1. Lecture 2 Data Representation in Computer Systems Lecture Duration: 2 Hours

2. Lecture Overview Introduction Positional Numbering System
Decimal to binary conversion
Signed integer representation
Floating-point representation

3. Some Notifications – A reminder (1/2)
Introduction
Some Notifications – A reminder (1/2)
Bit: The most basic unit of information in a digital computer (On/Off ; 0/1 state)
Byte: A set of 8bits
Word: two or more adjacent bytes that are manipulated collectively
Word size: The size of a word in bits depends on the computer organization (16, 32, 64 bits, …)
Nibbles (or nybbles): set of 4 bits – Usually a set of 8 bits is divided into two nibbles, a low order nibble and a high order nibble

4. Some notifications – A reminder (2/2)
Introduction
Some notifications – A reminder (2/2)
Example:
Most Significant bit
(MSB)
Least Significant bit
(LSB)
bit
bit
bit
bit
bit
bit
bit
bit
bit
bit
bit
bit
bit
bit
bit
bit
High Order nibble
Low Order nibble
High Order nibble
Low Order nibble
byte
byte
Word (16 bit)

5. Lecture Overview Introduction Positional Numbering System
Decimal to binary conversion
Signed integer representation
Floating-point representation

6. Positional Numbering System (1/3)
Any numeric value is represented through increasing powers of a radix (or base)
The set of valid numerals (digits) is equal in size to the radix of that system
The least numeral is 0 and the highest one in 1 smaller than the radix
Example:
In the decimal system (base 10)
The radix is 10
The number of valid numerals is 10 (equal to the radix)
The set of valid numerals is: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

7. Positional Numbering System (2/3)
The most important radices (bases) in computer science are:
Binary
Radix 2 or base 2
Numerals: {0 , 1}
Octal
Radix 8 or Base 8
Numerals: {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7}
Hexadecimal
Radix 16 or base 16
Numerals: {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , A , B , C , D , E , F}

8. Positional Numbering System (3/3)
Any numeric value is represented through increasing powers of a radix (or base)
Examples
= 2x x x x x10-2
2123 = 2x32 + 1x31 + 2x30 = 2310
= 1x24 + 0x23 + 1x22 + 1x21 + 0x20 + 0x x2-2=

9. Lecture Overview Introduction Positional Numbering System
Decimal to binary conversion
Converting Unsigned Whole Numbers
Converting fractions
Converting between Power-of-Two Radices
Signed integer representation
Floating-point representation

10. Some numbers to remember (1/1)
Decimal to binary conversion
Some numbers to remember (1/1)
Keep in mind the following tables or how to obtain them!

11. Converting Unsigned Whole Numbers (1/6)
Decimal to binary conversion
Converting Unsigned Whole Numbers (1/6)
A real number can take any value (ex ; )
Whole number: No fractions (ex: 10, 1231, 3543, …, -12, ,…)
Unsigned number: Only positive numbers (ex: , , 12357, …)
Unsigned whole numbers: No fraction and only positive numbers

12. Converting Unsigned Whole Numbers (2/6)
Decimal to binary conversion
Converting Unsigned Whole Numbers (2/6)
Convert the decimal number to binary: =
Method 1: Repeated subtraction
113
- 64 49
- 32 17
- 16 1
- 1 1 1
11310 = 1 1

13. Converting Unsigned Whole Numbers (3/6)
Decimal to binary conversion
Converting Unsigned Whole Numbers (3/6)
Method 2: Division-remainder
2 |113
2 |56
2 |28
2 |14
2 |7
2 |3
2 |1
Remainder 1
Remainder 0
LSB
11310 =
MSB

14. Converting Unsigned Whole Numbers (4/6)
Decimal to binary conversion
Converting Unsigned Whole Numbers (4/6)
A binary number with N bits can represent 2N unsigned integers from 0 to 2N-1
Example:
Having N=4 bits, we can represent 24 = 16 unsigned integers from 0 to 24-1=16-1=15
The number 16 CANNOT be represented with only 4 bits!!

15. Converting Unsigned Whole Numbers (5/6)
Decimal to binary conversion
Converting Unsigned Whole Numbers (5/6)
The subtraction method is cumbersome.
The subtraction method requires a familiarity with the powers of the radix being used.
The division-remainder method is faster and easier than the repeated subtraction method.
The division-remainder method can be used to convert from decimal to any other base system (not only to base 2).

16. Converting Unsigned Whole Numbers (6/6)
Decimal to binary conversion
Converting Unsigned Whole Numbers (6/6)
Example: Convert to base 3 using the division-remainder method.
3 |104
3 |34
3 |11
3 |3
3 |1
Remainder 2
Remainder 1
Remainder 0
10410 =

17. Lecture Overview Introduction Positional Numbering System
Decimal to binary conversion
Converting Unsigned Whole Numbers
Converting fractions
Converting between Power-of-Two Radices
Signed integer representation
Floating-point representation

18. Converting fractions (1/5)
Decimal to binary conversion
Converting fractions (1/5)
Fractions in a decimal system can be converted/approximated to fractions in any other radix system
Radix points separate the integer part of a number from its fractional part
Example of fractions (the integer part is italic and the fractional part is bold)
Base 10 :
Base 3 :
Base 2 :
The “radix point” is called a “decimal point” in a decimal system, a “binary point” in a binary system, and so on…

19. Converting fractions (2/5)
Decimal to binary conversion
Converting fractions (2/5)
To convert fractions from decimal to any other base system we repeatedly multiply by the destination radix
Example: Convert to base 5.
0.4304 x
The integer part is 2
0.1520
The integer part is 0
The integer part is 3
0.8000
The integer part is 4,
the fractional part is zero, we are done =

20. Converting fractions (3/5)
Decimal to binary conversion
Converting fractions (3/5)
Some fractions in one base could be indeterminate
Fractions that contain repeating strings of digits to the right of the radix point
Example: (2/3)10=(0.666…)10
An indeterminate fraction in one base could be determinate in another base (and vice-versa).
Example: (2/3)10=0.23=(0.666…)10
2/3 is indeterminate in base 10 but determinate in base 3.
When a fraction is indeterminate, an approximation is needed
We fix the number of digits to the right of the radix point
Also, approximation is needed due to the limited computing resources (example: limited size of the processor’s registers)

21. Converting fractions (4/5)
Decimal to binary conversion
Converting fractions (4/5)
Example: Convert to binary with 4 bits to the right of the binary point.
x 2
This is our fourth bit.
We will stop here. =

22. Converting fractions (5/5)
Decimal to binary conversion
Converting fractions (5/5)
Convert to binary: =
By using the methods just described we will have:
2610= and =
So =

23. Going back to positional numbering system (1/1)
Decimal to binary conversion
Going back to positional numbering system (1/1)
Any unsigned whole or fractional number could be converted to decimal by using the “Positional Numbering System” described previously
Examples:
=0x2-1+1x2-2+0x2-3+1x2-4 = =
= 1x52 + 3x51 + 4x50 + 2x x x x5-4 =

24. Lecture Overview Introduction Positional Numbering System
Decimal to binary conversion
Converting Unsigned Whole Numbers
Converting fractions
Converting between Power-of-Two Radices
Signed integer representation
Floating-point representation

25. Converting between Power-of-Two Radices (1/4)
Decimal to binary conversion
Converting between Power-of-Two Radices (1/4)
To convert between any base to any other base (different than base 10), it is easier to pass through base 10.
Example: 31214= ?
First step: = 3x43 + 1x42 + 2x41 + 1x40=21710
Second step: by using the division-remainder method: =
So 31214=220013
Working between bases that are powers of two is much more easier.

26. Converting between Power-of-Two Radices (2/4)
Decimal to binary conversion
Converting between Power-of-Two Radices (2/4)
The must famous power-of-two radices are: binary (base 2), octal (base 23 / base 8) and hexadecimal (base 24 / base 16).
Each octal digit is equivalent to a group of 3 binary digits called octet1
Each hexadecimal digit is equivalent to a group of 4 binary digits called hextet
We convert from binary to octal and from binary to hexadecimal by simply grouping bits
1 The term “Octet” could also be used in the literature to describe a set of 8 bits.

27. Converting between Power-of-Two Radices (3/4)
Decimal to binary conversion
Converting between Power-of-Two Radices (3/4)
Example: Convert to octal
Make Groups of 3 bits (from right to left):
Add zero(s) on the left to complete the last octet
Convert each octet to its corresponding octal digit
Finally: =

28. Converting between Power-of-Two Radices (4/4)
Decimal to binary conversion
Converting between Power-of-Two Radices (4/4)
Example: Convert to hexadecimal
Make Groups of 4 bits (from right to left):
Add zero(s) on the left to complete the last hextet
Convert each hextet to its corresponding hexadecimal digit
C D
Finally: = 2C9D16

29. Lecture Overview Introduction Positional Numbering System
Decimal to binary conversion
Signed integer representation
Signed Magnitude
Complement system
Floating-point representation

30. Signed integer representation
An integer is a whole number
Signed integers are the set of positive and negative whole numbers
How should we encode and deal with the actual sign of the number?
Two concepts are used
Signed Magnitude concept
Complement concept

31. Signed magnitude is the most intuitive method
Signed integer representation
Signed Magnitude (1/13)
Signed magnitude is the most intuitive method
The MSB (Most Significant Bit) of a binary number is kept as the “sign” of the number
MSB = 1: negative number
MSB = 0: positive number
The remaining bits represent the magnitude (or absolute value) of the numeric value

32. Signed integer representation
Signed Magnitude (2/13)
Example: In a 8 bit word signed magnitude system give the decimal representation of the following numbers ?
The MSB is 0: The number is positive
The remaining 7 bits are: = 110
The decimal number is +1 ?
The MSB is 1: The number is negative
The decimal number is -1

33. Signed integer representation
Signed Magnitude (3/13)
Example: In a 8 bit word signed magnitude system give the decimal representation of the following numbers ?
The MSB is 1: The number is negative
The remaining 7 bits are: = 910
The decimal number is -9 ?
The MSB is 0: The number is positive
The remaining 7 bits are: = 6510
The decimal number is +65

34. In a N bit word signed magnitude system
Signed integer representation
Signed Magnitude (4/13)
In a N bit word signed magnitude system
1 bit is used for the sign of the number
N-1 bits are used for the magnitude of the number
The largest integer is 2N-1 - 1
The smallest integer is -(2N-1 - 1)
Example: in a 8 bit word signed magnitude system
The largest integer is = 27-1 = 12710
The smallest integer is = -(27-1) =

35. Computers should be able to carry out mathematical operations
Signed integer representation
Signed Magnitude (5/13)
Computers should be able to carry out mathematical operations
Signed-magnitude arithmetic is carried out using essentially the same methods as humans
At first we look at the signs of the two operands
We arrange the operands in a certain way based on their signs
We perform the calculation without regard to the signs
Finally, we supply the sign as appropriate

36. Adding operands that have the same sign
Signed integer representation
Signed Magnitude (6/13)
Adding operands that have the same sign
Example: Add to using signed-magnitude arithmetic.
⇐ carries
(79)
(35)
(114)
We find = in signed-magnitude representation.
Sign

37. Example: Add 010000012 to 011000012 using signed-magnitude arithmetic
Signed integer representation
Signed Magnitude (7/13)
Overflow condition
In the last example, adding the seventh’ bits to the left gives no carry
If there is a carry, we say that we have an overflow condition and the carry is discarded, resulting in an incorrect sum.
Example: Add to using signed-magnitude arithmetic

38. The addition overflows The last carry is discarded
Signed integer representation
Signed Magnitude (8/13)
⇐ carries
(65)
(97)
The addition overflows
The last carry is discarded
The sum’s result is incorrect X
(34)

39. Signed integer representation
Signed Magnitude (9/13)
Signed-magnitude subtraction is carried out in a manner similar to pencil and paper decimal arithmetic
Example 1: Subtract (79) from (99) using signed-magnitude arithmetic.
⇐ borrows
(99)
(79)
(20)
We find = in signed-magnitude representation.

40. Signed integer representation
Signed Magnitude (10/13)
Example 2: Subtract (99) from (79) using signed-magnitude arithmetic.
Here the subtrahend, , is larger than the minuend,
With the result obtained in Example 2.12, we know that the difference of these two numbers is
Because the subtrahend is larger than the minuend, all that we need to do is change the sign of the difference.
So we find = in signed-magnitude representation

41. Signed integer representation
Signed Magnitude (11/13)
Example 3: Add (-19) to (+13) using signed-magnitude arithmetic.
The result is negative
We subtract 13 from 19
The result of the binary subtraction is: (-6)
Example 4: Subtract (-24) from (-43) using signed-magnitude arithmetic.
This is equivalent to adding -43 to 24
We subtract 24 from 43
The result of the binary subtraction is: (-19)

42. General rules when operands have different signs
Signed integer representation
Signed Magnitude (12/13)
General rules when operands have different signs
Determine which operand has the larger magnitude
The sign of the result is the same as the sign of the operand with the larger magnitude
the magnitude must be obtained by subtracting (not adding) the smaller one from the larger one

43. Problems related to signed magnitude
Signed integer representation
Signed Magnitude (13/13)
Problems related to signed magnitude
To much decisions to make (larger number? ; borrows? ; what signs?).
The number 0 could have two representations : and
Complicated method
Expensive circuits

44. Lecture Overview Introduction Positional Numbering System
Decimal to binary conversion
Signed integer representation
Signed Magnitude
Complement system
Floating-point representation

45. Complement system is used to represent/convert negative numbers only
Signed integer representation
Complement system (1/19)
Complement system is used to represent/convert negative numbers only
When using complement system the subtraction is converted to an addition
Advantages of complement system
Simplify computer arithmetic
No need to process sign bits separately
The sign of a number is easily checked by looking at its high-order bit (MSB).

46. In base 10, “Casting out 9s” was used to subtract numbers
Signed integer representation
Complement system (2/19)
In base 10, “Casting out 9s” was used to subtract numbers
Let’s say we wanted to find
At first, is calculated
999 – 52 = 947
947 is then added to 167 and the last carry is added to the sum:
167 – 52 = = = 115 a
Carries: 1 6 7 + 9 4

47. The last method uses a “diminished radix complement”
Signed integer representation
Complement system (3/19)
The last method uses a “diminished radix complement”
Working in base r (radix), the diminished radix is given by : r-1
Example: Base 10 ; r=10
The diminished radix is r-1 = = 9
We say that a negative number is converted to its 9’s complement
For example, is converted to its nine’s complement as follows: = = 7531C9

48. In a binary system r=2 The diminished radix complement is r-1 = 1
Signed integer representation
Complement system (4/19)
In a binary system r=2
The diminished radix complement is r-1 = 1
We say that we work in one’s complement (C1)
To convert a negative number to its one’s complement this number is subtracted from all ones
A positive number is directly converted to its binary representation
Example:
The one’s complement of is = 1010C1
It is nothing more than switching all of the 1s with 0s and vice versa!!

49. Example: Express 2310 and -910 in 8-bit binary one’s complement form.
Signed integer representation
Complement system (5/19)
Example: Express 2310 and -910 in 8-bit binary one’s complement form.
2310 = + ( ) = C1
-910 = - ( ) = C1

50. In one’s compliment the subtraction is converted into addition
Signed integer representation
Complement system (6/19)
In one’s compliment the subtraction is converted into addition
Example: 2310 – 910 = (-910)
Example: Add 2310 to -910 using 8-bit binary one’s complement arithmetic.
The result is C1 = +( ) = 1410
Carries: 1
2310 +
+ (-910)
1410

51. Signed integer representation
Complement system (7/19)
Example: Add 910 to using 8-bit binary one’s complement arithmetic.
-2310 = - ( )2 = C1
910 = + ( ) = C1
910 + (-2310) = C C1
Result: C1 = -( ) = -1410
Carries: 1
910 +
+ (-2310)
-1410

52. Computer engineers long ago stopped using one’s complement
Signed integer representation
Complement system (8/19)
In One’s complement, we still have two representations for zero: and
Computer engineers long ago stopped using one’s complement
A more efficient representation for binary numbers is the two’s complement

53. Two’s complement is an example of a radix complement
Signed integer representation
Complement system (9/19)
Two’s complement is an example of a radix complement
No need to subtract one from the radix r when working in a radix complement.
Example: Base 10 ; r=10
We say that a negative number is converted to its 10’s complement
For example, is converted to its ten’s complement as follows: = = 7532C10

54. In a binary system r=2 The diminished radix r = 2
Signed integer representation
Complement system (10/19)
In a binary system r=2
The diminished radix r = 2
We say that we work in two’s complement
Consider “d” is the number of digits
To convert a negative number “N” to its two’s complement this number is subtracted from rd = 2d : N10 = (2d – N)C2
A positive number is directly converted to its binary representation

55. Example: In a 4 bits system: d=4;
Signed integer representation
Complement system (11/19)
Example:
In a 4 bits system: d=4;
All negative numbers are converted by being subtracted from 2d = 24 = 1610 =
The two’s complement of is = 1101C2
It is nothing more than one’s complement incremented by 1!!

56. Signed integer representation
Complement system (12/19)
Example: Express 2310, -2310, and -910 in 8-bit binary two’s complement form.
2310 = + ( ) =
-2310 = -( ) = =
-910 = -( ) = =

57. Unlike C1 arithmetic, in C2 the last carry is discarded
Signed integer representation
Complement system (13/19)
Unlike C1 arithmetic, in C2 the last carry is discarded
Example 1: Add 910 to using two’s complement arithmetic.
The result is C2 = -( ) = -1410
Carries: 1
910 +
+ (-2310)
-1410

58. Note how a negative binary number in C2 is converted to decimal
Signed integer representation
Complement system (14/19)
Note how a negative binary number in C2 is converted to decimal
At first all 0 and 1 in the C2’s number are switched: →
A “1” is then added to the last number: =
So C2 = -( ) = -1410

59. Signed integer representation
Complement system (15/19)
Example 2: Find the sum of 2310 and -910 in binary using two’s complement arithmetic.
2310 = +( )2 = C2
-910 = -( ) = C2
(-910) = C C2
Result: C2 = +( ) = 1410
Carries: 1
2310 +
+ (-910)
-1410

60. Advantages of two’s complement
Signed integer representation
Complement system (16/19)
Advantages of two’s complement
It is the most popular choice for representing signed numbers
The algorithm for adding and subtracting is quite easy
It has the best representation for 0 (all 0 bits)
It is self-inverting
It is easily extended to larger numbers of bits.

61. Signed integer representation
Complement system (17/19)
Drawback
the asymmetry seen in the range of values that can be represented by N bits.
Examples:
With signed-magnitude, 4 bits allow us to represent the values -7 (11112) through +7 (01112).
Using two’s complement, we can represent the values: -8 (1000C2) through +7 (0111C2)

62. Overflow in complement systems (C1 and C2)
Signed integer representation
Complement system (18/19)
Overflow in complement systems (C1 and C2)
An overflow occurs if two positive numbers are added and the result is negative
or if two negative numbers are added and the result is positive.
It is not possible to have overflow when if a positive and a negative number are being added together.

63. Note that the last two carries are different
Signed integer representation
Complement system (19/19)
To Detect Overflow
Check the last two carries
If these are different: there is an overflow
If these are equal: there is no overflow
Example 1: Find the sum of and 810 in binary using two’s complement arithmetic.
The result is C2 = -( )2 = !!!
Note that the last two carries are different
Carries: 1
12610 +
-1410

64. Lecture Overview Introduction Positional Numbering System
Decimal to binary conversion
Signed integer representation
Floating-point representation
A simple model
Floating-point arithmetic
Floating point errors

65. Floating-point representation (1/1)
A computer is supposed to solve all problems
Huge and fractional numbers and complicated mathematical operations could be involved
An optimized solution to give a good ratio: “Biggest Number/word size” is the Floating point representation

66. Numbers written in scientific notation have three components:
Computers use a form of scientific notation for floating-point representation
Numbers written in scientific notation have three components:
Scientific notation in base 10:
Scientific notation in base 2:
0.579 x
107 + x 23 +

67. In digital computers, floating-point numbers consist of three parts:
Floating-point representation
A simple model (1/8)
In digital computers, floating-point numbers consist of three parts:
A sign bit,
an exponent part: representing the exponent on a power of 2,
a fractional part called a significand: which is a fancy word for a mantissa.

68. More bits used for the exponent increases the range of numbers
Floating-point representation
A simple model (2/8)
More bits used for the exponent increases the range of numbers
More bits used for the significant increases the precision
For simplicity, in all this course, we will use a simplified 14 bits model
Sign bit: 1 bit
Exponent: 5 bits
Significand: 8 bits

69. Floating-point representation
A simple model (3/8)
Exercise 1: Represent the number 17 in a 14 bits floating point representation
17 = 17.0 x 100 = 1.7 x 101 = 0.17 x 102
Analogically in binary:
1710 = x 20 = x 21= x 22 = x23 = x 24 = x 25 = x 26 = x 27 = ...
As a convention, we stop when the MSB of the significant is “1”: x 25
The exponent is 510 =
The significant is: →
So: 1

70. To solve such problems we use an excess-16 bias
Floating-point representation
A simple model (4/8)
The last floating point representation is not suitable for negative exponents
Example:
the number 0.25 = = 0.12 x 2-1
How to represent the negative exponent -1?!
To solve such problems we use an excess-16 bias
All negative and positive exponents are added by 16
We say that the real exponent is replaced by a biased exponent
All exponents are converted to positive biased exponents

71. Floating-point representation
A simple model (5/8)
With an excess-16 bias
Exponent values less than 16 will indicate negative exponent values
Exponent values more than 16 will indicate positive exponent values
exponents of all zeros or all ones are typically reserved for special numbers (such as zero or infinity).

72. Floating-point representation
A simple model (6/8)
Example 1: Represent the number 17 in a 14 bits floating point form with excess-16 bias
The number is positive: sign bit is “0”
1710 = x 25
The exponent is 510 → (5+16)10 = 2110 =
The significant is: →
So 17 in floating point form with excess-16 bias is: 1

73. Floating-point representation
A simple model (7/8)
Example 2: Represent the number in a 14 bits floating point form with excess-16 bias.
The number is positive: sign bit is “0”
0.25 = x 20 = 0.12 x 2-1
The exponent is -110 → (-1+16)10 = 1510 =
The significant is 1 →
So 0.25 in floating point form with excess-16 bias is: 1

74. Floating-point representation
A simple model (8/8)
Example 3: Express in normalized floating-point form with excess-16 bias.
The number is negative: sign bit is “1”
= = x20 = x2-1 = … = 0.1x2-4
The exponent is -410 → (-4+16)10 = 1210 =
The significant is 1 →
So in floating point form with excess-16 bias is: 1

75. Lecture Overview Introduction Positional Numbering System
Decimal to binary conversion
Signed integer representation
Floating-point representation
A simple model
Floating-point arithmetic
Floating point errors

76. Floating point arithmetic (1/2)
Floating-point representation
Floating point arithmetic (1/2)
To add/subtract two numbers in floating point form
Both numbers should have the same exponent
If exponents are different
we change one of the numbers so that both of them are expressed in the same power of the base
We add the binary numbers
We represent the result in a normalized floating point form

77. Floating point arithmetic (2/2)
Floating-point representation
Floating point arithmetic (2/2)
Example: Add the following binary numbers as represented in a normalized 14-bit format with an excess-16 bias.
The second number is x20
The first number is x22 = x20
Now :
The result is x 20 = x 22
In floating point form with excess-16
1810 → 210 1 +
1610 → 010 1

78. Lecture Overview Introduction Positional Numbering System
Decimal to binary conversion
Signed integer representation
Floating-point representation
A simple model
Floating-point arithmetic
Floating point errors

79. Floating Point Errors (1/2)
Floating-point representation
Floating Point Errors (1/2)
Computers are finite systems
When dealing with floating-point form, we are modeling the infinite system of real numbers in a finite system of integers
What we have, in truth, is an approximation of the real number system
The more bits we use, the better the approximation
However, there is always some element of error
Such errors can propagate through a lengthy calculation, causing substantial loss of precision

80. Floating Point Errors (2/2)
Floating-point representation
Floating Point Errors (2/2)
Example:
In our previous simple model
we are limited between x215 through x215.
we cannot store 2x-19 or 2128; they simply don’t fit.
Also, cannot be accurately stored even if it is well within our range
= = x28
The significant is expressed with more than 8 bits!
In practice we store only the first 8 bits:
We actually store 128 and not with an absolute error of 0.5
The relative error is : = = 0.39%.
128.5

81. End of lecture 2 Try to solve all exercises related to lecture 2