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.
1101 0011 1000 1001 0101 1001 0010 1101 0100 0010 0011 1111 0111 0011 0110 0110 - 1 = 1101 0011 1000 1001 0101 1001 0010 1101 0100 0010 0011 1111 0111 0011 0110 0101
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:
!(1101 0011 1000 1001 0101 1001 0010 1101 0100 0010 0011 1111 0111 0011 0110 0101) = 0010 1100 0111 0110 1010 0110 1101 0010 1011 1101 1100 0000 1000 1100 1001 1010
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
0 259
1 258
1 257
0 256
0 255
0 254
1 253
1 252
1 251
0 250
1 249
1 248
0 247
1 246
0 245
1 244
0 243
0 242
1 241
1 240
0 239
1 238
1 237
0 236
1 235
0 234
0 233
1 232
0 231
1 230
0 229
1 228
1 227
1 226
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
0 212
0 211
1 210
1 29
0 28
0 27
1 26
0 25
0 24
1 23
1 22
0 21
1 20
0
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0010 1100 0111 0110 1010 0110 1101 0010 1011 1101 1100 0000 1000 1100 1001 1010(2) =
(0 × 263 + 0 × 262 + 1 × 261 + 0 × 260 + 1 × 259 + 1 × 258 + 0 × 257 + 0 × 256 + 0 × 255 + 1 × 254 + 1 × 253 + 1 × 252 + 0 × 251 + 1 × 250 + 1 × 249 + 0 × 248 + 1 × 247 + 0 × 246 + 1 × 245 + 0 × 244 + 0 × 243 + 1 × 242 + 1 × 241 + 0 × 240 + 1 × 239 + 1 × 238 + 0 × 237 + 1 × 236 + 0 × 235 + 0 × 234 + 1 × 233 + 0 × 232 + 1 × 231 + 0 × 230 + 1 × 229 + 1 × 228 + 1 × 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 + 0 × 213 + 0 × 212 + 1 × 211 + 1 × 210 + 0 × 29 + 0 × 28 + 1 × 27 + 0 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20)(10) =
(0 + 0 + 2 305 843 009 213 693 952 + 0 + 576 460 752 303 423 488 + 288 230 376 151 711 744 + 0 + 0 + 0 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 0 + 1 125 899 906 842 624 + 562 949 953 421 312 + 0 + 140 737 488 355 328 + 0 + 35 184 372 088 832 + 0 + 0 + 4 398 046 511 104 + 2 199 023 255 552 + 0 + 549 755 813 888 + 274 877 906 944 + 0 + 68 719 476 736 + 0 + 0 + 8 589 934 592 + 0 + 2 147 483 648 + 0 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 0 + 16 777 216 + 8 388 608 + 4 194 304 + 0 + 0 + 0 + 0 + 0 + 0 + 32 768 + 0 + 0 + 0 + 2 048 + 1 024 + 0 + 0 + 128 + 0 + 0 + 16 + 8 + 0 + 2 + 0)(10) =
(2 305 843 009 213 693 952 + 576 460 752 303 423 488 + 288 230 376 151 711 744 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 1 125 899 906 842 624 + 562 949 953 421 312 + 140 737 488 355 328 + 35 184 372 088 832 + 4 398 046 511 104 + 2 199 023 255 552 + 549 755 813 888 + 274 877 906 944 + 68 719 476 736 + 8 589 934 592 + 2 147 483 648 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 16 777 216 + 8 388 608 + 4 194 304 + 32 768 + 2 048 + 1 024 + 128 + 16 + 8 + 2)(10) =
3 203 931 608 977 542 298(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1101 0011 1000 1001 0101 1001 0010 1101 0100 0010 0011 1111 0111 0011 0110 0110(2) = -3 203 931 608 977 542 298(10)
The signed binary number in two's complement representation 1101 0011 1000 1001 0101 1001 0010 1101 0100 0010 0011 1111 0111 0011 0110 0110(2) converted and written as an integer in decimal system (base ten):
1101 0011 1000 1001 0101 1001 0010 1101 0100 0010 0011 1111 0111 0011 0110 0110(2) = -3 203 931 608 977 542 298(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.