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.
1110 0110 0000 1110 0010 1000 0001 0000 1000 1010 0000 0010 0001 0000 0000 0011 - 1 = 1110 0110 0000 1110 0010 1000 0001 0000 1000 1010 0000 0010 0001 0000 0000 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:
!(1110 0110 0000 1110 0010 1000 0001 0000 1000 1010 0000 0010 0001 0000 0000 0010) = 0001 1001 1111 0001 1101 0111 1110 1111 0111 0101 1111 1101 1110 1111 1111 1101
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
1 259
1 258
0 257
0 256
1 255
1 254
1 253
1 252
1 251
0 250
0 249
0 248
1 247
1 246
1 245
0 244
1 243
0 242
1 241
1 240
1 239
1 238
1 237
1 236
0 235
1 234
1 233
1 232
1 231
0 230
1 229
1 228
1 227
0 226
1 225
0 224
1 223
1 222
1 221
1 220
1 219
1 218
1 217
0 216
1 215
1 214
1 213
1 212
0 211
1 210
1 29
1 28
1 27
1 26
1 25
1 24
1 23
1 22
1 21
0 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0001 1001 1111 0001 1101 0111 1110 1111 0111 0101 1111 1101 1110 1111 1111 1101(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 1 × 260 + 1 × 259 + 0 × 258 + 0 × 257 + 1 × 256 + 1 × 255 + 1 × 254 + 1 × 253 + 1 × 252 + 0 × 251 + 0 × 250 + 0 × 249 + 1 × 248 + 1 × 247 + 1 × 246 + 0 × 245 + 1 × 244 + 0 × 243 + 1 × 242 + 1 × 241 + 1 × 240 + 1 × 239 + 1 × 238 + 1 × 237 + 0 × 236 + 1 × 235 + 1 × 234 + 1 × 233 + 1 × 232 + 0 × 231 + 1 × 230 + 1 × 229 + 1 × 228 + 0 × 227 + 1 × 226 + 0 × 225 + 1 × 224 + 1 × 223 + 1 × 222 + 1 × 221 + 1 × 220 + 1 × 219 + 1 × 218 + 0 × 217 + 1 × 216 + 1 × 215 + 1 × 214 + 1 × 213 + 0 × 212 + 1 × 211 + 1 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20)(10) =
(0 + 0 + 0 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 0 + 0 + 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 + 0 + 0 + 0 + 281 474 976 710 656 + 140 737 488 355 328 + 70 368 744 177 664 + 0 + 17 592 186 044 416 + 0 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 0 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 0 + 1 073 741 824 + 536 870 912 + 268 435 456 + 0 + 67 108 864 + 0 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 0 + 65 536 + 32 768 + 16 384 + 8 192 + 0 + 2 048 + 1 024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 0 + 1)(10) =
(1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 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 + 281 474 976 710 656 + 140 737 488 355 328 + 70 368 744 177 664 + 17 592 186 044 416 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 1 073 741 824 + 536 870 912 + 268 435 456 + 67 108 864 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 65 536 + 32 768 + 16 384 + 8 192 + 2 048 + 1 024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 1)(10) =
1 869 512 743 812 198 397(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1110 0110 0000 1110 0010 1000 0001 0000 1000 1010 0000 0010 0001 0000 0000 0011(2) = -1 869 512 743 812 198 397(10)
The signed binary number in two's complement representation 1110 0110 0000 1110 0010 1000 0001 0000 1000 1010 0000 0010 0001 0000 0000 0011(2) converted and written as an integer in decimal system (base ten):
1110 0110 0000 1110 0010 1000 0001 0000 1000 1010 0000 0010 0001 0000 0000 0011(2) = -1 869 512 743 812 198 397(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.