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
0 254
0 253
0 252
0 251
1 250
1 249
1 248
0 247
0 246
1 245
0 244
0 243
1 242
0 241
1 240
0 239
1 238
0 237
0 236
0 235
1 234
0 233
0 232
1 231
0 230
1 229
0 228
0 227
1 226
0 225
1 224
0 223
0 222
1 221
0 220
1 219
0 218
0 217
1 216
1 215
0 214
0 213
0 212
1 211
1 210
0 29
0 28
1 27
0 26
1 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 0000 1110 0100 1010 1000 1001 0100 1010 0101 0011 0001 1001 0110 0110(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 0 × 257 + 0 × 256 + 0 × 255 + 0 × 254 + 0 × 253 + 0 × 252 + 1 × 251 + 1 × 250 + 1 × 249 + 0 × 248 + 0 × 247 + 1 × 246 + 0 × 245 + 0 × 244 + 1 × 243 + 0 × 242 + 1 × 241 + 0 × 240 + 1 × 239 + 0 × 238 + 0 × 237 + 0 × 236 + 1 × 235 + 0 × 234 + 0 × 233 + 1 × 232 + 0 × 231 + 1 × 230 + 0 × 229 + 0 × 228 + 1 × 227 + 0 × 226 + 1 × 225 + 0 × 224 + 0 × 223 + 1 × 222 + 0 × 221 + 1 × 220 + 0 × 219 + 0 × 218 + 1 × 217 + 1 × 216 + 0 × 215 + 0 × 214 + 0 × 213 + 1 × 212 + 1 × 211 + 0 × 210 + 0 × 29 + 1 × 28 + 0 × 27 + 1 × 26 + 1 × 25 + 0 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 562 949 953 421 312 + 0 + 0 + 70 368 744 177 664 + 0 + 0 + 8 796 093 022 208 + 0 + 2 199 023 255 552 + 0 + 549 755 813 888 + 0 + 0 + 0 + 34 359 738 368 + 0 + 0 + 4 294 967 296 + 0 + 1 073 741 824 + 0 + 0 + 134 217 728 + 0 + 33 554 432 + 0 + 0 + 4 194 304 + 0 + 1 048 576 + 0 + 0 + 131 072 + 65 536 + 0 + 0 + 0 + 4 096 + 2 048 + 0 + 0 + 256 + 0 + 64 + 32 + 0 + 0 + 4 + 2 + 0)(10) =
(2 251 799 813 685 248 + 1 125 899 906 842 624 + 562 949 953 421 312 + 70 368 744 177 664 + 8 796 093 022 208 + 2 199 023 255 552 + 549 755 813 888 + 34 359 738 368 + 4 294 967 296 + 1 073 741 824 + 134 217 728 + 33 554 432 + 4 194 304 + 1 048 576 + 131 072 + 65 536 + 4 096 + 2 048 + 256 + 64 + 32 + 4 + 2)(10) =
4 022 603 191 884 134(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0000 0000 1110 0100 1010 1000 1001 0100 1010 0101 0011 0001 1001 0110 0110(2) = 4 022 603 191 884 134(10)
The signed binary number in two's complement representation 0000 0000 0000 1110 0100 1010 1000 1001 0100 1010 0101 0011 0001 1001 0110 0110(2) converted and written as an integer in decimal system (base ten):
0000 0000 0000 1110 0100 1010 1000 1001 0100 1010 0101 0011 0001 1001 0110 0110(2) = 4 022 603 191 884 134(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.