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Why does 2’s complement arithmetic work?

COMP 40: Machine Structure and Assembly Language Programming (Fall 2013). Why does 2’s complement arithmetic work?. Noah Mendelsohn Tufts University Email: noah@cs.tufts.edu Web: http://www.cs.tufts.edu/~noah. Goals for this presentation.

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Why does 2’s complement arithmetic work?

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  1. COMP 40: Machine Structure and Assembly Language Programming (Fall 2013) Why does 2’s complement arithmetic work? Noah Mendelsohn Tufts UniversityEmail: noah@cs.tufts.edu Web: http://www.cs.tufts.edu/~noah

  2. Goals for this presentation • Develop your intuitionabout why 2’s complement arithmetic works

  3. The problem: • I get why: 2 is encoded as 00000010 • I have no clue why: -2 is encoded as 11111110

  4. Odometer ArithmeticModular Number Systems

  5. Modular arithmetic What happens when this goes past 999999? 999999 + 1 = 0 So, what should 0-1 be?

  6. Modular arithmetic – mod 100 0 99 1 98 2 75 50

  7. Modular arithmetic – mod 256 255 0 1 254 2 192 128

  8. Unsigned modular arithmetic – mod 256 00000000 11111111 255 0 00000001 1 254 11111110 00000010 2 192 11000000 128 10000000

  9. Signed modular arithmetic – mod 256 00000000 11111111 00000001 -1 0 1 11111110 -2 00000010 2 -64 11000000 -128 10000000

  10. The 2s complement magic The hardware does the same thing, only the interpretatation is different!

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