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
0 247
0 246
0 245
0 244
0 243
0 242
0 241
0 240
0 239
0 238
1 237
1 236
0 235
1 234
0 233
0 232
1 231
1 230
0 229
0 228
1 227
0 226
1 225
0 224
1 223
1 222
1 221
0 220
1 219
1 218
1 217
0 216
1 215
0 214
0 213
1 212
0 211
1 210
0 29
1 28
0 27
0 26
0 25
1 24
0 23
0 22
1 21
0 20
0
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 0000 0000 0000 0000 0000 0110 1001 1001 0101 1101 1101 0010 1010 0010 0100(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 + 1 × 238 + 1 × 237 + 0 × 236 + 1 × 235 + 0 × 234 + 0 × 233 + 1 × 232 + 1 × 231 + 0 × 230 + 0 × 229 + 1 × 228 + 0 × 227 + 1 × 226 + 0 × 225 + 1 × 224 + 1 × 223 + 1 × 222 + 0 × 221 + 1 × 220 + 1 × 219 + 1 × 218 + 0 × 217 + 1 × 216 + 0 × 215 + 0 × 214 + 1 × 213 + 0 × 212 + 1 × 211 + 0 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 0 × 23 + 1 × 22 + 0 × 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 + 274 877 906 944 + 137 438 953 472 + 0 + 34 359 738 368 + 0 + 0 + 4 294 967 296 + 2 147 483 648 + 0 + 0 + 268 435 456 + 0 + 67 108 864 + 0 + 16 777 216 + 8 388 608 + 4 194 304 + 0 + 1 048 576 + 524 288 + 262 144 + 0 + 65 536 + 0 + 0 + 8 192 + 0 + 2 048 + 0 + 512 + 0 + 0 + 0 + 32 + 0 + 0 + 4 + 0 + 0)(10) =
(274 877 906 944 + 137 438 953 472 + 34 359 738 368 + 4 294 967 296 + 2 147 483 648 + 268 435 456 + 67 108 864 + 16 777 216 + 8 388 608 + 4 194 304 + 1 048 576 + 524 288 + 262 144 + 65 536 + 8 192 + 2 048 + 512 + 32 + 4)(10) =
453 485 865 508(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0000 0000 0000 0000 0000 0110 1001 1001 0101 1101 1101 0010 1010 0010 0100(2) = 453 485 865 508(10)
The signed binary number in two's complement representation 0000 0000 0000 0000 0000 0000 0110 1001 1001 0101 1101 1101 0010 1010 0010 0100(2) converted and written as an integer in decimal system (base ten):
0000 0000 0000 0000 0000 0000 0110 1001 1001 0101 1101 1101 0010 1010 0010 0100(2) = 453 485 865 508(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.