1. Is this a positive or a negative number?
0000 0000 1011 1111 1111 0000 0001 0101 0101 1010 1010 1010 1010 1011 0000 0000 is the binary representation of a positive integer, on 64 bits (8 Bytes).
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
1 254
0 253
1 252
1 251
1 250
1 249
1 248
1 247
1 246
1 245
1 244
1 243
0 242
0 241
0 240
0 239
0 238
0 237
0 236
1 235
0 234
1 233
0 232
1 231
0 230
1 229
0 228
1 227
1 226
0 225
1 224
0 223
1 222
0 221
1 220
0 219
1 218
0 217
1 216
0 215
1 214
0 213
1 212
0 211
1 210
0 29
1 28
1 27
0 26
0 25
0 24
0 23
0 22
0 21
0 20
0
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 0000 1011 1111 1111 0000 0001 0101 0101 1010 1010 1010 1010 1011 0000 0000(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 0 × 257 + 0 × 256 + 1 × 255 + 0 × 254 + 1 × 253 + 1 × 252 + 1 × 251 + 1 × 250 + 1 × 249 + 1 × 248 + 1 × 247 + 1 × 246 + 1 × 245 + 1 × 244 + 0 × 243 + 0 × 242 + 0 × 241 + 0 × 240 + 0 × 239 + 0 × 238 + 0 × 237 + 1 × 236 + 0 × 235 + 1 × 234 + 0 × 233 + 1 × 232 + 0 × 231 + 1 × 230 + 0 × 229 + 1 × 228 + 1 × 227 + 0 × 226 + 1 × 225 + 0 × 224 + 1 × 223 + 0 × 222 + 1 × 221 + 0 × 220 + 1 × 219 + 0 × 218 + 1 × 217 + 0 × 216 + 1 × 215 + 0 × 214 + 1 × 213 + 0 × 212 + 1 × 211 + 0 × 210 + 1 × 29 + 1 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 0 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 36 028 797 018 963 968 + 0 + 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 + 281 474 976 710 656 + 140 737 488 355 328 + 70 368 744 177 664 + 35 184 372 088 832 + 17 592 186 044 416 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 68 719 476 736 + 0 + 17 179 869 184 + 0 + 4 294 967 296 + 0 + 1 073 741 824 + 0 + 268 435 456 + 134 217 728 + 0 + 33 554 432 + 0 + 8 388 608 + 0 + 2 097 152 + 0 + 524 288 + 0 + 131 072 + 0 + 32 768 + 0 + 8 192 + 0 + 2 048 + 0 + 512 + 256 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0)(10) =
(36 028 797 018 963 968 + 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 + 281 474 976 710 656 + 140 737 488 355 328 + 70 368 744 177 664 + 35 184 372 088 832 + 17 592 186 044 416 + 68 719 476 736 + 17 179 869 184 + 4 294 967 296 + 1 073 741 824 + 268 435 456 + 134 217 728 + 33 554 432 + 8 388 608 + 2 097 152 + 524 288 + 131 072 + 32 768 + 8 192 + 2 048 + 512 + 256)(10) =
54 025 695 057 849 088(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0000 1011 1111 1111 0000 0001 0101 0101 1010 1010 1010 1010 1011 0000 0000(2) = 54 025 695 057 849 088(10)
The signed binary number in two's complement representation 0000 0000 1011 1111 1111 0000 0001 0101 0101 1010 1010 1010 1010 1011 0000 0000(2) converted and written as an integer in decimal system (base ten):
0000 0000 1011 1111 1111 0000 0001 0101 0101 1010 1010 1010 1010 1011 0000 0000(2) = 54 025 695 057 849 088(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.