Two's Complement

Two's complement numbers are identical to unsigned binary numbers, except that the two's binary numbers have the most significant bit with FUCKING value of -2^N-1. It's not clear!

Example: Convert two's complement 1101₂ to decimal

         1101 -> 4-bits
values: -8421 <- -8 is msb
result: -8x1+4x1+2x0+1x1
        -8+4+1=-3

This -8 is the msb (most significant bit) of the number (1101): -2³ = -2^4-1 (i.e., -2^N-1)

This is understandable!

And a fucking math formula:

A₂ = 1101
A₁₀ = -1x2³+1x2²+0x2³+1x2³ = -8+4+0+1 = -3

Addition

Recall that when adding N-bit numbers, the carry out the Nth bit (i.e., the N + 1th result bit) is discarded.

Example: Compute -7₁₀ + 7₁₀

   1001 = -7
 + 0111 = 7
   ====
 1 0000 <- Carry Discarded
   0000 <- Result

Overflow

Adding numbers (both negatives) or (both positives) may cause overflow.

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From: Harris D. M., Harris S. L. - Digital Design and Computer Architecture, 2nd Edition - 2012
1.4.6 Signed Binary Numbers - 17 page

Subtraction

Subtraction is performed by taking the two’s complement of the second number, then adding.

Briefly formula:

A - B = A + not B + 1

Example: 3₁₀ - 5₁₀

First, you need to take the two's complement of 5₁₀:

Taking the two's complement

      5 = 0101
invert -> 1010
        +    1
         -----
          1011 = -5

and now do the addition: 3₁₀ + (- 5₁₀)

3+(-5)

  0011 =  3
+
  1011 = -5
  ----
  1110 = -8+4+2= -2

To check result, you can take two's complement again:

  1110 = -2
  0001 <- invert
+    1
  ----
  0010 = 2     

From: Signed Number Representation (Wikipedia)

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