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.
1101 0110 0001 1101 1111 1010 1100 0000 0111 0101 0111 0011 1001 0110 1100 1101 - 1 = 1101 0110 0001 1101 1111 1010 1100 0000 0111 0101 0111 0011 1001 0110 1100 1100
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:
!(1101 0110 0001 1101 1111 1010 1100 0000 0111 0101 0111 0011 1001 0110 1100 1100) = 0010 1001 1110 0010 0000 0101 0011 1111 1000 1010 1000 1100 0110 1001 0011 0011
4. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
263
0 262
0 261
1 260
0 259
1 258
0 257
0 256
1 255
1 254
1 253
1 252
0 251
0 250
0 249
1 248
0 247
0 246
0 245
0 244
0 243
0 242
1 241
0 240
1 239
0 238
0 237
1 236
1 235
1 234
1 233
1 232
1 231
1 230
0 229
0 228
0 227
1 226
0 225
1 224
0 223
1 222
0 221
0 220
0 219
1 218
1 217
0 216
0 215
0 214
1 213
1 212
0 211
1 210
0 29
0 28
1 27
0 26
0 25
1 24
1 23
0 22
0 21
1 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0010 1001 1110 0010 0000 0101 0011 1111 1000 1010 1000 1100 0110 1001 0011 0011(2) =
(0 × 263 + 0 × 262 + 1 × 261 + 0 × 260 + 1 × 259 + 0 × 258 + 0 × 257 + 1 × 256 + 1 × 255 + 1 × 254 + 1 × 253 + 0 × 252 + 0 × 251 + 0 × 250 + 1 × 249 + 0 × 248 + 0 × 247 + 0 × 246 + 0 × 245 + 0 × 244 + 0 × 243 + 1 × 242 + 0 × 241 + 1 × 240 + 0 × 239 + 0 × 238 + 1 × 237 + 1 × 236 + 1 × 235 + 1 × 234 + 1 × 233 + 1 × 232 + 1 × 231 + 0 × 230 + 0 × 229 + 0 × 228 + 1 × 227 + 0 × 226 + 1 × 225 + 0 × 224 + 1 × 223 + 0 × 222 + 0 × 221 + 0 × 220 + 1 × 219 + 1 × 218 + 0 × 217 + 0 × 216 + 0 × 215 + 1 × 214 + 1 × 213 + 0 × 212 + 1 × 211 + 0 × 210 + 0 × 29 + 1 × 28 + 0 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 1 × 20)(10) =
(0 + 0 + 2 305 843 009 213 693 952 + 0 + 576 460 752 303 423 488 + 0 + 0 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 0 + 0 + 0 + 562 949 953 421 312 + 0 + 0 + 0 + 0 + 0 + 0 + 4 398 046 511 104 + 0 + 1 099 511 627 776 + 0 + 0 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 0 + 0 + 0 + 134 217 728 + 0 + 33 554 432 + 0 + 8 388 608 + 0 + 0 + 0 + 524 288 + 262 144 + 0 + 0 + 0 + 16 384 + 8 192 + 0 + 2 048 + 0 + 0 + 256 + 0 + 0 + 32 + 16 + 0 + 0 + 2 + 1)(10) =
(2 305 843 009 213 693 952 + 576 460 752 303 423 488 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 562 949 953 421 312 + 4 398 046 511 104 + 1 099 511 627 776 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 134 217 728 + 33 554 432 + 8 388 608 + 524 288 + 262 144 + 16 384 + 8 192 + 2 048 + 256 + 32 + 16 + 2 + 1)(10) =
3 017 980 470 757 189 939(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1101 0110 0001 1101 1111 1010 1100 0000 0111 0101 0111 0011 1001 0110 1100 1101(2) = -3 017 980 470 757 189 939(10)
The signed binary number in two's complement representation 1101 0110 0001 1101 1111 1010 1100 0000 0111 0101 0111 0011 1001 0110 1100 1101(2) converted and written as an integer in decimal system (base ten):
1101 0110 0001 1101 1111 1010 1100 0000 0111 0101 0111 0011 1001 0110 1100 1101(2) = -3 017 980 470 757 189 939(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.