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 0000 0000 1000 0101 1001 0101 0000 0000 0000 1011 = 000 0000 0000 0000 0000 0000 0000 0000 1000 0101 1001 0101 0000 0000 0000 1011
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
0 236
0 235
0 234
0 233
0 232
0 231
1 230
0 229
0 228
0 227
0 226
1 225
0 224
1 223
1 222
0 221
0 220
1 219
0 218
1 217
0 216
1 215
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
0 21
1 20
1
4. Multiply each bit by its corresponding power of 2 and add all the terms up.
000 0000 0000 0000 0000 0000 0000 0000 1000 0101 1001 0101 0000 0000 0000 1011(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 + 0 × 237 + 0 × 236 + 0 × 235 + 0 × 234 + 0 × 233 + 0 × 232 + 1 × 231 + 0 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 1 × 226 + 0 × 225 + 1 × 224 + 1 × 223 + 0 × 222 + 0 × 221 + 1 × 220 + 0 × 219 + 1 × 218 + 0 × 217 + 1 × 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 + 0 × 22 + 1 × 21 + 1 × 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 + 0 + 0 + 0 + 0 + 0 + 0 + 2 147 483 648 + 0 + 0 + 0 + 0 + 67 108 864 + 0 + 16 777 216 + 8 388 608 + 0 + 0 + 1 048 576 + 0 + 262 144 + 0 + 65 536 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 8 + 0 + 2 + 1)(10) =
(2 147 483 648 + 67 108 864 + 16 777 216 + 8 388 608 + 1 048 576 + 262 144 + 65 536 + 8 + 2 + 1)(10) =
2 241 134 603(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 0000 0000 1000 0101 1001 0101 0000 0000 0000 1011(2) = 2 241 134 603(10)
The number 0000 0000 0000 0000 0000 0000 0000 0000 1000 0101 1001 0101 0000 0000 0000 1011(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
0000 0000 0000 0000 0000 0000 0000 0000 1000 0101 1001 0101 0000 0000 0000 1011(2) = 2 241 134 603(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.