Todays Agenda

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Todays Agenda

• Representation of Binary Numbers
• Unsigned, Signed Magnitude, 1s Compliment Done
  !!!
• 2s Complement
• Base Conversion
• Binary to Decimal and Decimal to Binary Done !!!
• Decimal to Octal and Octal to Decimal
• Octal to Binary and Binary to Octal
• Binary to Hexadecimal and Hexadecimal to Binary

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Twos Complement Representation

• If number is positive or zero
• Normal binary representation
• If number is negative
• Start with positive number
• Flip every bit (i.e., take the ones complement)
• Then add one
• Example

00101 (5) 01001 (9) 11010 (1s comp)
10110 (1s comp) 1 1 11011 (-5) 10
111 (-9)
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Twos Complement Shortcut

• To take the twos complement of a number
• Copy bits from right to left until (and
  including) the first 1
• Flip remaining bits to the left

011010000 011010000 100101111 (1s
comp) 1 100110000 100110000
(copy)
(flip)
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Advantage of 2s complement

• Twos complement representation developed to
  makecircuits easy for arithmetic.
• For each positive number (X), assign value to its
  negative (-X),such that X (-X) 0 with
  normal addition, ignoring carry out

00101 (5) 01001 (9) 11011 (-5) 10111 (-9)
00000 (0) 00000 (0)
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Example

• 2(-3) ??
• Express -3 in 2s complement form
• 3 ? 00011
• 1s Complement of 3 ? 11100
• 2s Complement of 3 ? 11101
  i.e. -3
• 2 ? 00010
• -3 ? 11101
• 11111
• Look at MSB ? 1 So take 2s complement again
• 00001

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Twos Complement Signed Integers

• MS bit is sign bit it has weight 2n-1.
• Range of an n-bit number -2n-1 through 2n-1 1.
• The most negative number (-2n-1) has no positive
  counterpart.

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What is Base?

• Decimal Number System
• Base is 10
• All numbers are represented by 0 to 9
• Binary Number System
• Base is 2
• All numbers are represented by 0 and 1
• Octal Number System
• Base is 8
• All numbers are represented by 0 to 7
• Inference
• Maximum digit value in any number system is Base
  -1

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Converting Binary (2s C) to Decimal

• If leading bit is one, take twos complement to
  get a positive number.
• Add powers of 2 that have 1 in
  thecorresponding bit positions.
• If original number was negative,add a minus
  sign.

Assuming 8-bit 2s complement numbers.
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Example Binary to Decimal Conversion

• X 01101000two
• Step 1 Leading bit is 0 no need to take 2s
  complement
• X 01101000
• Step 2 Add powers of 2 that have 1 in the
  corresponding bit positions.
• 262523 64328
• X 104ten
• Step 3 Not required (Number is positive)

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More Examples
X 00100111two 25222120
32421 X 39ten
X 11100110two -X 00011010 242321
1682 26ten X -26ten
Assuming 8-bit 2s complement numbers.
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Converting Decimal to Binary (2s C)

• First Method Division
• Find magnitude of decimal number. (Always ive)
• Divide by two remainder is least significant
  bit.
• Keep dividing by two until answer is zero,
  writing remainders from right to left.
• Append a zero as the MS bit.
• If original number was negative, take twos
  complement.

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Example

• X 104ten
• 104/2 52 r 0 bit 0
• 52/2 26 r 0 bit 1
• 26/2 13 r 0 bit 2
• 13/2 06 r 1 bit 3
• 6/2 03 r 0 bit 4
• 3/2 01 r 1 bit 5
• 1/2 0 r 1 bit 6
• X 01101000two

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Converting Decimal to Binary (2s C)

• Second Method Subtract Powers of Two
• Find magnitude of decimal number.
• Subtract largest power of two less than or equal
  to number.
• Put a 1 in the corresponding bit position.
• Keep subtracting until result is zero.
• Append a zero as MS bit if original was
  negative, take twos complement.
• Put a 0 in the other bit positions

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Example

• X 104ten 104 64 40 bit 7
• 40 32 8 bit 6
• 8 8 0 bit 4
• X 01101000two

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Decimal to Octal

• Find magnitude of decimal number. (Always ive)
• Divide by Eight remainder is least significant
  bit.
• Keep dividing by Eight until answer is zero,
  writing remainders from right to left.
• Example Convert 40410 to its octal equivalent.
• Octal to Decimal
• Sum the multiplication of each digit with its
  position value
• Example Convert 6248 to its decimal
  equivalent.
• 6 x 82 2 x 81 4 x 80 40410

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Octal to Binary

• First Method
• First convert Octal number into its Decimal
  Equivalent and then From Decimal to Binary.
• Second Method
• Write 3-bit binary equivalent for each octal
  digit in the given number. Resultant number will
  be binary equivalent.
• Example Binary equivalent of 6248
• 110010100
• Question How to convert a Binary number to its
  Octal equivalent?

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Hexadecimal Notation

• It is often convenient to write binary (base-2)
  numbers as hexadecimal (base-16) numbers instead.
• Why?
• Fewer digits -- four bits per hex digit
• Less error prone -- easy to corrupt long string
  of 1s and 0s

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Converting from Binary to Hexadecimal

• Every four bits is a hex digit.
• Start grouping from right-hand side
• Example

011101010001111010011010111
7
D
4
F
8
A
3
This is not a new machine representation,just a
convenient way to write the number.
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Exercise?

• How we convert a Hexadecimal Number into Binary
  Number?
• Try these
• ADF6
• DB69
• CA36
• 5678
• DABC