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.
1100 1001 1001 0100 1001 0001 0100 1010 0100 1001 0100 1010 1000 0010 1001 0000 - 1 = 1100 1001 1001 0100 1001 0001 0100 1010 0100 1001 0100 1010 1000 0010 1000 1111
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:
!(1100 1001 1001 0100 1001 0001 0100 1010 0100 1001 0100 1010 1000 0010 1000 1111) = 0011 0110 0110 1011 0110 1110 1011 0101 1011 0110 1011 0101 0111 1101 0111 0000
4. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
263
0 262
0 261
1 260
1 259
0 258
1 257
1 256
0 255
0 254
1 253
1 252
0 251
1 250
0 249
1 248
1 247
0 246
1 245
1 244
0 243
1 242
1 241
1 240
0 239
1 238
0 237
1 236
1 235
0 234
1 233
0 232
1 231
1 230
0 229
1 228
1 227
0 226
1 225
1 224
0 223
1 222
0 221
1 220
1 219
0 218
1 217
0 216
1 215
0 214
1 213
1 212
1 211
1 210
1 29
0 28
1 27
0 26
1 25
1 24
1 23
0 22
0 21
0 20
0
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0011 0110 0110 1011 0110 1110 1011 0101 1011 0110 1011 0101 0111 1101 0111 0000(2) =
(0 × 263 + 0 × 262 + 1 × 261 + 1 × 260 + 0 × 259 + 1 × 258 + 1 × 257 + 0 × 256 + 0 × 255 + 1 × 254 + 1 × 253 + 0 × 252 + 1 × 251 + 0 × 250 + 1 × 249 + 1 × 248 + 0 × 247 + 1 × 246 + 1 × 245 + 0 × 244 + 1 × 243 + 1 × 242 + 1 × 241 + 0 × 240 + 1 × 239 + 0 × 238 + 1 × 237 + 1 × 236 + 0 × 235 + 1 × 234 + 0 × 233 + 1 × 232 + 1 × 231 + 0 × 230 + 1 × 229 + 1 × 228 + 0 × 227 + 1 × 226 + 1 × 225 + 0 × 224 + 1 × 223 + 0 × 222 + 1 × 221 + 1 × 220 + 0 × 219 + 1 × 218 + 0 × 217 + 1 × 216 + 0 × 215 + 1 × 214 + 1 × 213 + 1 × 212 + 1 × 211 + 1 × 210 + 0 × 29 + 1 × 28 + 0 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 0 × 20)(10) =
(0 + 0 + 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 + 0 + 0 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 0 + 2 251 799 813 685 248 + 0 + 562 949 953 421 312 + 281 474 976 710 656 + 0 + 70 368 744 177 664 + 35 184 372 088 832 + 0 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 0 + 549 755 813 888 + 0 + 137 438 953 472 + 68 719 476 736 + 0 + 17 179 869 184 + 0 + 4 294 967 296 + 2 147 483 648 + 0 + 536 870 912 + 268 435 456 + 0 + 67 108 864 + 33 554 432 + 0 + 8 388 608 + 0 + 2 097 152 + 1 048 576 + 0 + 262 144 + 0 + 65 536 + 0 + 16 384 + 8 192 + 4 096 + 2 048 + 1 024 + 0 + 256 + 0 + 64 + 32 + 16 + 0 + 0 + 0 + 0)(10) =
(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 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 2 251 799 813 685 248 + 562 949 953 421 312 + 281 474 976 710 656 + 70 368 744 177 664 + 35 184 372 088 832 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 549 755 813 888 + 137 438 953 472 + 68 719 476 736 + 17 179 869 184 + 4 294 967 296 + 2 147 483 648 + 536 870 912 + 268 435 456 + 67 108 864 + 33 554 432 + 8 388 608 + 2 097 152 + 1 048 576 + 262 144 + 65 536 + 16 384 + 8 192 + 4 096 + 2 048 + 1 024 + 256 + 64 + 32 + 16)(10) =
3 921 349 627 289 632 112(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1100 1001 1001 0100 1001 0001 0100 1010 0100 1001 0100 1010 1000 0010 1001 0000(2) = -3 921 349 627 289 632 112(10)
The signed binary number in two's complement representation 1100 1001 1001 0100 1001 0001 0100 1010 0100 1001 0100 1010 1000 0010 1001 0000(2) converted and written as an integer in decimal system (base ten):
1100 1001 1001 0100 1001 0001 0100 1010 0100 1001 0100 1010 1000 0010 1001 0000(2) = -3 921 349 627 289 632 112(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.