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
1 253
0 252
1 251
0 250
0 249
1 248
0 247
1 246
1 245
0 244
1 243
1 242
0 241
0 240
0 239
1 238
1 237
0 236
1 235
1 234
1 233
1 232
0 231
1 230
1 229
1 228
0 227
1 226
1 225
0 224
0 223
1 222
1 221
0 220
0 219
1 218
0 217
1 216
0 215
1 214
1 213
1 212
1 211
0 210
1 29
0 28
0 27
1 26
1 25
0 24
0 23
0 22
1 21
0 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 0000 1101 0010 1101 1000 1101 1110 1110 1100 1100 1010 1111 0100 1100 0101(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 0 × 257 + 0 × 256 + 1 × 255 + 1 × 254 + 0 × 253 + 1 × 252 + 0 × 251 + 0 × 250 + 1 × 249 + 0 × 248 + 1 × 247 + 1 × 246 + 0 × 245 + 1 × 244 + 1 × 243 + 0 × 242 + 0 × 241 + 0 × 240 + 1 × 239 + 1 × 238 + 0 × 237 + 1 × 236 + 1 × 235 + 1 × 234 + 1 × 233 + 0 × 232 + 1 × 231 + 1 × 230 + 1 × 229 + 0 × 228 + 1 × 227 + 1 × 226 + 0 × 225 + 0 × 224 + 1 × 223 + 1 × 222 + 0 × 221 + 0 × 220 + 1 × 219 + 0 × 218 + 1 × 217 + 0 × 216 + 1 × 215 + 1 × 214 + 1 × 213 + 1 × 212 + 0 × 211 + 1 × 210 + 0 × 29 + 0 × 28 + 1 × 27 + 1 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 1 × 22 + 0 × 21 + 1 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 0 + 4 503 599 627 370 496 + 0 + 0 + 562 949 953 421 312 + 0 + 140 737 488 355 328 + 70 368 744 177 664 + 0 + 17 592 186 044 416 + 8 796 093 022 208 + 0 + 0 + 0 + 549 755 813 888 + 274 877 906 944 + 0 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 0 + 2 147 483 648 + 1 073 741 824 + 536 870 912 + 0 + 134 217 728 + 67 108 864 + 0 + 0 + 8 388 608 + 4 194 304 + 0 + 0 + 524 288 + 0 + 131 072 + 0 + 32 768 + 16 384 + 8 192 + 4 096 + 0 + 1 024 + 0 + 0 + 128 + 64 + 0 + 0 + 0 + 4 + 0 + 1)(10) =
(36 028 797 018 963 968 + 18 014 398 509 481 984 + 4 503 599 627 370 496 + 562 949 953 421 312 + 140 737 488 355 328 + 70 368 744 177 664 + 17 592 186 044 416 + 8 796 093 022 208 + 549 755 813 888 + 274 877 906 944 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 2 147 483 648 + 1 073 741 824 + 536 870 912 + 134 217 728 + 67 108 864 + 8 388 608 + 4 194 304 + 524 288 + 131 072 + 32 768 + 16 384 + 8 192 + 4 096 + 1 024 + 128 + 64 + 4 + 1)(10) =
59 348 197 076 300 997(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0000 1101 0010 1101 1000 1101 1110 1110 1100 1100 1010 1111 0100 1100 0101(2) = 59 348 197 076 300 997(10)
The signed binary number in two's complement representation 0000 0000 1101 0010 1101 1000 1101 1110 1110 1100 1100 1010 1111 0100 1100 0101(2) converted and written as an integer in decimal system (base ten):
0000 0000 1101 0010 1101 1000 1101 1110 1110 1100 1100 1010 1111 0100 1100 0101(2) = 59 348 197 076 300 997(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.