Review Twos complement

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Review Twos complement
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Negate a Binary Number in 2C

• Negation means to find the corresponding negative
  number with the same absolute value (in decimal
  the Negation of 10 is 10)
• The method is the same whether you negate a
  positive to a negative or vice versa and doing it
  twice should give you the same result -(-a) a
• Method
• Invert the binary number (flip all bits)
• Inverse (01001101) 10110010
• Add to the result binary 1 (append 0s on the
  left)
• 10110010 1101100100000000110110011

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Convert Decimal to 2C

• Decimal to Binary (word length k 8)
• If positive Fraction number into a sum of powers
  of 2
• bin(13)841232220 00001101
• If Negative Fraction the negated number and
  negate binary number (invert and add 1)
• bin(-13) Negate(bin(13)) Negate(00001101)
• Invert(00001101)000000011111001
  000000001
• 11110011

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Convert 2C to Decimal

• Binary to Decimal (word length k8)
• If most significant bit (left most) is 0 the
  number is positive Sum the powers of 2
• Dec(00101101)027 026125 024 123 122
  021 120
• 112806413201618
  140210
• 173
• If most significant bit (left most) is 1 the
  number is negative, that means you can not just
  add the powers of 2
• Find the decimal value of the negated binary
  number and negate it back in decimal (putting a
  in front)
• Dec(10101101) - (Dec(Negate(10101101))
• - (Dec(Invert(10101101
  )0000001))
• - (Dec(01010010000000
  1))
• - (Dec(01010011))
  
• - now it is a positive number and we can sum
  the powers of 2!
• -(641621) - 83

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Operations on 2C

• Addition bitwise including carry as in decimal
• Negation Invert and add 1
• Subtraction Addition of Negated operand
• A B A ( B)
• Multiplication by 2 Shift left
• 00101101 2 01011010
• Division by 2 Shift right
• 00101101 2 00010110

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Issues Binary Numbers

• Signed Extension
• Sometimes we are lazy and dont write out all
  bits. In 2C you can only add 0s to the left if
  the number is positive, otherwise you make a
  negative number positive
• -3 1101 for k4
• 11111101 for k8
• For negative number you have to add 1sto the
  left!
• Overflow
• If the result of an addition/subtraction is
  bigger than the biggest or smaller than the
  smallest possible number of word length k
• Can never happen if you add a positive and a
  negative number

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Hexadecimal Numbers

• Same principle as decimal and binary but with a
  base of 16
• Number has a little x to the side xF5
• Motivation Word size is either 32 or 64, that
  are very long binary numbers, in Hex you can save
  space
• Conversion from Hex to binary is simple in blocks
  of 4(this works because 1624)
• x5B2F 0101 1011 0010 1111
• To decimal (if positive)
• 5163 11162 2161 15160
• 20480 2816 32 15
• 23343

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LOGICAL OPERATORS

• AND is only 1 if all inputs are 1
• OR is 1 if any of the inputs is 1
• XOR is 1 if either input is 1
• NOT is the inverse (flip the bit)

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Logic on Words and Multiple Input

• 00101101
• AND 11101010
• 00101000
• 00101101
• 10011000
• AND 11101010
• 00001000

00101101 OR 11101010 11101111
00101101 10011000 OR 11101010 11111111
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Text Data ASCII Code

• ASCII is a 7 bit encoding scheme for all keyboard
  symbols and some additional ones
• Digits x30 to x39
• Uppercase x41 to x5A
• Lowercase x61 to x7A
• Conversion Table
• Question How can you mathematically convert a
  lowercase letter into uppercase?