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
1 245
0 244
1 243
0 242
1 241
0 240
1 239
0 238
1 237
0 236
1 235
0 234
1 233
0 232
1 231
0 230
1 229
0 228
1 227
0 226
1 225
0 224
1 223
0 222
1 221
0 220
0 219
1 218
1 217
0 216
1 215
1 214
0 213
1 212
0 211
1 210
0 29
0 28
0 27
0 26
1 25
1 24
0 23
1 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 0101 0101 0101 0101 0101 0101 0100 1101 1010 1000 0110 1100(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 + 1 × 246 + 0 × 245 + 1 × 244 + 0 × 243 + 1 × 242 + 0 × 241 + 1 × 240 + 0 × 239 + 1 × 238 + 0 × 237 + 1 × 236 + 0 × 235 + 1 × 234 + 0 × 233 + 1 × 232 + 0 × 231 + 1 × 230 + 0 × 229 + 1 × 228 + 0 × 227 + 1 × 226 + 0 × 225 + 1 × 224 + 0 × 223 + 1 × 222 + 0 × 221 + 0 × 220 + 1 × 219 + 1 × 218 + 0 × 217 + 1 × 216 + 1 × 215 + 0 × 214 + 1 × 213 + 0 × 212 + 1 × 211 + 0 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 1 × 26 + 1 × 25 + 0 × 24 + 1 × 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 + 70 368 744 177 664 + 0 + 17 592 186 044 416 + 0 + 4 398 046 511 104 + 0 + 1 099 511 627 776 + 0 + 274 877 906 944 + 0 + 68 719 476 736 + 0 + 17 179 869 184 + 0 + 4 294 967 296 + 0 + 1 073 741 824 + 0 + 268 435 456 + 0 + 67 108 864 + 0 + 16 777 216 + 0 + 4 194 304 + 0 + 0 + 524 288 + 262 144 + 0 + 65 536 + 32 768 + 0 + 8 192 + 0 + 2 048 + 0 + 0 + 0 + 0 + 64 + 32 + 0 + 8 + 4 + 0 + 0)(10) =
(70 368 744 177 664 + 17 592 186 044 416 + 4 398 046 511 104 + 1 099 511 627 776 + 274 877 906 944 + 68 719 476 736 + 17 179 869 184 + 4 294 967 296 + 1 073 741 824 + 268 435 456 + 67 108 864 + 16 777 216 + 4 194 304 + 524 288 + 262 144 + 65 536 + 32 768 + 8 192 + 2 048 + 64 + 32 + 8 + 4)(10) =
93 824 991 733 868(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0000 0000 0000 0101 0101 0101 0101 0101 0101 0100 1101 1010 1000 0110 1100(2) = 93 824 991 733 868(10)
The signed binary number in two's complement representation 0000 0000 0000 0000 0101 0101 0101 0101 0101 0101 0100 1101 1010 1000 0110 1100(2) converted and written as an integer in decimal system (base ten):
0000 0000 0000 0000 0101 0101 0101 0101 0101 0101 0100 1101 1010 1000 0110 1100(2) = 93 824 991 733 868(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.