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.
* Not the case - the number is positive *
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:
* Not the case - the number is positive *
4. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
263
0 262
0 261
0 260
0 259
0 258
0 257
0 256
0 255
0 254
0 253
0 252
0 251
0 250
0 249
0 248
1 247
0 246
0 245
0 244
0 243
0 242
1 241
0 240
1 239
0 238
1 237
0 236
1 235
0 234
0 233
0 232
0 231
0 230
0 229
0 228
0 227
0 226
0 225
0 224
0 223
0 222
0 221
0 220
0 219
0 218
0 217
0 216
1 215
0 214
0 213
0 212
0 211
0 210
0 29
1 28
0 27
0 26
1 25
0 24
0 23
1 22
0 21
1 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 0000 0000 0001 0000 0101 0101 0000 0000 0000 0000 0001 0000 0010 0100 1011(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 0 × 257 + 0 × 256 + 0 × 255 + 0 × 254 + 0 × 253 + 0 × 252 + 0 × 251 + 0 × 250 + 0 × 249 + 1 × 248 + 0 × 247 + 0 × 246 + 0 × 245 + 0 × 244 + 0 × 243 + 1 × 242 + 0 × 241 + 1 × 240 + 0 × 239 + 1 × 238 + 0 × 237 + 1 × 236 + 0 × 235 + 0 × 234 + 0 × 233 + 0 × 232 + 0 × 231 + 0 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 0 × 224 + 0 × 223 + 0 × 222 + 0 × 221 + 0 × 220 + 0 × 219 + 0 × 218 + 0 × 217 + 1 × 216 + 0 × 215 + 0 × 214 + 0 × 213 + 0 × 212 + 0 × 211 + 0 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 1 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 1 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 281 474 976 710 656 + 0 + 0 + 0 + 0 + 0 + 4 398 046 511 104 + 0 + 1 099 511 627 776 + 0 + 274 877 906 944 + 0 + 68 719 476 736 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 65 536 + 0 + 0 + 0 + 0 + 0 + 0 + 512 + 0 + 0 + 64 + 0 + 0 + 8 + 0 + 2 + 1)(10) =
(281 474 976 710 656 + 4 398 046 511 104 + 1 099 511 627 776 + 274 877 906 944 + 68 719 476 736 + 65 536 + 512 + 64 + 8 + 2 + 1)(10) =
287 316 132 299 339(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0000 0000 0001 0000 0101 0101 0000 0000 0000 0000 0001 0000 0010 0100 1011(2) = 287 316 132 299 339(10)
The signed binary number in two's complement representation 0000 0000 0000 0001 0000 0101 0101 0000 0000 0000 0000 0001 0000 0010 0100 1011(2) converted and written as an integer in decimal system (base ten):
0000 0000 0000 0001 0000 0101 0101 0000 0000 0000 0000 0001 0000 0010 0100 1011(2) = 287 316 132 299 339(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.