1. Is this a positive or a negative number?
0000 0000 0000 0000 0101 0101 1000 0001 0000 0000 0000 0000 0000 0000 0100 0111 is the binary representation of a positive 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:
* Not the case - the number is positive *
3. 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
1 238
0 237
0 236
0 235
0 234
0 233
0 232
1 231
0 230
0 229
0 228
0 227
0 226
0 225
0 224
0 223
0 222
0 221
0 220
0 219
0 218
0 217
0 216
0 215
0 214
0 213
0 212
0 211
0 210
0 29
0 28
0 27
0 26
1 25
0 24
0 23
0 22
1 21
1 20
1
4. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 0000 0000 0000 0101 0101 1000 0001 0000 0000 0000 0000 0000 0000 0100 0111(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 + 1 × 239 + 0 × 238 + 0 × 237 + 0 × 236 + 0 × 235 + 0 × 234 + 0 × 233 + 1 × 232 + 0 × 231 + 0 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 0 × 224 + 0 × 223 + 0 × 222 + 0 × 221 + 0 × 220 + 0 × 219 + 0 × 218 + 0 × 217 + 0 × 216 + 0 × 215 + 0 × 214 + 0 × 213 + 0 × 212 + 0 × 211 + 0 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 1 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 1 × 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 + 549 755 813 888 + 0 + 0 + 0 + 0 + 0 + 0 + 4 294 967 296 + 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 + 64 + 0 + 0 + 0 + 4 + 2 + 1)(10) =
(70 368 744 177 664 + 17 592 186 044 416 + 4 398 046 511 104 + 1 099 511 627 776 + 549 755 813 888 + 4 294 967 296 + 64 + 4 + 2 + 1)(10) =
94 012 539 142 215(10)
5. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0000 0000 0000 0101 0101 1000 0001 0000 0000 0000 0000 0000 0000 0100 0111(2) = 94 012 539 142 215(10)
The signed binary number in one's complement representation 0000 0000 0000 0000 0101 0101 1000 0001 0000 0000 0000 0000 0000 0000 0100 0111(2) converted and written as an integer in decimal system (base ten):
0000 0000 0000 0000 0101 0101 1000 0001 0000 0000 0000 0000 0000 0000 0100 0111(2) = 94 012 539 142 215(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.