What are the steps to convert the signed binary in two's (2's) complement representation to an integer in decimal system (in base ten)?
1. Is this a positive or a negative number?
0000 0000 1000 0011 0111 1101 0100 1111 1110 0000 1000 0000 0111 0110 0011 0001 is the binary representation of a positive integer, on 64 bits (8 Bytes).
- 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 = 01; 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
0 253
0 252
0 251
0 250
0 249
1 248
1 247
0 246
1 245
1 244
1 243
1 242
1 241
0 240
1 239
0 238
1 237
0 236
0 235
1 234
1 233
1 232
1 231
1 230
1 229
1 228
0 227
0 226
0 225
0 224
0 223
1 222
0 221
0 220
0 219
0 218
0 217
0 216
0 215
0 214
1 213
1 212
1 211
0 210
1 29
1 28
0 27
0 26
0 25
1 24
1 23
0 22
0 21
0 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 0000 1000 0011 0111 1101 0100 1111 1110 0000 1000 0000 0111 0110 0011 0001(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 0 × 257 + 0 × 256 + 1 × 255 + 0 × 254 + 0 × 253 + 0 × 252 + 0 × 251 + 0 × 250 + 1 × 249 + 1 × 248 + 0 × 247 + 1 × 246 + 1 × 245 + 1 × 244 + 1 × 243 + 1 × 242 + 0 × 241 + 1 × 240 + 0 × 239 + 1 × 238 + 0 × 237 + 0 × 236 + 1 × 235 + 1 × 234 + 1 × 233 + 1 × 232 + 1 × 231 + 1 × 230 + 1 × 229 + 0 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 0 × 224 + 1 × 223 + 0 × 222 + 0 × 221 + 0 × 220 + 0 × 219 + 0 × 218 + 0 × 217 + 0 × 216 + 0 × 215 + 1 × 214 + 1 × 213 + 1 × 212 + 0 × 211 + 1 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 1 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 36 028 797 018 963 968 + 0 + 0 + 0 + 0 + 0 + 562 949 953 421 312 + 281 474 976 710 656 + 0 + 70 368 744 177 664 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 0 + 1 099 511 627 776 + 0 + 274 877 906 944 + 0 + 0 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 1 073 741 824 + 536 870 912 + 0 + 0 + 0 + 0 + 0 + 8 388 608 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 16 384 + 8 192 + 4 096 + 0 + 1 024 + 512 + 0 + 0 + 0 + 32 + 16 + 0 + 0 + 0 + 1)(10) =
(36 028 797 018 963 968 + 562 949 953 421 312 + 281 474 976 710 656 + 70 368 744 177 664 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 1 099 511 627 776 + 274 877 906 944 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 1 073 741 824 + 536 870 912 + 8 388 608 + 16 384 + 8 192 + 4 096 + 1 024 + 512 + 32 + 16 + 1)(10) =
37 011 003 971 499 569(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0000 1000 0011 0111 1101 0100 1111 1110 0000 1000 0000 0111 0110 0011 0001(2) = 37 011 003 971 499 569(10)
The number 0000 0000 1000 0011 0111 1101 0100 1111 1110 0000 1000 0000 0111 0110 0011 0001(2), signed binary in two's (2's) complement representation, converted and written as an integer in decimal system (base ten):
0000 0000 1000 0011 0111 1101 0100 1111 1110 0000 1000 0000 0111 0110 0011 0001(2) = 37 011 003 971 499 569(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.