1. Is this a positive or a negative number?
1101 1111 0111 1110 1101 1101 0010 0011 1000 0000 0000 1001 0000 0101 1011 0000 is the binary representation of a negative integer, on 64 bits (8 Bytes).
In a signed binary, the first bit (the leftmost) is reserved for the sign,
1 = negative, 0 = positive. This bit does not count when calculating the absolute value.
2. Construct the unsigned binary number.
Exclude the first bit (the leftmost), that is reserved for the sign:
1101 1111 0111 1110 1101 1101 0010 0011 1000 0000 0000 1001 0000 0101 1011 0000 = 101 1111 0111 1110 1101 1101 0010 0011 1000 0000 0000 1001 0000 0101 1011 0000
3. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
262
1 261
0 260
1 259
1 258
1 257
1 256
1 255
0 254
1 253
1 252
1 251
1 250
1 249
1 248
0 247
1 246
1 245
0 244
1 243
1 242
1 241
0 240
1 239
0 238
0 237
1 236
0 235
0 234
0 233
1 232
1 231
1 230
0 229
0 228
0 227
0 226
0 225
0 224
0 223
0 222
0 221
0 220
0 219
1 218
0 217
0 216
1 215
0 214
0 213
0 212
0 211
0 210
1 29
0 28
1 27
1 26
0 25
1 24
1 23
0 22
0 21
0 20
0
4. Multiply each bit by its corresponding power of 2 and add all the terms up.
101 1111 0111 1110 1101 1101 0010 0011 1000 0000 0000 1001 0000 0101 1011 0000(2) =
(1 × 262 + 0 × 261 + 1 × 260 + 1 × 259 + 1 × 258 + 1 × 257 + 1 × 256 + 0 × 255 + 1 × 254 + 1 × 253 + 1 × 252 + 1 × 251 + 1 × 250 + 1 × 249 + 0 × 248 + 1 × 247 + 1 × 246 + 0 × 245 + 1 × 244 + 1 × 243 + 1 × 242 + 0 × 241 + 1 × 240 + 0 × 239 + 0 × 238 + 1 × 237 + 0 × 236 + 0 × 235 + 0 × 234 + 1 × 233 + 1 × 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 + 1 × 219 + 0 × 218 + 0 × 217 + 1 × 216 + 0 × 215 + 0 × 214 + 0 × 213 + 0 × 212 + 0 × 211 + 1 × 210 + 0 × 29 + 1 × 28 + 1 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 0 × 20)(10) =
(4 611 686 018 427 387 904 + 0 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 0 + 18 014 398 509 481 984 + 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 + 0 + 140 737 488 355 328 + 70 368 744 177 664 + 0 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 0 + 1 099 511 627 776 + 0 + 0 + 137 438 953 472 + 0 + 0 + 0 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 524 288 + 0 + 0 + 65 536 + 0 + 0 + 0 + 0 + 0 + 1 024 + 0 + 256 + 128 + 0 + 32 + 16 + 0 + 0 + 0 + 0)(10) =
(4 611 686 018 427 387 904 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 18 014 398 509 481 984 + 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 + 140 737 488 355 328 + 70 368 744 177 664 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 1 099 511 627 776 + 137 438 953 472 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 524 288 + 65 536 + 1 024 + 256 + 128 + 32 + 16)(10) =
6 881 180 425 210 365 360(10)
5. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1101 1111 0111 1110 1101 1101 0010 0011 1000 0000 0000 1001 0000 0101 1011 0000(2) = -6 881 180 425 210 365 360(10)
The number 1101 1111 0111 1110 1101 1101 0010 0011 1000 0000 0000 1001 0000 0101 1011 0000(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
1101 1111 0111 1110 1101 1101 0010 0011 1000 0000 0000 1001 0000 0101 1011 0000(2) = -6 881 180 425 210 365 360(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.