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.
1000 1000 0000 0101 0100 0011 1000 1000 1111 1111 1111 1111 1111 1111 1101 0011 - 1 = 1000 1000 0000 0101 0100 0011 1000 1000 1111 1111 1111 1111 1111 1111 1101 0010
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:
!(1000 1000 0000 0101 0100 0011 1000 1000 1111 1111 1111 1111 1111 1111 1101 0010) = 0111 0111 1111 1010 1011 1100 0111 0111 0000 0000 0000 0000 0000 0000 0010 1101
4. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
263
0 262
1 261
1 260
1 259
0 258
1 257
1 256
1 255
1 254
1 253
1 252
1 251
1 250
0 249
1 248
0 247
1 246
0 245
1 244
1 243
1 242
1 241
0 240
0 239
0 238
1 237
1 236
1 235
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
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
1 24
0 23
1 22
1 21
0 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0111 0111 1111 1010 1011 1100 0111 0111 0000 0000 0000 0000 0000 0000 0010 1101(2) =
(0 × 263 + 1 × 262 + 1 × 261 + 1 × 260 + 0 × 259 + 1 × 258 + 1 × 257 + 1 × 256 + 1 × 255 + 1 × 254 + 1 × 253 + 1 × 252 + 1 × 251 + 0 × 250 + 1 × 249 + 0 × 248 + 1 × 247 + 0 × 246 + 1 × 245 + 1 × 244 + 1 × 243 + 1 × 242 + 0 × 241 + 0 × 240 + 0 × 239 + 1 × 238 + 1 × 237 + 1 × 236 + 0 × 235 + 1 × 234 + 1 × 233 + 1 × 232 + 0 × 231 + 0 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 0 × 224 + 0 × 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 + 1 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20)(10) =
(0 + 4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 0 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 0 + 562 949 953 421 312 + 0 + 140 737 488 355 328 + 0 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 0 + 0 + 0 + 274 877 906 944 + 137 438 953 472 + 68 719 476 736 + 0 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 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 + 32 + 0 + 8 + 4 + 0 + 1)(10) =
(4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 562 949 953 421 312 + 140 737 488 355 328 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 274 877 906 944 + 137 438 953 472 + 68 719 476 736 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 32 + 8 + 4 + 1)(10) =
8 645 429 653 978 218 541(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1000 1000 0000 0101 0100 0011 1000 1000 1111 1111 1111 1111 1111 1111 1101 0011(2) = -8 645 429 653 978 218 541(10)
The signed binary number in two's complement representation 1000 1000 0000 0101 0100 0011 1000 1000 1111 1111 1111 1111 1111 1111 1101 0011(2) converted and written as an integer in decimal system (base ten):
1000 1000 0000 0101 0100 0011 1000 1000 1111 1111 1111 1111 1111 1111 1101 0011(2) = -8 645 429 653 978 218 541(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.