What are the steps to convert the signed binary in one's (1's) complement representation to an integer in decimal system (in base ten)?
1. Is this a positive or a negative number?
0100 0000 0110 0011 0001 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 is the binary representation of a positive integer, on 64 bits (8 Bytes).
- In a signed binary in one's complement representation, the first bit (the leftmost) indicates the sign, 1 = negative, 0 = positive.
2. 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 *
3. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
263
0 262
1 261
0 260
0 259
0 258
0 257
0 256
0 255
0 254
1 253
1 252
0 251
0 250
0 249
1 248
1 247
0 246
0 245
0 244
1 243
1 242
0 241
0 240
0 239
0 238
0 237
0 236
0 235
0 234
0 233
0 232
0 231
0 230
0 229
0 228
0 227
0 226
0 225
0 224
0 223
0 222
0 221
0 220
0 219
0 218
0 217
0 216
0 215
0 214
0 213
0 212
0 211
0 210
0 29
0 28
0 27
0 26
0 25
0 24
0 23
0 22
0 21
1 20
0
4. Multiply each bit by its corresponding power of 2 and add all the terms up.
0100 0000 0110 0011 0001 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010(2) =
(0 × 263 + 1 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 0 × 257 + 0 × 256 + 0 × 255 + 1 × 254 + 1 × 253 + 0 × 252 + 0 × 251 + 0 × 250 + 1 × 249 + 1 × 248 + 0 × 247 + 0 × 246 + 0 × 245 + 1 × 244 + 1 × 243 + 0 × 242 + 0 × 241 + 0 × 240 + 0 × 239 + 0 × 238 + 0 × 237 + 0 × 236 + 0 × 235 + 0 × 234 + 0 × 233 + 0 × 232 + 0 × 231 + 0 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 0 × 224 + 0 × 223 + 0 × 222 + 0 × 221 + 0 × 220 + 0 × 219 + 0 × 218 + 0 × 217 + 0 × 216 + 0 × 215 + 0 × 214 + 0 × 213 + 0 × 212 + 0 × 211 + 0 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 0 × 20)(10) =
(0 + 4 611 686 018 427 387 904 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 0 + 0 + 0 + 562 949 953 421 312 + 281 474 976 710 656 + 0 + 0 + 0 + 17 592 186 044 416 + 8 796 093 022 208 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 2 + 0)(10) =
(4 611 686 018 427 387 904 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 562 949 953 421 312 + 281 474 976 710 656 + 17 592 186 044 416 + 8 796 093 022 208 + 2)(10) =
4 639 578 429 400 809 474(10)
5. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0100 0000 0110 0011 0001 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010(2) = 4 639 578 429 400 809 474(10)
The number 0100 0000 0110 0011 0001 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010(2), signed binary in one's (1's) complement representation, converted and written as an integer in decimal system (base ten):
0100 0000 0110 0011 0001 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010(2) = 4 639 578 429 400 809 474(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.