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
1 255
1 254
0 253
0 252
0 251
0 250
0 249
0 248
0 247
0 246
1 245
1 244
0 243
1 242
1 241
0 240
0 239
1 238
0 237
1 236
0 235
0 234
0 233
1 232
0 231
0 230
1 229
0 228
0 227
0 226
0 225
0 224
0 223
0 222
1 221
1 220
1 219
0 218
1 217
0 216
0 215
0 214
1 213
1 212
0 211
0 210
0 29
1 28
1 27
1 26
0 25
1 24
1 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 0001 1000 0000 0110 1100 1010 0010 0100 0000 0111 0100 0110 0011 1011 0110(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 0 × 257 + 1 × 256 + 1 × 255 + 0 × 254 + 0 × 253 + 0 × 252 + 0 × 251 + 0 × 250 + 0 × 249 + 0 × 248 + 0 × 247 + 1 × 246 + 1 × 245 + 0 × 244 + 1 × 243 + 1 × 242 + 0 × 241 + 0 × 240 + 1 × 239 + 0 × 238 + 1 × 237 + 0 × 236 + 0 × 235 + 0 × 234 + 1 × 233 + 0 × 232 + 0 × 231 + 1 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 0 × 224 + 0 × 223 + 1 × 222 + 1 × 221 + 1 × 220 + 0 × 219 + 1 × 218 + 0 × 217 + 0 × 216 + 0 × 215 + 1 × 214 + 1 × 213 + 0 × 212 + 0 × 211 + 0 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 0 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 70 368 744 177 664 + 35 184 372 088 832 + 0 + 8 796 093 022 208 + 4 398 046 511 104 + 0 + 0 + 549 755 813 888 + 0 + 137 438 953 472 + 0 + 0 + 0 + 8 589 934 592 + 0 + 0 + 1 073 741 824 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 4 194 304 + 2 097 152 + 1 048 576 + 0 + 262 144 + 0 + 0 + 0 + 16 384 + 8 192 + 0 + 0 + 0 + 512 + 256 + 128 + 0 + 32 + 16 + 0 + 4 + 2 + 0)(10) =
(72 057 594 037 927 936 + 36 028 797 018 963 968 + 70 368 744 177 664 + 35 184 372 088 832 + 8 796 093 022 208 + 4 398 046 511 104 + 549 755 813 888 + 137 438 953 472 + 8 589 934 592 + 1 073 741 824 + 4 194 304 + 2 097 152 + 1 048 576 + 262 144 + 16 384 + 8 192 + 512 + 256 + 128 + 32 + 16 + 4 + 2)(10) =
108 205 835 178 763 190(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0001 1000 0000 0110 1100 1010 0010 0100 0000 0111 0100 0110 0011 1011 0110(2) = 108 205 835 178 763 190(10)
The signed binary number in two's complement representation 0000 0001 1000 0000 0110 1100 1010 0010 0100 0000 0111 0100 0110 0011 1011 0110(2) converted and written as an integer in decimal system (base ten):
0000 0001 1000 0000 0110 1100 1010 0010 0100 0000 0111 0100 0110 0011 1011 0110(2) = 108 205 835 178 763 190(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.