Hexadecimal

1
Hexadecimal

• Base 16
• Uses 16 symbols to represent all 16 hex digits
• 0 .. 9 and A, B, C, D, E, F (for values 10 to
  15)
• Decimal Hex Binary (4 bits)
• 0 0 0000
• 1 1 0001
• 2 2 0010
• 3 3 0011
• 4 4 0100
• 5 5 0101
• 6 6 0110 (contd.)

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Hexadecimal

• Decimal Hex Binary (4 bits)
• 7 7 0111
• 8 8 1000
• 9 9 1001
• 10 A 1010
• 11 B 1011
• 12 C 1100 (contd.)

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Hexadecimal

• Decimal Hex Binary (4 bits)
• 13 D 1101
• 14 E 1110
• 15 F 1111

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Conversions between Binary and Hexadecimal

• Hexadecimal ? Binary
• Expand each Hex. Symbol to four binary bits
• Examples
• 3 F 9 5 0 C
• 0011 1111 1001 0101 0000 1100

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Binary ? Hexadecimal

• Pack (from right to left) each group of 4-bit
  binary into one Hex. Symbol
• Examples
• 100 1101 11102 1010 0000 00112
• 4 D E16 A 0 316

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In Class Exercises

• Convert the following Binary to Hex.
• 1101101102
• 11010101002
• 1111111112
• Convert the following Hex. To Binary
• 83416
• 50D16
• 7A016

1 B 6 3 5 4 1 F F
1000 0011 0100 0101 0000 1101 0111 1010 0000
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Adding Binary Numbers

• Binary addition facts
• 1 1 0 0
• 1 0 1 0
• 10 1 1 0
• Example 01011100
• 00110110
• 10010010

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In Class Exercises

• Perform the following binary additions
• 10101110 11001100
• 00000011 00110011
• 10110001 11111111
• 11001111 11110001
• 00000001 00001001
• 11010000 11111010

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Representing Negative Values

• Negative values can be represented in Binary for
  use by computers.
• Several different ways of representing negative
  values can be used.
• We will discussed two only
• Sign Magnitude
• Twos Complement

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Sign-Magnitude

• Sign Bit Absolute value
• 1 for negative
• 0 for positive
• Therefore, the above binary represents -5

0
0
1
0
0
0
1
1
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Problem with sign-magnitude

• To perform math., adjustments are needed to
  obtain the correct answers
• 7 00000111
• 5 10000101
• 2 10001100
• Which means -12

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Twos Complement

• negative decimal integer ?
• Twos complement binary number
• 1. Take absolute value of negative decimal
  integer
• 2. Convert it to binary number
• 3. Change all the 1s to 0 s and 0 s to 1s
• (1s complement)
• 4 Add 1 to the result of step 3
• (2s complement)

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2s Complement Examples

• To represent 5 in binary using 2s complement
• ? binary for positive 5 00000101
• ? 1s complement 11111010
• ? 2s complement (adding 1 to result from step
  2) 11111010
• 1
• Answer 11111011

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In Class Exercises

• Represent the following values in binary bits
  using ?Sign-Magnitude and ? Twos Complement
• Sign-Magnitude Twos Complement
• 1810 100100102 111011102
• 7010 110001102 101110102
• 4210 001010102 001010102

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Advantages of using 2s Comp.

• No additional adjustments needed when performing
  arithmetic operations.
• 12 12 00001100
• 5 ? (-5) ? 11111011
• 7 7 100000111
• Overflow 7

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Interpreting Negative Binaries

• Sign-magnitude
• Check the sign bit, combine the sign with the
  absolute value calculated from the rest of the
  bits.
• Twos-Complement
• Check the sign bit, if negative, perform the 2s
  complement to compute the absolute value. If
  positive, perform the regular binary to decimal
  conversion.

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In Class Exercises

• Find the value for the following sign-magnitude
  binary numbers
• 10001001
• 00011100
• Find the values for the following twos
  complement binary numbers
• 10001001
• 00011100

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