1. Is this a positive or a negative number?
0000 0000 0000 0000 0000 1010 0111 1100 0101 0000 0111 1000 0000 1111 1010 1011 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
0 254
0 253
0 252
0 251
0 250
0 249
0 248
0 247
0 246
0 245
0 244
0 243
1 242
0 241
1 240
0 239
0 238
1 237
1 236
1 235
1 234
1 233
0 232
0 231
0 230
1 229
0 228
1 227
0 226
0 225
0 224
0 223
0 222
1 221
1 220
1 219
1 218
0 217
0 216
0 215
0 214
0 213
0 212
0 211
1 210
1 29
1 28
1 27
1 26
0 25
1 24
0 23
1 22
0 21
1 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 0000 0000 0000 0000 1010 0111 1100 0101 0000 0111 1000 0000 1111 1010 1011(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 + 1 × 243 + 0 × 242 + 1 × 241 + 0 × 240 + 0 × 239 + 1 × 238 + 1 × 237 + 1 × 236 + 1 × 235 + 1 × 234 + 0 × 233 + 0 × 232 + 0 × 231 + 1 × 230 + 0 × 229 + 1 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 0 × 224 + 0 × 223 + 1 × 222 + 1 × 221 + 1 × 220 + 1 × 219 + 0 × 218 + 0 × 217 + 0 × 216 + 0 × 215 + 0 × 214 + 0 × 213 + 0 × 212 + 1 × 211 + 1 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 1 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 8 796 093 022 208 + 0 + 2 199 023 255 552 + 0 + 0 + 274 877 906 944 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 0 + 0 + 0 + 1 073 741 824 + 0 + 268 435 456 + 0 + 0 + 0 + 0 + 0 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 2 048 + 1 024 + 512 + 256 + 128 + 0 + 32 + 0 + 8 + 0 + 2 + 1)(10) =
(8 796 093 022 208 + 2 199 023 255 552 + 274 877 906 944 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 1 073 741 824 + 268 435 456 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 2 048 + 1 024 + 512 + 256 + 128 + 32 + 8 + 2 + 1)(10) =
11 529 042 268 075(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 1010 0111 1100 0101 0000 0111 1000 0000 1111 1010 1011(2) = 11 529 042 268 075(10)
The signed binary number in two's complement representation 0000 0000 0000 0000 0000 1010 0111 1100 0101 0000 0111 1000 0000 1111 1010 1011(2) converted and written as an integer in decimal system (base ten):
0000 0000 0000 0000 0000 1010 0111 1100 0101 0000 0111 1000 0000 1111 1010 1011(2) = 11 529 042 268 075(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.