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
1 256
0 255
1 254
1 253
1 252
1 251
1 250
1 249
1 248
0 247
0 246
0 245
0 244
1 243
1 242
1 241
0 240
1 239
1 238
0 237
1 236
0 235
0 234
1 233
1 232
0 231
0 230
0 229
0 228
1 227
0 226
1 225
0 224
1 223
0 222
1 221
1 220
1 219
1 218
1 217
1 216
0 215
0 214
0 213
0 212
1 211
1 210
1 29
0 28
1 27
1 26
1 25
0 24
1 23
1 22
0 21
1 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 0010 1111 1110 0001 1101 1010 0110 0001 0101 0111 1110 0001 1101 1101 1011(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 1 × 257 + 0 × 256 + 1 × 255 + 1 × 254 + 1 × 253 + 1 × 252 + 1 × 251 + 1 × 250 + 1 × 249 + 0 × 248 + 0 × 247 + 0 × 246 + 0 × 245 + 1 × 244 + 1 × 243 + 1 × 242 + 0 × 241 + 1 × 240 + 1 × 239 + 0 × 238 + 1 × 237 + 0 × 236 + 0 × 235 + 1 × 234 + 1 × 233 + 0 × 232 + 0 × 231 + 0 × 230 + 0 × 229 + 1 × 228 + 0 × 227 + 1 × 226 + 0 × 225 + 1 × 224 + 0 × 223 + 1 × 222 + 1 × 221 + 1 × 220 + 1 × 219 + 1 × 218 + 1 × 217 + 0 × 216 + 0 × 215 + 0 × 214 + 0 × 213 + 1 × 212 + 1 × 211 + 1 × 210 + 0 × 29 + 1 × 28 + 1 × 27 + 1 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 1 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 144 115 188 075 855 872 + 0 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 562 949 953 421 312 + 0 + 0 + 0 + 0 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 0 + 1 099 511 627 776 + 549 755 813 888 + 0 + 137 438 953 472 + 0 + 0 + 17 179 869 184 + 8 589 934 592 + 0 + 0 + 0 + 0 + 268 435 456 + 0 + 67 108 864 + 0 + 16 777 216 + 0 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 0 + 0 + 0 + 0 + 4 096 + 2 048 + 1 024 + 0 + 256 + 128 + 64 + 0 + 16 + 8 + 0 + 2 + 1)(10) =
(144 115 188 075 855 872 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 562 949 953 421 312 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 1 099 511 627 776 + 549 755 813 888 + 137 438 953 472 + 17 179 869 184 + 8 589 934 592 + 268 435 456 + 67 108 864 + 16 777 216 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 4 096 + 2 048 + 1 024 + 256 + 128 + 64 + 16 + 8 + 2 + 1)(10) =
215 642 431 322 725 851(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0010 1111 1110 0001 1101 1010 0110 0001 0101 0111 1110 0001 1101 1101 1011(2) = 215 642 431 322 725 851(10)
The signed binary number in two's complement representation 0000 0010 1111 1110 0001 1101 1010 0110 0001 0101 0111 1110 0001 1101 1101 1011(2) converted and written as an integer in decimal system (base ten):
0000 0010 1111 1110 0001 1101 1010 0110 0001 0101 0111 1110 0001 1101 1101 1011(2) = 215 642 431 322 725 851(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.