Data Representation

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Data Representation

• Lecture 21
• 5.3.2001.

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• 1001011101101
• 5
• 1001001100101
• 12ed

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Signed Integer Representation

• Many of the numerical data items that are used in
  a program are signed (positive or negative).
• Question How to represent sign?
• Three possible approaches
• Sign-magnitude representation
• Ones complement representation
• Twos complement representation

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Sign-magnitude Representation

• For an n-bit number representation
• The most significant bit (MSB) indicates sign
• 0 ? positive
• 1 ? negative
• The remaining n-1 bits represent magnitude.

b0
bn-1
bn-2
b1
Sign
Magnitude
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Signed Magnitude scheme

• Range of numbers that can be represented
• Maximum (2n-1 1)
• Minimum ? (2n-1 1)
• A problem
• Two different representations of zero.
• 0 ? 0 000.0
• -0 ? 1 000.0

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Ones Complement Representation

• Basic idea
• Positive numbers are represented exactly as in
  sign-magnitude form.
• Negative numbers are represented in 1s
  complement form.
• How to compute the 1s complement of a number?
• Complement every bit of the number (1?0 and 0?1).
• MSB will indicate the sign of the number.
• 0 ? positive
• 1 ? negative

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n4

• 1000 ? -7
• 1001 ? -6
• 1010 ? -5
• 1011 ? -4
• 1100 ? -3
• 1101 ? -2
• 1110 ? -1
• 1111 ? -0

• 0000 ? 0
• 0001 ? 1
• 0010 ? 2
• 0011 ? 3
• 0100 ? 4
• 0101 ? 5
• 0110 ? 6
• 0111 ? 7

To find the representation of, say, -4, first
note that 4 0100 -4 1s
complement of 0100 1011
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• Range of numbers that can be represented
• Maximum (2n-1 1)
• Minimum ? (2n-1 1)
• A problem
• Two different representations of zero.
• 0 ? 0 000.0
• -0 ? 1 111.1
• Advantage of 1s complement representation
• Subtraction can be done using addition.
• Leads to substantial saving in circuitry.

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Twos Complement Representation

• Basic idea
• Positive numbers are represented exactly as in
  sign-magnitude form.
• Negative numbers are represented in 2s
  complement form.
• How to compute the 2s complement of a number?
• Complement every bit of the number (1?0 and 0?1),
  and then add one to the resulting number.
• MSB will indicate the sign of the number.
• 0 ? positive
• 1 ? negative

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• In C
• short int
• 16 bits ? (215-1) to -215
• int
• 32 bits ? (231-1) to -231
• long int
• 64 bits ? (263-1) to -263

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Twos complement

• Consider 8 bit numbers. In this system if the
  value of the MSB is
• zero then the rest of the seven bits represent
  normal binary numbers (0 to 127).
• one then this represents -128 which is added to
  the total of the other seven bits that represent
  a normal binary number.

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Example n8

• 00000001 ? 1
• 00000101 ? 5
• 10000001 ? -127
• 10000101 ? -123
• 11011111 ? -33

To find the representation of, say, -4, first
note that 4 00000100 -4
2s complement of 00000100
111110111 11111100
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• Range of numbers that can be represented
• Maximum (2n-1 1)
• Minimum ? 2n-1
• Advantage
• Unique representation of zero.
• Subtraction can be done using addition.
• Leads to substantial saving in circuitry.
• Most computers today use the 2s complement
  representation for storing negative numbers.

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Twos complement arithmetic

• Twos complement is very important when we wish
  to perform a subtraction between two binary
  numbers.
• Adding one binary number to the 2s complement of
  another binary number is equivalent to a
  subtraction operation.

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Negative numbers in 2s complement

• 1. Flip all the binary digits within the
  corresponding positive number (1s complement).
• 2. Add 1 to the result.
• 13 00001101
• 1s complement 11110010
• 2s complement 11110011
• This represents -13
• (-1286432160021)

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2s complement arithmetic

• Example 1 18-11 ?
• 18 is represented as 00010010
• 11 is represented as 00001011
• 1s complement of 11 is 11110100
• 2s complement of 11 is 11110101
• Add 18 to 2s complement of 11

00010010 11110101 ----------------
00000111 (with a carry of 1
which is ignored)
00000111 is binary for 7.
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2s complement arithmetic

• Example 2 7 - 9 ?
• 7 is represented as 00000111
• 9 is represented as 00001001
• 1s complement of 9 is 11110110
• 2s complement of 9 is 11110111
• Add 7 to 2s complement of 9

00000111 11110111 ----------------
11111110 (with a carry of 0
which is ignored)
00000111 is binary for -2.
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Overflow

• Overflow condition carry in and carry out from
  MSB are different.

(64) 01000000 ( 4) 00000100
-------------- (68) 01000100
(64) 01000000 (96) 01100000
-------------- (-96) 10100000
carry (out)(in) 0 0
carry out in 0 1
differ overflow
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Real Numbersfloating point representation

• 1.5, 36.415 how to represent these numbers ?
• In floating point representation, a number is
  represented in two parts
• 1. mantissa (M) F M x BE
• 2. exponent (E) B exponent base
• 1903810 1.9038 x 104
• 0.0032510 3.25 x 10-3

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Floating point binary numbers

• All of the mantissa appears to the right of the
  decimal (binary?) point.
• 1101.00112 becomes 0.110100112 x 24
• 0.0001012 becomes 0.1012 x 2-3

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Storage of floating point numbers

• Two pieces of information need to be stored
• 1. mantissa more bits for mantissa means more
  precision
• 2. exponent more bits in the exponent provides
  a greater range of numbers.

1. M represents a 2s complement fraction 1
gt M gt -1 2. E represents the exponent in 2s
complement
M
E
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8
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• In C
• 4 bytes for single precision float
• 8 bytes for double
• Different computer systems use different data
  representations for floating point numbers.

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Example
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Range of values

• 4 bytes are used to represent a floating point
  number.
• Mantissa (3 bytes) and
• exponent (1 byte)
• are in 2s complement form.
• What would be the smallest, greatest, and the
  smallest positive fraction that could be handled
  by this system ?

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• 1. The largest single precision number is
  01111111 11111111 11111111 01111111
• 2. The smallest number (most -ve) would be
  10000000 00000000 00000000 01111111
• 3. The smallest fraction would be 00000000
  00000000 00000001 10000000

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Representation of Characters

• Many applications have to deal with non-numerical
  data.
• Characters and strings.
• There must be a standard mechanism to represent
  alphanumeric and other characters in memory.
• Two standards in use
• Extended Binary Coded Decimal Interchange Code
  (EBCDIC) -- Used in older IBM machines
• American Standard Code for Information
  Interchange (ASCII)
• Most widely used today

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ASCII code

• Each character (letters, numerals, punctuation
  marks, control characters and other symbols) is
  assigned a numeric code (7 or 8 bits).
• The binary encoding of the characters follow a
  regular ordering.
• Digits are ordered consecutively in their proper
  numerical sequence (0 to 9).
• Letters (uppercase and lowercase) are arranged
  consecutively in their proper alphabetic order.

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Some Common ASCII Codes

• 0 30 (H) 48 (D)
• 1 31 (H) 49 (D)
• ..
• 9 39 (H) 57 (D)
• ( 28 (H) 40 (D)
• 2B (H) 43 (D)
• ? 3F (H) 63 (D)
• \n 0A (H) 10 (D)
• \0 00 (H) 00 (D)

• A 41 (H) 65 (D)
• B 42 (H) 66 (D)
• ..
• Z 5A (H) 90 (D)
• a 61 (H) 97 (D)
• b 62 (H) 98 (D)
• ..
• z 7A (H) 122 (D)

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Character Strings

• Two ways of representing a sequence of characters
  in memory.
• The first location contains the number of
  characters in the string, followed by the actual
  characters.
• The characters follow one another, and is
  terminated by a special delimiter.
• In C, the second approach is used.
• The \0 character is used as the string
  delimiter.
• Example Hello ?

e
H
l
o
0
l