Data Representation

1
Data Representation

• Assembly Language Programming
• University of Akron
• Dr. Tim Margush

2
Binary Numerals

• 1101101b
• Bits are numbered from the right b6b5b4b3b2b1b0
• Subscripts represent the place value
• b6 has place value 26
• Conversion to decimal is done by evaluating the
  polynomial
• b626 b525 b424 b323 b222 b121 b020
• In this case, 6432841 109d

3
Decimal to Binary

• 109d is converted to binary by repeated divisions
  by 2
• make note of the remainders
• Stop when you reach 0
• Concatenate the remainders to form the binary
  representation
• First remainder is least significant bit
• 109d 1101101b

• 109 / 2 54 r 1
• 54 / 2 27 r 0
• 27 / 2 13 r 1
• 13 / 2 6 r 1
• 6 / 2 3 r 0
• 3 / 2 1 r 1
• 1 / 2 0 r 1

4
Binary and Hexadecimal

• Binary to Hex
• Group bits by fours (starting with least
  significant bits)
• Add leading zeros as necessary to complete the
  last group
• Convert each group to the equivalent hex digit

• Hex to Binary
• Expand each hex digit to the equivalent 4-bit
  binary form
• You may omit leading zeros of leftmost digit
• 0100 1110b 4Eh
• 37h 0011 0111b
• (or 110111b)

5
Character Data

• American Standard Code for Information
  Interchange (ASCII)
• a 7-bit binary code for a set of 128 characters
• usually occupy a byte of storage
• leading bit may be ignored or used differently by
  various programs
• a sequence of bytes containing ASCII codes is
  referred to as an ASCII string

6
Numeric Data

• Binary storage
• twos complement, ones complement, sign and
  magnitude, or biased representations
• ASCII storage
• sequence of ASCII bytes representing the digits
  of the number expressed in some radix
• Binary Coded Decimal
• sequence of nybbles representing digits 0-9 of
  the number

7
Binary Storage

• A pre-arranged storage size is used
• typically byte, word, doubleword, or quadword
• Represent a number in base two and encode the
  bits
• 197d is 11000101b
• at least 8 bits will be required to store this
  number (leading zeros are added if necessary to
  fill additional bits for larger storage sizes)

8
Signed vs Unsigned Codes

• Unsigned Byte
• all 8 bits used to represent the magnitude of the
  number
• Minimum 0 (zero) is coded as 00000000b
• Maximum 255 is coded as 11111111b

• Signed Byte
• twos complement code is most common
• only 7 bits are used for the magnitude
• Minimum -128 is coded as 10000000b
• Maximum 127 is coded as 01111111b
• Zero is 00000000b

9
Understanding Twos Complement Code

• Codes for positive numbers are identical to
  unsigned codes
• at least one leading zero is required
• Negative numbers are coded by adding 256 and
  writing the unsigned code of the result

• Code for 107
• 107d 1101011b
• code 01101011 (6Bh)
• Code for -107
• -107 256 149
• 149d 10010101b
• code 10010101b (95h)
• What is 6Bh 95h?

10
Twos Complement Sign Change

• The 2s Complement codes for x and -x add to a
  power of 2
• 8-bit code c(-c)28
• 16-bit code c(-c)216
• (-c)2n-c(2n-1)-c1
• Note that (2n-1) is 1111..1b, making subtraction
  a cinch!
• Roles of (-c) and c can be reversed

• Change Sign Rule I
• Subtract from 2n
• Change Sign Rule II
• Flip all the bits
• Add 1
• Change Sign Rule III
• Scan right to left to the first bit with value 1
• Flip all bits to its left

11
Decoding Twos Complement Data

• 0001011010111000b
• Leading zero indicates value is positive
• simply convert to decimal
• 15B8h 5560d
• Data 5560

• 1110100101001000b
• Leading one indicates value is negative
• apply change sign rule, then convert to decimal
• 0001011010111000b
• last 4 bits unchanged
• 15B8h 5560d
• Data -5560

Dont forget the minus!
12
Storage Sizes and Ranges

• Unsigned Integer
• Byte
• 0 to 255
• Word
• 0 to 65,535
• Doubleword
• 0 to 4,294,967,295
• Quadword
• 0 to 18,446,744,073,709,551,615

• Signed (2s Complement Code)
• Byte
• -128 to 127
• Word
• -32,768 to 32,767
• Doubleword
• -2,147,483,648 to 2,147,483,647
• Quadword
• -9,223,372,036,854,775,808 to 9,223,372,036,854,77
  5,807

13
Basic Addition

• Numbers can be added in any number base
• Use the same algorithm you practiced in second
  grade!
• Binary Example
• c c c c
• 10101
• 1111
• 100100

• Hex Example
• c c
• 3CF02
• 435C9
• 804CB
• A carry occurs when the sum is sixteen or greater

Watch Carries
00 0 01 1 10 1 11 10 111 11
14
Basic Subtraction

• Just remember how to borrow.
• Take into account the place values!
• Hex example
• b b
• FCF02
• -435C9
• B9939

• Binary subtraction is easy, but tricky!
• Borrows are worth two but sometimes you need to
  borrow from afar!
• b b b b b
• 1101000011
• - 1101001
• 1011011010

A borrow adds sixteen
15
Twos Complement Arithmetic

• Adding or subtracting integers represented by
  n-bit codes gives an n-bit result
• These operations use the coded representations,
  treating them as unsigned values
• Addition is accomplished by adding the codes,
  ignoring any final carry
• Subtraction is simplest if you change the sign
  and add

16
Example of Twos Complement Arithmetic

• 111100001101
• 011101101001
• 011001110110
• These are 12-bit, twos complement codes
• A negative and positive quantity are being added
• The result is positive
• Decode and check!

• F23C F23C
• - 2CF0 D310
• ? C54C
• These are 16-bit, twos complement codes
• The equivalent addition problem adds two
  negatives and the result is negative
• Decode and check!

17
Example Decoded!

• Binary Dec
• 111100001101 -243
• 011101101001 1897
• 011001110110 1654
• Decoding the negative number
• 111100001101
• -(000011110011)
• -243

• Hex Decimal
• F23C (-3524)
• - 2CF0 - 11504
• C54C - 15028
• Decoding F23C
• 1111001000111100
• -(0000110111000100)
• -3524