signed-integer

2’s complement

0 1 2 3 -4 -3 -2 -1 
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• naming rationale: for n-bit integer, sum (x + -x) = 2^n, which is all 0s in n-bit.
  • -x is a complement of x with respect to 2^n
• representation: sX = -s * 2^(n - 1) + X
  • first bit represents -2^(n - 1) (s)
  • all other bits are normal binary representation (X)
• conversion: 0000 1011
  • flip all bits 1111 0100
  • then add 1 1111 0101

  0000 1011 = 15
  1111 0101 = -15 = -2^7 + 117 (=111 0101)
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1|0000 0000 = 0 (= 2^8 in unsigned)

• Pros:
  • arithmetic behaves normally (since there’s only 1 zero)