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Understanding Negative Integer Representation in Binary

Explore different schemes for representing negative numbers in binary, including Signed Magnitude, 1's Complement, 2's Complement, and Excess Representation. Learn how to perform operations and exercises using these representations.

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Understanding Negative Integer Representation in Binary

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  1. Department of Computer and Information Science,School of Science, IUPUI CSCI N305 Information Representation: Negative Integer Representation

  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. 1’s Complement • One’s 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 - 410using 1’s Complement Q: 0000 0000 = ? 1111 1111 = ? Solution: 410 = 0 0000100b -410 = 1 1111011b

  4. 2’s Complement • 2’s Complement • Sign bit same as above • Magnitude is Complemented first and a “1” is added to the Complemented digits • Example: • 210 0 000 0010b • 1’s Complement  1 1111101b • +1 • -210 1 111 1110b • Exercise: try -710using 2’s Complement • 710 • 1’s Complement  • +1 • -710  0000 0111b 11111000b • 1111 1001b

  5. 2’s Complement 710 = 0000 0111b 310 = 0000 0011b 1’s complement 1111 1100b 2’s complement 1111 1101b -310 7+(-3)  0000 0111 + 1111 1101 • Example: 7+(-3) [hint]: A – B = A + (~B) +1 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 Typesin C(assume 2’s 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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