Number Systems: Negative Integers and Floating Point

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Department of Computer and Information
Science,School of Science, IUPUI
CSCI 230
Information Representation
Negative Integer Representation
Dale Roberts, Lecturer IUPUI droberts_at_cs.iupui.edu
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Negative Numbers in Binary

• Four different representation schemes are used
  for negative numbers
• Signed Magnitude
• Left most bit (LMB) is the sign bit
• 0 ? positive ()
• 1 ? negative (-)
• Remaining bits hold absolute magnitude
• Example
• 210 ? 0000 0010b
• -210 ? 1000 0010b
• Q 0000 0000 ?
• 1000 0000 ?

Try, 1000 0100b
-410
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1s Complement

• Ones Complement
• Left most bit is the sign bit
• 0 ? positive ()
• 1 ? negative (-)
• The magnitude is Complemented
• Example
• 210 ? 0 000 0010b
• -210 ? 1 111 1101b
• Exercise try - 410 using 1s Complement
• Q 0000 0000 ?
• 1111 1111 ?

Solution 410 0 000 0100 b -410
111 1011 b
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2s Complement

• 2s Complement
• Sign bit same as above
• Magnitude is Complemented first and a 1 is
  added to the Complemented digits
• Example
• 210 ? 0 000 0010b
• 1s Complement ? 1 111 1101b
• 1
• -210 ? 1 111 1110b

Exercise try -710 using 2s Complement
710 ? 1s Complement ? 1
-710 ?
0000 0111b
1111 1000b
1111 1001b
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2s Complement
Example 7(-3) hint A B A (B) 1

• 710 0000 0111b
• 310 0000 0011b
• 1s complement 1111 1100b
• 2s complement 1111 1101b ? -310
• 7(-3) ? 0000 0111
• 1111 1101

1 1111 111 carry
ignore 1 0000 0100 ? 0000 0100 ? 410
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Three Representation of Signed Integer
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Negative Numbers in Binary (cont.)

• Excess Representation
• For a given fixed number of bits the range is
  remapped such that roughly half the numbers are
  negative and half are positive.
• Example (as left)
• Excess 8 notation for 4 bit numbers
• Binary value 8 excess-8 value
• MSB can be used as a sign bit, but
• If MSB 1, positive number
• If MSB 0, negative number
• Excess Representation is also called bias

Numbers Binary Value Notation Excess 8 Value
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
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Fundamental Data Types

• With vs. without using sign bit
• For a 16 bit binary pattern

2 byte unsigned (Default type is int) 2 byte int
0000 0000 0000 0000 ( ? 0D) 0000 0000 0000 0001 ( ? 1D ) 0000 0000 0000 0010 ( ? 2D ) . 0111 1111 1111 1111 ( ? 32767D ? 215 -1) 1000 0000 0000 0000 ( ? 32768D ? 215) . 1111 1111 1111 1111 (? 216 1) 1000 0000 0000 0000 ( ? -32768D ? - 215 ) 1000 0000 0000 0001 ( ? -32767D ? - 215 1) . 1111 1111 1111 1110 ( ? - 2D ) 1111 1111 1111 1111 ( ? - 1D ) 0000 0000 0000 0000 ( ? 0D ) 0000 0000 0000 0001 ( ? 1D ) 0000 0000 0000 0010 ( ? 2D ) . 0111 1111 1111 1111 ( ? 32767D ? 215 -1)
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Fundamental Data Types

• Four Data Types in C (assume 2s complement, byte
  machine)

Data Type Data Type Abbreviation Size (byte) Range
char char 1 -128 127
char unsigned char 1 0 255
int int 2 or 4 -215 215-1 or -231 231-1
int unsigned int unsigned 2 or 4 0 65535 or 0 232-1
int short int short 2 -32768 32767
int unsigned short int unsigned short 2 0 65535
int long int long 4 -231 231-1
int unsigned long int unsigned long 4 0 232-1
float float 4
double double 8
Note 27 128, 215 32768, 215
2147483648 Complex and double complex are not
available
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Acknowledgements

• These slides where originally prepared by Dr.
  Jeffrey Huang, updated by Dale Roberts.