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.
1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0111 1001 1101 1001 0111 0110 - 1 = 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0111 1001 1101 1001 0111 0101
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:
!(1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0111 1001 1101 1001 0111 0101) = 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1000 0110 0010 0110 1000 1010
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
0 247
0 246
0 245
0 244
0 243
0 242
0 241
0 240
0 239
0 238
0 237
0 236
0 235
0 234
0 233
0 232
0 231
0 230
0 229
0 228
0 227
0 226
0 225
0 224
1 223
1 222
0 221
0 220
0 219
0 218
1 217
1 216
0 215
0 214
0 213
1 212
0 211
0 210
1 29
1 28
0 27
1 26
0 25
0 24
0 23
1 22
0 21
1 20
0
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1000 0110 0010 0110 1000 1010(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 + 0 × 248 + 0 × 247 + 0 × 246 + 0 × 245 + 0 × 244 + 0 × 243 + 0 × 242 + 0 × 241 + 0 × 240 + 0 × 239 + 0 × 238 + 0 × 237 + 0 × 236 + 0 × 235 + 0 × 234 + 0 × 233 + 0 × 232 + 0 × 231 + 0 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 1 × 224 + 1 × 223 + 0 × 222 + 0 × 221 + 0 × 220 + 0 × 219 + 1 × 218 + 1 × 217 + 0 × 216 + 0 × 215 + 0 × 214 + 1 × 213 + 0 × 212 + 0 × 211 + 1 × 210 + 1 × 29 + 0 × 28 + 1 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 16 777 216 + 8 388 608 + 0 + 0 + 0 + 0 + 262 144 + 131 072 + 0 + 0 + 0 + 8 192 + 0 + 0 + 1 024 + 512 + 0 + 128 + 0 + 0 + 0 + 8 + 0 + 2 + 0)(10) =
(16 777 216 + 8 388 608 + 262 144 + 131 072 + 8 192 + 1 024 + 512 + 128 + 8 + 2)(10) =
25 568 906(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0111 1001 1101 1001 0111 0110(2) = -25 568 906(10)
The signed binary number in two's complement representation 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0111 1001 1101 1001 0111 0110(2) converted and written as an integer in decimal system (base ten):
1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0111 1001 1101 1001 0111 0110(2) = -25 568 906(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.