1. Is this a positive or a negative number?
1000 1000 0101 0101 0111 1001 1111 0001 1001 1010 1000 0000 0010 1101 1001 1011 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:
!(1000 1000 0101 0101 0111 1001 1111 0001 1001 1010 1000 0000 0010 1101 1001 1011) = 0111 0111 1010 1010 1000 0110 0000 1110 0110 0101 0111 1111 1101 0010 0110 0100
3. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
263
0 262
1 261
1 260
1 259
0 258
1 257
1 256
1 255
1 254
0 253
1 252
0 251
1 250
0 249
1 248
0 247
1 246
0 245
0 244
0 243
0 242
1 241
1 240
0 239
0 238
0 237
0 236
0 235
1 234
1 233
1 232
0 231
0 230
1 229
1 228
0 227
0 226
1 225
0 224
1 223
0 222
1 221
1 220
1 219
1 218
1 217
1 216
1 215
1 214
1 213
0 212
1 211
0 210
0 29
1 28
0 27
0 26
1 25
1 24
0 23
0 22
1 21
0 20
0
4. Multiply each bit by its corresponding power of 2 and add all the terms up.
0111 0111 1010 1010 1000 0110 0000 1110 0110 0101 0111 1111 1101 0010 0110 0100(2) =
(0 × 263 + 1 × 262 + 1 × 261 + 1 × 260 + 0 × 259 + 1 × 258 + 1 × 257 + 1 × 256 + 1 × 255 + 0 × 254 + 1 × 253 + 0 × 252 + 1 × 251 + 0 × 250 + 1 × 249 + 0 × 248 + 1 × 247 + 0 × 246 + 0 × 245 + 0 × 244 + 0 × 243 + 1 × 242 + 1 × 241 + 0 × 240 + 0 × 239 + 0 × 238 + 0 × 237 + 0 × 236 + 1 × 235 + 1 × 234 + 1 × 233 + 0 × 232 + 0 × 231 + 1 × 230 + 1 × 229 + 0 × 228 + 0 × 227 + 1 × 226 + 0 × 225 + 1 × 224 + 0 × 223 + 1 × 222 + 1 × 221 + 1 × 220 + 1 × 219 + 1 × 218 + 1 × 217 + 1 × 216 + 1 × 215 + 1 × 214 + 0 × 213 + 1 × 212 + 0 × 211 + 0 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 1 × 26 + 1 × 25 + 0 × 24 + 0 × 23 + 1 × 22 + 0 × 21 + 0 × 20)(10) =
(0 + 4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 0 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 0 + 9 007 199 254 740 992 + 0 + 2 251 799 813 685 248 + 0 + 562 949 953 421 312 + 0 + 140 737 488 355 328 + 0 + 0 + 0 + 0 + 4 398 046 511 104 + 2 199 023 255 552 + 0 + 0 + 0 + 0 + 0 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 0 + 0 + 1 073 741 824 + 536 870 912 + 0 + 0 + 67 108 864 + 0 + 16 777 216 + 0 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 65 536 + 32 768 + 16 384 + 0 + 4 096 + 0 + 0 + 512 + 0 + 0 + 64 + 32 + 0 + 0 + 4 + 0 + 0)(10) =
(4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 9 007 199 254 740 992 + 2 251 799 813 685 248 + 562 949 953 421 312 + 140 737 488 355 328 + 4 398 046 511 104 + 2 199 023 255 552 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 1 073 741 824 + 536 870 912 + 67 108 864 + 16 777 216 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 65 536 + 32 768 + 16 384 + 4 096 + 512 + 64 + 32 + 4)(10) =
8 622 851 832 944 775 780(10)
5. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1000 1000 0101 0101 0111 1001 1111 0001 1001 1010 1000 0000 0010 1101 1001 1011(2) = -8 622 851 832 944 775 780(10)
The signed binary number in one's complement representation 1000 1000 0101 0101 0111 1001 1111 0001 1001 1010 1000 0000 0010 1101 1001 1011(2) converted and written as an integer in decimal system (base ten):
1000 1000 0101 0101 0111 1001 1111 0001 1001 1010 1000 0000 0010 1101 1001 1011(2) = -8 622 851 832 944 775 780(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.