DATA REPRESENTATION

1
DATA REPRESENTATION

• Digital computers store information in binary
• Binary allows two states
• On yes true 1
• Off no false 0
• Each digit in a binary number is called a bit
• 0 1 1 0
• bit
• 0ff 0n 0n 0ff
• Binary number system how numbers can be
  processed in binary

2
Binary Numbers

• Digits are 1 and 0
• 1 true
• 0 false
• MSB most significant bit
• LSB least significant bit
• Bit numbering

3
Number system

• A number system of base (radix) r is a system
  that uses distinct symbol for r digits.
• Numbers are represented by a string of digit
  symbols.

4
Number system

• The base of a number is usually specified as a
  subscript, e.g.
• (01000011)2,
• (71203)8,
• (FF078ABC)16, ...etc.
• Or a letter indicating the base (d for decimal, b
  for binary, o for octal and h for hexadecimal) is
  appended to the number, e.g.
• 01000011b,
• 71203o,
• FF078ABCh, ...etc.

5
Number system
Attribute Binary Octal Decimal Hexadecimal
Base 2 8 10 16
Lowest Digit 0 0 0 0
Highest Digit 1 7 9 F
6
Number system

• The value of a digit depends not only on its
  value but also on its position within the number
  ? gives the power of the radix by which it is
  multiplied.

7
Positional values in the decimal number system
Decimal Digit 9 3 7
Position Name Hundreds Tens Ones
Positional Value 100 10 1
Positional Value as a power of the base (10) 102 101 100
8
Positional values in the binary number system
Binary Digit 1 0 1
Position Name Fours Twos Ones
Positional Value 4 2 1
Positional Value as a power of the base (2) 22 21 20
9
Example

• 2
  1 0 -1
  -2
• 379.25 310 710 910210510
• n
  n-1
• (dndn-1...d0.f1f2fm)r dn ( r ) dn-1 ( r
  ) .
• 0
  -1 -2
  -m
• d0 ( r ) f1 ( r ) f2 ( r ) fm ( r
  )
• Converting to decimal

10
Examples on converting from different bases to
Decimal

• Convert the following to Decimal
• (1001001)2
• (203)8
• (FA07)16
• Solution
• (1001001)2 1 021 022 123 024 025
  126 73d
• (203)8 3 081 282 131d
• (FA07)16 7 0161 10162 15163 64007d

11
Conversion from decimal

• The conversion of a decimal integer into a base r
  is done by
• Whole numbers divisions by r
• Fractions repeated multiplication by r.

12
Conversion from decimal

• To convert from decimal to any numbering system
  with base r
• The decimal number is divided by r,
• Keeping the remainder aside, the result is
  further divided by r, and the new remainder is
  kept aside,
• The new result is divided again by r, and so on
  till the result is less than r and this would be
  the last remainder,
• The remainders make up the equivalent base-r
  number, with the last remainder being the
  most-significant digit and the first remainder
  being the least-significant digit.

13
Conversion to binaryWhole number

• Ex. (67)10 (?)2
• 2 67
• 2 33 rem 1 LSB
• 2 16 rem 1
• 2 8 rem 0
  
• 2 4 rem 0
  67d 10000112
• 2 2 rem 0
• 2 1 rem 0
• 0 rem 1 MSB

14
Fraction

• Repeat multiplication by 2 until the fractional
  product is 0
• Ex. (0.3125)10 (?)2
• 
  Carry
• 0.3125 2 0.625 0 MSB
• 0.625 2 1.25 1
  (0.3125)10 (0.0101)2
• 0.25 2 0.5 0
• 0.5 2 1.00 1
  LSB

15
Conversion to base r

• (50) 10 (?)8
• 8 50
• 8 6 2 8
  0 6 (50)10 (62)8

16
Octal and Hexadecimal number systems

• Binary numbers are long.
• On average, it takes about 3.3 times as many
  digits to represent a value in binary as it does
  to represent the same value in decimal.

17
Octal and Hexadecimal number systems

• If a base R1 is an integral power of
• 
  
  d
• another base R2 i.e. R1 R2 ?
• Group of d digits in base R2 maps directly
  into one digit in base R1
• And each digit in base R1 maps directly into d
  digits in base R2.

18
Octal number systems

• 3
• Base 8 2 ? d 3
• Digits 0 ? 7
• Binary to Octal conversion
• Group every 3 bits (starting from the right)
• Replace them by their corresponding octal digit
• Ex. ( 0 1 0 1 1 0 1 0 1 )2 ( 265 )8
• ( 1 0 0 0 1 )2 ( 21 )8

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Octal to binary conversion

• Replace each digit by its 3-bit binary
  equivalent.
• Ex. (476)8 (100111110)2

20
Octal to binary conversion
21
Binary Octal conversion table

• Binary Octal Binary Octal
• 000 0 001000
  10
• 001 1 010000
  20
• 010 2
• 011 3
• 100 4
• 101 5
• 110 6
• 111 7

22
Hexadecimal number systems (Hex)

• 
  4
• Base 16 2 ? d 4
• Digits 0,1,2,..,9,A,B,C,D,E,F
• Where A 10, B 11, C 12, D 13, E 14, F
  15

23
Binary to Hex conversion

• Group each 4 bits and replace them by their
  corresponding hex digit.
• Ex.
• ( 1 0 1 1 0 1 0 1 )2 ( B5 )16

24
Hex to binary conversion

• Replace each digit by their 4 bit binary
  equivalent
• Ex.
• ( 1D9C )16 (0001110110011100 )2

25
Hex to binary conversion
26
Binary Hexadecimal conversion table

• Binary Hex Binary
  Hex
• 0000 0 1000
  8
• 0001 1 1001
  9
• 0010 2 1010
  A
• 0011 3 1011
  B
• 0100 4 1100
  C
• 0101 5 1101
  D
• 0110 6 1110
  E
• 0111 7 1111
  F
• (00010100)2 ? (14)16 ? ( 20 )10
• (12.8)16 ? ( 00010010.1000)2 ? (18.5)10

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Binary Arithmetic operations

• Addition
• 00 0
• 01 1
• 10 1
• 11 10 (carry)

28
Binary Arithmetic operations

• Subtraction
• 0 - 0 0
• 0 - 1 1 (after borrowing)
• 1 - 0 1
• 1 - 1 0

29
Binary Arithmetic operations

• Multiplication
• 00 0
• 01 0
• 10 0
• 11 1

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ADDITION

• Like decimal numbers, two numbers can be added by
  adding each pair of digits together with carry
  propagation.

31
Example
Carries
32
SUBTRACTION

• Two numbers can be subtracted by subtracting each
  pair of digits together with borrowing, where
  needed.

33
SUBTRACTION
34
Representation of signed binary numbers

• Given a fixed number of bit positions (n), we
• 
  n
• can represent 2 patterns From (00..0)2 to
  (11..1)2.
• Using unsigned numbers we can represent
• n
• 0 ? 2 -1
• n
• Using signed numbers ? 2 positive negative
  patterns

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Complementary notations

• The most commonly used way of representing signed
  numbers. It greatly simplifies all arithmetic
  operations.
• Positive number is represented as it is (like an
  unsigned positive number).
• Negative numbers are represented by the
  complement of unsigned number.

36
Complementary notations

• The value used for complementation is a power of
  the base or less one than a power of the base.
• For binary numbers we have 2s complement and
  1s complement.
• For decimals, we have 10s complement and 9s
  complement.
• For hex. Numbers we have 16s complement and Fs
  (15s) complement.

37
Ones complement

• Is obtained by subtracting the number from
  (11..1)2 (n 1s) ? reversing each bit changing
  every 1 to 0 and every 0 to 1.
• Ex. represent (14d), -(14d) in 8-bit binary
  number
• (14)10 ( 00001110)2 (00001110)1s
• (-14)10 -( 00001110)2 (11110001)1s

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For 8-bits number system

• Largest positive number 0 1111111 (127)10
• smallest negative number 1 0000000 - (127)10
• Zeroes 0 0000000
• 1
  1111111
• Range
• m-1
  m-1
• - (2 -1) to 2 -1 (m the
  number of bits)
• -(127)10 to (127)10
• The most significant bit represents the sign
• 0 ve 1 -ve.

39
2s complement

• Take the 1s complement and add 1 ? invert all
  the bits and add 1.
• Ex. 2s complement of (01110100)2
• 1s complement (10001011)2 add 1 ?
• 2s complement (10001100)2

40
2s complement

• 2s complement of (00000000)2
• 1s complement (11111111)2 add 1 ?
  (100000000)2
• 2s complement of n-bit number include only the
  rightmost n bits
• 2s complement of (00000000)2
  (00000000)2

41
2s complement

• - For 8-bit binary numbers 2s complement of (5)d
  (00000101)2
• 1s complement (11111010)2 add 1?
  (11111011)2
• 510 (-5)10 ? 000001012 111110112
  1000000002
• Carry
• 2s complement of any N integer represents N
• N (-N) 0

42
2s complement

• To find the 2s complement of a binary number,
  proceeding from right to left, leave all bits
  unchanged up to and including the first 1.
  Reverse all the remaining bits.
• Range of signed value that can be represented in
  2s complemented notation is
• m-1 m-1
• - 2 to 2 -1

43
For 8-bits number system

• Largest positive number
• 0 1111111 (127)10
• smallest negative number
• 1 0000000 - (128)10
• Zero 0 0000000
  
• Range -(128)10 to (127)10
• The most significant bit represents the sign
• 0 ve 1 -ve.

44
2s complement in hexadecimal

• Each group of four bits corresponds to a single
  hexadecimal digit.
• Subtract each hexadecimal digit from F and then
  add 1 (equivalent of taking the 16s complement).

45
Ex. 2s complement of (3A6E)16

• 1s complement FFFF
• - 3A6E
• C591
• 1
• 2s complement (C592)16

46
Comparing 2s complement numbers

• Check the signs
• If they differ, they determine the order
• If they are the same, the order of the numbers is
  the same as that of their representations.

47
Ex. (01001100)2 and (10100101)2

• Different signs ? the 1st is ve
• the 2nd is ve
• ? 1st gt 2nd

48
(10110010)2 and (10111001)2

• Both numbers are negative 1st lt 2nd

49
Signed binary arithmetic

• Overflow
• Signed binary numbers are of a fixed range.
• If the result of addition/subtraction goes beyond
  this range, overflow occurs.
• Two conditions under which overflow can occur
  are
• (i) positive add positive gives
  negative
• (ii) negative add negative gives
  positive

50
2s complement addition

• Algorithm
• Perform binary addition on the two numbers.
• Ignore the carry out of the MSB.
• Check for overflow Overflow occurs if the
  carriers into and out of the MSB are different.

51
2s complement subtraction

• Algorithm for performing A - B
• A-B A (-B)
• Take 2s complement of B by inverting all the bits
  and adding 1
• Add the 2s complement of B to A

52
2s complement addition/Subtraction

• Examples (4-bits system )

3 0011 4 0100
---- ------- 7 0111
---- -------
-2 1110 -6 1010
---- ------- -8
11000 ---- -------
53
2s complement addition/Subtraction
4 0100 -7 1001
---- ------- -3
1101 ---- -------
6 0110 -3 1101
---- ------- 3
10011 ---- -------
54

Examples Overflow in 2s Addition/Subtraction4-bi
ts system
-3 1101 -6 1010
---- ------- -9 10111
---- -------
5 0101 6 0110
---- ------- 11
1011 ---- -------
7
-5
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Examples

• 1)Add 45 and 18 (use 8-bit 2s complement
  representation
• (45)d B00101101
• (18)d B00010010 ? (-18)d B11101110
• B00101101
• B11101110
• B100011011 ? 27
• Discard carry

56
Examples

• 2)Add 38 and 57
• 38 B00100110
• 57B00111001 ? -57 B11000111
• B 00100110 B 11000111 B11101101 ?
  -19

57
Examples

• 3)Add 110 and 75
• 110 75 185
• 
  010011100 carries
• 110 B01101110 B01101110
  
• 75B01001011 B01001011
• 
  B10111001 negative result ? overflow the result
  is too big. 8 bits ? -128lt I lt 127

58
Examples

• 4)Subtraction
• B01101011 -B00101001
• 2s complement of B00101001 is
• 11010111
• B01101011 B11010111 B101000010

59
Examples

• Note In addition subtraction, if the operands
  have different length, the shortest must be sign
  extended.
• Ex.
• 1) Add B1011 with B01101101
• B11111011 B01101101
• B101101000

60
Examples

• 2) subtract XA3BC from X34A10BEE
• XA3BC sign extended ? FFFFA3BC
• 2s complement ? FFFFFFFF-FFFFA3BC 1
  00005C44
• X34A10BEE X00005C44
• X34A16832

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CODING

• The most fundamental problem of coding is to get
  as much information as possible in a fixed-length
  representation.
• In binary How many different binary numbers we
  can write with n bits for a fixed
• n
• n? ? 2 numbers.

62
Binary Coded Decimal (BCD)

• Decimal numbers are more natural to humans.
  Binary numbers are natural to computers. Quite
  expensive to convert between the two.
• If little calculation is involved, we can use
  some coding schemes for decimal numbers.

63
Binary Coded Decimal (BCD)

• One such scheme is BCD, also known as the 8421
  code.
• Represent each decimal digit as a 4-bit binary
  code.

64
Binary Coded Decimal (BCD)

• Some codes are unused, eg (1010)BCD, (1011)BCD,
  , (1111)BCD. These codes are considered as
  errors.
• Easy to convert, but arithmetic operations are
  more complicated.

65
Binary Coded Decimal (BCD)

• Examples
• (234)10 (0010 0011 0100)BCD
• (7093)10 (0111 0000 1001 0011)BCD
• (1000 0110)BCD (86)10
• (1001 0100 0111 0010)BCD (9472)10
• Note
• BCD is not equivalent to binary.
• Example (234)10 (11101010)2

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Character data type

• Apart from numbers, computers also handle textual
  data.
• Internally all computers represent characters by
  storing them as binary data
• Character set frequently used includes
• alphabets A .. Z, and a .. z
• digits 0 .. 9
• special symbols , ., ,, _at_, ,
• non-printable line feed, bell,

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Character data type

• Usually, these characters can be represented
  using 7 or 8 bits.
• Two important issues concerning the character
  codes
• - The size i.e. the of bits used to
  represent each character.
• - The collating sequence.

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Character data type

• Standard encoding of character data is 8-bit.
• Each character was to be represented as a string
  of 8-binary digits and there are a total
• 8
• of 2 (256) unique characters available.
• widely used standardASCII (American Standard
  Code for Information Interchange) 7-bit
  the code fits into a byte, with the most
  significant bit set to 0

69
ASCII

• ASCII code is really 7-bit code ? 128 ASCII codes
• 2 calsses of characters
• - Printable characters ( code 2016 (408)?7E16
  (1768) ) displayed on screen, or printed on
  printer. They consist of
• . Special characters e.g. , , , lt, gt,
• . Numbers 0 - 9
• . Uppercase characters A - Z
• . Lowercase characters a - z

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ASCII

• Control characters (0 - 1F16 (378) , 7F16 (1778)
  ). These do not cause a character to be printed
  but instead initiate some action, control the
  operation of devices (e.g. tab, LF, CR, ring
  bell) or convey some status information.

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ASCII

• Example
• What is the internal ASCII code for the character
  string 123?
• 1 2 3
• 0618 0628 0638
• 3116 3216 3316
• 00110001 00110010 00110011
• ? 001100010011001000110011

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ASCII

• This example points up that the representation of
  the character string 123 is quit different from
  the representation of the signed integer 123.
• Note The character representation takes 24 bits
  (38).
• The signed integer representation in 16-bits is
• (0000000001111011)2

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ASCII
74
ASCII code table book page 32
CHAR HEX DEC CHAR HEX DEC
_at_ 40 64 ltCCgt 00 0
A 41 65 ltCCgt 01 1
B 42 66 ltCCgt 02 2