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Number Systems: Negative Integers and Floating Point

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Dale Roberts. Department of Computer and Information Science, School of Science, IUPUI ... IUPUI. droberts_at_cs.iupui.edu. Dale Roberts. Negative Numbers in Binary ... – PowerPoint PPT presentation

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Title: Number Systems: Negative Integers and Floating Point


1
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
2
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
3
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
1
4
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
5
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
6
Three Representation of Signed Integer
7
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

8
Fundamental Data Types
  • With vs. without using sign bit
  • For a 16 bit binary pattern

9
Fundamental Data Types
  • Four Data Types in C (assume 2s complement, byte
    machine)

Note 27 128, 215 32768, 215
2147483648 Complex and double complex are not
available
10
Acknowledgements
  • These slides where originally prepared by Dr.
    Jeffrey Huang, updated by Dale Roberts.
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