1. Is this a positive or a negative number?
1011 0111 1000 0001 0100 1111 0100 1111 0111 1110 0111 1111 0111 1011 1000 0111 is the binary representation of a negative integer, on 64 bits (8 Bytes).
In a signed binary in one's complement representation, the first bit (the leftmost) indicates the sign, 1 = negative, 0 = positive.
2. 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:
!(1011 0111 1000 0001 0100 1111 0100 1111 0111 1110 0111 1111 0111 1011 1000 0111) = 0100 1000 0111 1110 1011 0000 1011 0000 1000 0001 1000 0000 1000 0100 0111 1000
3. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
263
0 262
1 261
0 260
0 259
1 258
0 257
0 256
0 255
0 254
1 253
1 252
1 251
1 250
1 249
1 248
0 247
1 246
0 245
1 244
1 243
0 242
0 241
0 240
0 239
1 238
0 237
1 236
1 235
0 234
0 233
0 232
0 231
1 230
0 229
0 228
0 227
0 226
0 225
0 224
1 223
1 222
0 221
0 220
0 219
0 218
0 217
0 216
0 215
1 214
0 213
0 212
0 211
0 210
1 29
0 28
0 27
0 26
1 25
1 24
1 23
1 22
0 21
0 20
0
4. Multiply each bit by its corresponding power of 2 and add all the terms up.
0100 1000 0111 1110 1011 0000 1011 0000 1000 0001 1000 0000 1000 0100 0111 1000(2) =
(0 × 263 + 1 × 262 + 0 × 261 + 0 × 260 + 1 × 259 + 0 × 258 + 0 × 257 + 0 × 256 + 0 × 255 + 1 × 254 + 1 × 253 + 1 × 252 + 1 × 251 + 1 × 250 + 1 × 249 + 0 × 248 + 1 × 247 + 0 × 246 + 1 × 245 + 1 × 244 + 0 × 243 + 0 × 242 + 0 × 241 + 0 × 240 + 1 × 239 + 0 × 238 + 1 × 237 + 1 × 236 + 0 × 235 + 0 × 234 + 0 × 233 + 0 × 232 + 1 × 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 + 0 × 218 + 0 × 217 + 0 × 216 + 1 × 215 + 0 × 214 + 0 × 213 + 0 × 212 + 0 × 211 + 1 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 0 × 20)(10) =
(0 + 4 611 686 018 427 387 904 + 0 + 0 + 576 460 752 303 423 488 + 0 + 0 + 0 + 0 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 562 949 953 421 312 + 0 + 140 737 488 355 328 + 0 + 35 184 372 088 832 + 17 592 186 044 416 + 0 + 0 + 0 + 0 + 549 755 813 888 + 0 + 137 438 953 472 + 68 719 476 736 + 0 + 0 + 0 + 0 + 2 147 483 648 + 0 + 0 + 0 + 0 + 0 + 0 + 16 777 216 + 8 388 608 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 32 768 + 0 + 0 + 0 + 0 + 1 024 + 0 + 0 + 0 + 64 + 32 + 16 + 8 + 0 + 0 + 0)(10) =
(4 611 686 018 427 387 904 + 576 460 752 303 423 488 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 562 949 953 421 312 + 140 737 488 355 328 + 35 184 372 088 832 + 17 592 186 044 416 + 549 755 813 888 + 137 438 953 472 + 68 719 476 736 + 2 147 483 648 + 16 777 216 + 8 388 608 + 32 768 + 1 024 + 64 + 32 + 16 + 8)(10) =
5 223 806 889 929 770 104(10)
5. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1011 0111 1000 0001 0100 1111 0100 1111 0111 1110 0111 1111 0111 1011 1000 0111(2) = -5 223 806 889 929 770 104(10)
The signed binary number in one's complement representation 1011 0111 1000 0001 0100 1111 0100 1111 0111 1110 0111 1111 0111 1011 1000 0111(2) converted and written as an integer in decimal system (base ten):
1011 0111 1000 0001 0100 1111 0100 1111 0111 1110 0111 1111 0111 1011 1000 0111(2) = -5 223 806 889 929 770 104(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.