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.
* Not the case - the number is positive *
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:
* Not the case - the number is positive *
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
0 259
0 258
0 257
0 256
0 255
1 254
0 253
0 252
1 251
1 250
0 249
1 248
0 247
1 246
0 245
0 244
0 243
1 242
0 241
1 240
0 239
1 238
0 237
1 236
0 235
0 234
1 233
1 232
0 231
1 230
0 229
0 228
1 227
0 226
0 225
0 224
0 223
1 222
0 221
1 220
0 219
1 218
0 217
1 216
0 215
1 214
0 213
0 212
0 211
1 210
1 29
1 28
0 27
0 26
0 25
1 24
0 23
0 22
1 21
1 20
0
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 0000 1001 1010 1000 1010 1010 0110 1001 0000 1010 1010 1000 1110 0010 0110(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 0 × 257 + 0 × 256 + 1 × 255 + 0 × 254 + 0 × 253 + 1 × 252 + 1 × 251 + 0 × 250 + 1 × 249 + 0 × 248 + 1 × 247 + 0 × 246 + 0 × 245 + 0 × 244 + 1 × 243 + 0 × 242 + 1 × 241 + 0 × 240 + 1 × 239 + 0 × 238 + 1 × 237 + 0 × 236 + 0 × 235 + 1 × 234 + 1 × 233 + 0 × 232 + 1 × 231 + 0 × 230 + 0 × 229 + 1 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 0 × 224 + 1 × 223 + 0 × 222 + 1 × 221 + 0 × 220 + 1 × 219 + 0 × 218 + 1 × 217 + 0 × 216 + 1 × 215 + 0 × 214 + 0 × 213 + 0 × 212 + 1 × 211 + 1 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 36 028 797 018 963 968 + 0 + 0 + 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 + 0 + 0 + 8 796 093 022 208 + 0 + 2 199 023 255 552 + 0 + 549 755 813 888 + 0 + 137 438 953 472 + 0 + 0 + 17 179 869 184 + 8 589 934 592 + 0 + 2 147 483 648 + 0 + 0 + 268 435 456 + 0 + 0 + 0 + 0 + 8 388 608 + 0 + 2 097 152 + 0 + 524 288 + 0 + 131 072 + 0 + 32 768 + 0 + 0 + 0 + 2 048 + 1 024 + 512 + 0 + 0 + 0 + 32 + 0 + 0 + 4 + 2 + 0)(10) =
(36 028 797 018 963 968 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 562 949 953 421 312 + 140 737 488 355 328 + 8 796 093 022 208 + 2 199 023 255 552 + 549 755 813 888 + 137 438 953 472 + 17 179 869 184 + 8 589 934 592 + 2 147 483 648 + 268 435 456 + 8 388 608 + 2 097 152 + 524 288 + 131 072 + 32 768 + 2 048 + 1 024 + 512 + 32 + 4 + 2)(10) =
43 499 594 409 741 862(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0000 1001 1010 1000 1010 1010 0110 1001 0000 1010 1010 1000 1110 0010 0110(2) = 43 499 594 409 741 862(10)
The signed binary number in two's complement representation 0000 0000 1001 1010 1000 1010 1010 0110 1001 0000 1010 1010 1000 1110 0010 0110(2) converted and written as an integer in decimal system (base ten):
0000 0000 1001 1010 1000 1010 1010 0110 1001 0000 1010 1010 1000 1110 0010 0110(2) = 43 499 594 409 741 862(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.