1. Is this a positive or a negative number?
1111 0010 1111 0011 1111 0000 1111 0110 1111 1110 1111 1100 1111 1001 0111 0001 is the binary representation of a negative 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.
1111 0010 1111 0011 1111 0000 1111 0110 1111 1110 1111 1100 1111 1001 0111 0001 - 1 = 1111 0010 1111 0011 1111 0000 1111 0110 1111 1110 1111 1100 1111 1001 0111 0000
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:
!(1111 0010 1111 0011 1111 0000 1111 0110 1111 1110 1111 1100 1111 1001 0111 0000) = 0000 1101 0000 1100 0000 1111 0000 1001 0000 0001 0000 0011 0000 0110 1000 1111
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
1 258
1 257
0 256
1 255
0 254
0 253
0 252
0 251
1 250
1 249
0 248
0 247
0 246
0 245
0 244
0 243
1 242
1 241
1 240
1 239
0 238
0 237
0 236
0 235
1 234
0 233
0 232
1 231
0 230
0 229
0 228
0 227
0 226
0 225
0 224
1 223
0 222
0 221
0 220
0 219
0 218
0 217
1 216
1 215
0 214
0 213
0 212
0 211
0 210
1 29
1 28
0 27
1 26
0 25
0 24
0 23
1 22
1 21
1 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 1101 0000 1100 0000 1111 0000 1001 0000 0001 0000 0011 0000 0110 1000 1111(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 0 × 260 + 1 × 259 + 1 × 258 + 0 × 257 + 1 × 256 + 0 × 255 + 0 × 254 + 0 × 253 + 0 × 252 + 1 × 251 + 1 × 250 + 0 × 249 + 0 × 248 + 0 × 247 + 0 × 246 + 0 × 245 + 0 × 244 + 1 × 243 + 1 × 242 + 1 × 241 + 1 × 240 + 0 × 239 + 0 × 238 + 0 × 237 + 0 × 236 + 1 × 235 + 0 × 234 + 0 × 233 + 1 × 232 + 0 × 231 + 0 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 1 × 224 + 0 × 223 + 0 × 222 + 0 × 221 + 0 × 220 + 0 × 219 + 0 × 218 + 1 × 217 + 1 × 216 + 0 × 215 + 0 × 214 + 0 × 213 + 0 × 212 + 0 × 211 + 1 × 210 + 1 × 29 + 0 × 28 + 1 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20)(10) =
(0 + 0 + 0 + 0 + 576 460 752 303 423 488 + 288 230 376 151 711 744 + 0 + 72 057 594 037 927 936 + 0 + 0 + 0 + 0 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 0 + 0 + 0 + 0 + 0 + 0 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 0 + 0 + 0 + 0 + 34 359 738 368 + 0 + 0 + 4 294 967 296 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 16 777 216 + 0 + 0 + 0 + 0 + 0 + 0 + 131 072 + 65 536 + 0 + 0 + 0 + 0 + 0 + 1 024 + 512 + 0 + 128 + 0 + 0 + 0 + 8 + 4 + 2 + 1)(10) =
(576 460 752 303 423 488 + 288 230 376 151 711 744 + 72 057 594 037 927 936 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 34 359 738 368 + 4 294 967 296 + 16 777 216 + 131 072 + 65 536 + 1 024 + 512 + 128 + 8 + 4 + 2 + 1)(10) =
940 142 953 559 688 847(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1111 0010 1111 0011 1111 0000 1111 0110 1111 1110 1111 1100 1111 1001 0111 0001(2) = -940 142 953 559 688 847(10)
The signed binary number in two's complement representation 1111 0010 1111 0011 1111 0000 1111 0110 1111 1110 1111 1100 1111 1001 0111 0001(2) converted and written as an integer in decimal system (base ten):
1111 0010 1111 0011 1111 0000 1111 0110 1111 1110 1111 1100 1111 1001 0111 0001(2) = -940 142 953 559 688 847(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.