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.
* Not the case - the number is positive *
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:
* Not the case - the number is positive *
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
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
1 244
0 243
1 242
0 241
0 240
1 239
1 238
1 237
0 236
0 235
1 234
0 233
0 232
1 231
0 230
1 229
1 228
0 227
1 226
0 225
0 224
1 223
0 222
1 221
0 220
1 219
0 218
0 217
1 216
1 215
0 214
1 213
0 212
1 211
0 210
0 29
1 28
0 27
0 26
1 25
1 24
1 23
0 22
1 21
1 20
0
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 0000 0000 0000 0010 1001 1100 1001 0110 1001 0101 0011 0101 0010 0111 0110(2) =
(0 × 263 + 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 + 1 × 245 + 0 × 244 + 1 × 243 + 0 × 242 + 0 × 241 + 1 × 240 + 1 × 239 + 1 × 238 + 0 × 237 + 0 × 236 + 1 × 235 + 0 × 234 + 0 × 233 + 1 × 232 + 0 × 231 + 1 × 230 + 1 × 229 + 0 × 228 + 1 × 227 + 0 × 226 + 0 × 225 + 1 × 224 + 0 × 223 + 1 × 222 + 0 × 221 + 1 × 220 + 0 × 219 + 0 × 218 + 1 × 217 + 1 × 216 + 0 × 215 + 1 × 214 + 0 × 213 + 1 × 212 + 0 × 211 + 0 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 35 184 372 088 832 + 0 + 8 796 093 022 208 + 0 + 0 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 0 + 0 + 34 359 738 368 + 0 + 0 + 4 294 967 296 + 0 + 1 073 741 824 + 536 870 912 + 0 + 134 217 728 + 0 + 0 + 16 777 216 + 0 + 4 194 304 + 0 + 1 048 576 + 0 + 0 + 131 072 + 65 536 + 0 + 16 384 + 0 + 4 096 + 0 + 0 + 512 + 0 + 0 + 64 + 32 + 16 + 0 + 4 + 2 + 0)(10) =
(35 184 372 088 832 + 8 796 093 022 208 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 34 359 738 368 + 4 294 967 296 + 1 073 741 824 + 536 870 912 + 134 217 728 + 16 777 216 + 4 194 304 + 1 048 576 + 131 072 + 65 536 + 16 384 + 4 096 + 512 + 64 + 32 + 16 + 4 + 2)(10) =
45 945 032 233 590(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0000 0000 0000 0010 1001 1100 1001 0110 1001 0101 0011 0101 0010 0111 0110(2) = 45 945 032 233 590(10)
The signed binary number in two's complement representation 0000 0000 0000 0000 0010 1001 1100 1001 0110 1001 0101 0011 0101 0010 0111 0110(2) converted and written as an integer in decimal system (base ten):
0000 0000 0000 0000 0010 1001 1100 1001 0110 1001 0101 0011 0101 0010 0111 0110(2) = 45 945 032 233 590(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.