity
Not a Number (NaN): exponents all 1s,
A good reference is available at

37. Bit Patterns and Logical OperationsOR
NOT
Exclusive OR
They are bit-wise operations.

38. AND Operation OR Operation
x AND y is true, if and only if both bits x and y are true.
A =
B =
C=A AND B =
OR Operation
x OR y is true, if either bits x or y is true.
A =
B =
C=A OR B =

39. EOR (Exclusive OR) Operation
NOT Operation
A 1 becomes 0, and a 0 becomes 1.
A =
C = NOT A =
EOR (Exclusive OR) Operation
It is true, if and only if just one of inputs is true.
A =
B =
C = A EOR B =