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

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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 UniversityEmail: [email protected]

Web: http://www.cs.tufts.edu/~noah


Goals for this presentation
Goals for this presentation

  • Develop your intuitionabout why 2’s complement arithmetic works


The problem
The problem:

  • I get why:

    2 is encoded as 00000010

  • I have no clue why:

    -2 is encoded as 11111110


Odometer arithmetic modular number systems
Odometer ArithmeticModular Number Systems


Modular arithmetic
Modular arithmetic

What happens when this goes past 999999?

999999 + 1 = 0

So, what should 0-1 be?



Modular arithmetic mod 256
Modular arithmetic – mod 256

255

0

1

254

2

192

128


Unsigned modular arithmetic mod 256
Unsigned modular arithmetic – mod 256

00000000

11111111

255

0

00000001

1

254

11111110

00000010

2

192

11000000

128

10000000


Signed m odular arithmetic mod 256
Signed modular arithmetic – mod 256

00000000

11111111

00000001

-1

0

1

11111110

-2

00000010

2

-64

11000000

-128

10000000


The 2s complement magic
The 2s complement magic

The hardware does the same thing, only the interpretatation is different!


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