In a signed binary in two's complement representation, the first bit (the leftmost) indicates the sign, 1 = negative, 0 = positive.
2. Get the binary representation in one's complement.
* Run this step only if the number is negative *
Note on binary subtraction rules:
11 - 1 = 10; 10 - 1 = 1; 1 - 0 = 1; 1 - 1 = 0.
Subtract 1 from the initial binary number.
1010 1000 0101 0111 0111 0101 0111 1000 - 1 = 1010 1000 0101 0111 0111 0101 0111 0111
3. Get the binary representation of the positive (unsigned) number.
* Run this step only if the number is negative *
Flip all the bits of the signed binary in one's complement representation (reverse the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s:
!(1010 1000 0101 0111 0111 0101 0111 0111) = 0101 0111 1010 1000 1000 1010 1000 1000
4. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
231
0 230
1 229
0 228
1 227
0 226
1 225
1 224
1 223
1 222
0 221
1 220
0 219
1 218
0 217
0 216
0 215
1 214
0 213
0 212
0 211
1 210
0 29
1 28
0 27
1 26
0 25
0 24
0 23
1 22
0 21
0 20
0
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0101 0111 1010 1000 1000 1010 1000 1000(2) =
(0 × 231 + 1 × 230 + 0 × 229 + 1 × 228 + 0 × 227 + 1 × 226 + 1 × 225 + 1 × 224 + 1 × 223 + 0 × 222 + 1 × 221 + 0 × 220 + 1 × 219 + 0 × 218 + 0 × 217 + 0 × 216 + 1 × 215 + 0 × 214 + 0 × 213 + 0 × 212 + 1 × 211 + 0 × 210 + 1 × 29 + 0 × 28 + 1 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 0 × 20)(10) =
(0 + 1 073 741 824 + 0 + 268 435 456 + 0 + 67 108 864 + 33 554 432 + 16 777 216 + 8 388 608 + 0 + 2 097 152 + 0 + 524 288 + 0 + 0 + 0 + 32 768 + 0 + 0 + 0 + 2 048 + 0 + 512 + 0 + 128 + 0 + 0 + 0 + 8 + 0 + 0 + 0)(10) =
(1 073 741 824 + 268 435 456 + 67 108 864 + 33 554 432 + 16 777 216 + 8 388 608 + 2 097 152 + 524 288 + 32 768 + 2 048 + 512 + 128 + 8)(10) =
1 470 663 304(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1010 1000 0101 0111 0111 0101 0111 1000(2) = -1 470 663 304(10)
The signed binary number in two's complement representation 1010 1000 0101 0111 0111 0101 0111 1000(2) converted and written as an integer in decimal system (base ten):
1010 1000 0101 0111 0111 0101 0111 1000(2) = -1 470 663 304(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.