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.
1010 1010 1010 1010 1010 1010 1010 0101 0110 1010 1001 0101 0101 0101 0101 0100 - 1 = 1010 1010 1010 1010 1010 1010 1010 0101 0110 1010 1001 0101 0101 0101 0101 0011
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:
!(1010 1010 1010 1010 1010 1010 1010 0101 0110 1010 1001 0101 0101 0101 0101 0011) = 0101 0101 0101 0101 0101 0101 0101 1010 1001 0101 0110 1010 1010 1010 1010 1100
4. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
263
0 262
1 261
0 260
1 259
0 258
1 257
0 256
1 255
0 254
1 253
0 252
1 251
0 250
1 249
0 248
1 247
0 246
1 245
0 244
1 243
0 242
1 241
0 240
1 239
0 238
1 237
0 236
1 235
1 234
0 233
1 232
0 231
1 230
0 229
0 228
1 227
0 226
1 225
0 224
1 223
0 222
1 221
1 220
0 219
1 218
0 217
1 216
0 215
1 214
0 213
1 212
0 211
1 210
0 29
1 28
0 27
1 26
0 25
1 24
0 23
1 22
1 21
0 20
0
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0101 0101 0101 0101 0101 0101 0101 1010 1001 0101 0110 1010 1010 1010 1010 1100(2) =
(0 × 263 + 1 × 262 + 0 × 261 + 1 × 260 + 0 × 259 + 1 × 258 + 0 × 257 + 1 × 256 + 0 × 255 + 1 × 254 + 0 × 253 + 1 × 252 + 0 × 251 + 1 × 250 + 0 × 249 + 1 × 248 + 0 × 247 + 1 × 246 + 0 × 245 + 1 × 244 + 0 × 243 + 1 × 242 + 0 × 241 + 1 × 240 + 0 × 239 + 1 × 238 + 0 × 237 + 1 × 236 + 1 × 235 + 0 × 234 + 1 × 233 + 0 × 232 + 1 × 231 + 0 × 230 + 0 × 229 + 1 × 228 + 0 × 227 + 1 × 226 + 0 × 225 + 1 × 224 + 0 × 223 + 1 × 222 + 1 × 221 + 0 × 220 + 1 × 219 + 0 × 218 + 1 × 217 + 0 × 216 + 1 × 215 + 0 × 214 + 1 × 213 + 0 × 212 + 1 × 211 + 0 × 210 + 1 × 29 + 0 × 28 + 1 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 0 × 20)(10) =
(0 + 4 611 686 018 427 387 904 + 0 + 1 152 921 504 606 846 976 + 0 + 288 230 376 151 711 744 + 0 + 72 057 594 037 927 936 + 0 + 18 014 398 509 481 984 + 0 + 4 503 599 627 370 496 + 0 + 1 125 899 906 842 624 + 0 + 281 474 976 710 656 + 0 + 70 368 744 177 664 + 0 + 17 592 186 044 416 + 0 + 4 398 046 511 104 + 0 + 1 099 511 627 776 + 0 + 274 877 906 944 + 0 + 68 719 476 736 + 34 359 738 368 + 0 + 8 589 934 592 + 0 + 2 147 483 648 + 0 + 0 + 268 435 456 + 0 + 67 108 864 + 0 + 16 777 216 + 0 + 4 194 304 + 2 097 152 + 0 + 524 288 + 0 + 131 072 + 0 + 32 768 + 0 + 8 192 + 0 + 2 048 + 0 + 512 + 0 + 128 + 0 + 32 + 0 + 8 + 4 + 0 + 0)(10) =
(4 611 686 018 427 387 904 + 1 152 921 504 606 846 976 + 288 230 376 151 711 744 + 72 057 594 037 927 936 + 18 014 398 509 481 984 + 4 503 599 627 370 496 + 1 125 899 906 842 624 + 281 474 976 710 656 + 70 368 744 177 664 + 17 592 186 044 416 + 4 398 046 511 104 + 1 099 511 627 776 + 274 877 906 944 + 68 719 476 736 + 34 359 738 368 + 8 589 934 592 + 2 147 483 648 + 268 435 456 + 67 108 864 + 16 777 216 + 4 194 304 + 2 097 152 + 524 288 + 131 072 + 32 768 + 8 192 + 2 048 + 512 + 128 + 32 + 8 + 4)(10) =
6 148 914 713 786 493 612(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1010 1010 1010 1010 1010 1010 1010 0101 0110 1010 1001 0101 0101 0101 0101 0100(2) = -6 148 914 713 786 493 612(10)
The signed binary number in two's complement representation 1010 1010 1010 1010 1010 1010 1010 0101 0110 1010 1001 0101 0101 0101 0101 0100(2) converted and written as an integer in decimal system (base ten):
1010 1010 1010 1010 1010 1010 1010 0101 0110 1010 1001 0101 0101 0101 0101 0100(2) = -6 148 914 713 786 493 612(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.