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
1 256
0 255
1 254
0 253
0 252
0 251
1 250
1 249
1 248
1 247
0 246
1 245
0 244
1 243
1 242
1 241
0 240
0 239
0 238
0 237
1 236
0 235
1 234
0 233
0 232
0 231
1 230
1 229
1 228
1 227
0 226
1 225
0 224
1 223
1 222
1 221
0 220
0 219
0 218
0 217
0 216
0 215
1 214
0 213
1 212
1 211
1 210
0 29
1 28
0 27
1 26
1 25
0 24
1 23
1 22
0 21
0 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 0010 1000 1111 0101 1100 0010 1000 1111 0101 1100 0000 1011 1010 1101 1001(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 1 × 257 + 0 × 256 + 1 × 255 + 0 × 254 + 0 × 253 + 0 × 252 + 1 × 251 + 1 × 250 + 1 × 249 + 1 × 248 + 0 × 247 + 1 × 246 + 0 × 245 + 1 × 244 + 1 × 243 + 1 × 242 + 0 × 241 + 0 × 240 + 0 × 239 + 0 × 238 + 1 × 237 + 0 × 236 + 1 × 235 + 0 × 234 + 0 × 233 + 0 × 232 + 1 × 231 + 1 × 230 + 1 × 229 + 1 × 228 + 0 × 227 + 1 × 226 + 0 × 225 + 1 × 224 + 1 × 223 + 1 × 222 + 0 × 221 + 0 × 220 + 0 × 219 + 0 × 218 + 0 × 217 + 0 × 216 + 1 × 215 + 0 × 214 + 1 × 213 + 1 × 212 + 1 × 211 + 0 × 210 + 1 × 29 + 0 × 28 + 1 × 27 + 1 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 1 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 144 115 188 075 855 872 + 0 + 36 028 797 018 963 968 + 0 + 0 + 0 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 562 949 953 421 312 + 281 474 976 710 656 + 0 + 70 368 744 177 664 + 0 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 0 + 0 + 0 + 0 + 137 438 953 472 + 0 + 34 359 738 368 + 0 + 0 + 0 + 2 147 483 648 + 1 073 741 824 + 536 870 912 + 268 435 456 + 0 + 67 108 864 + 0 + 16 777 216 + 8 388 608 + 4 194 304 + 0 + 0 + 0 + 0 + 0 + 0 + 32 768 + 0 + 8 192 + 4 096 + 2 048 + 0 + 512 + 0 + 128 + 64 + 0 + 16 + 8 + 0 + 0 + 1)(10) =
(144 115 188 075 855 872 + 36 028 797 018 963 968 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 562 949 953 421 312 + 281 474 976 710 656 + 70 368 744 177 664 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 137 438 953 472 + 34 359 738 368 + 2 147 483 648 + 1 073 741 824 + 536 870 912 + 268 435 456 + 67 108 864 + 16 777 216 + 8 388 608 + 4 194 304 + 32 768 + 8 192 + 4 096 + 2 048 + 512 + 128 + 64 + 16 + 8 + 1)(10) =
184 467 440 736 975 577(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0010 1000 1111 0101 1100 0010 1000 1111 0101 1100 0000 1011 1010 1101 1001(2) = 184 467 440 736 975 577(10)
The signed binary number in two's complement representation 0000 0010 1000 1111 0101 1100 0010 1000 1111 0101 1100 0000 1011 1010 1101 1001(2) converted and written as an integer in decimal system (base ten):
0000 0010 1000 1111 0101 1100 0010 1000 1111 0101 1100 0000 1011 1010 1101 1001(2) = 184 467 440 736 975 577(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.