1. Is this a positive or a negative number?
0000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1100 is the binary representation of a positive 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:
0000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1100 = 000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1100
3. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
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
0 242
0 241
0 240
0 239
0 238
0 237
1 236
1 235
1 234
1 233
1 232
0 231
1 230
0 229
1 228
0 227
0 226
0 225
1 224
1 223
1 222
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
0 25
0 24
0 23
1 22
1 21
0 20
0
4. Multiply each bit by its corresponding power of 2 and add all the terms up.
000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1100(2) =
(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 + 0 × 242 + 0 × 241 + 0 × 240 + 0 × 239 + 0 × 238 + 1 × 237 + 1 × 236 + 1 × 235 + 1 × 234 + 1 × 233 + 0 × 232 + 1 × 231 + 0 × 230 + 1 × 229 + 0 × 228 + 0 × 227 + 0 × 226 + 1 × 225 + 1 × 224 + 1 × 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 + 0 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 0 × 20)(10) =
(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 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 0 + 2 147 483 648 + 0 + 536 870 912 + 0 + 0 + 0 + 33 554 432 + 16 777 216 + 8 388 608 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 8 + 4 + 0 + 0)(10) =
(137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 2 147 483 648 + 536 870 912 + 33 554 432 + 16 777 216 + 8 388 608 + 8 + 4)(10) =
269 031 047 180(10)
5. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1100(2) = 269 031 047 180(10)
The number 0000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1100(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
0000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1100(2) = 269 031 047 180(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.