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 0000 1101 0001 1011 0010 0000 0001 0010 0001 0010 0110 0011 1010 1000 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
0 254
0 253
0 252
0 251
1 250
1 249
0 248
1 247
0 246
0 245
0 244
1 243
1 242
0 241
1 240
1 239
0 238
0 237
1 236
0 235
0 234
0 233
0 232
0 231
0 230
0 229
0 228
1 227
0 226
0 225
1 224
0 223
0 222
0 221
0 220
1 219
0 218
0 217
1 216
0 215
0 214
1 213
1 212
0 211
0 210
0 29
1 28
1 27
1 26
0 25
1 24
0 23
1 22
0 21
0 20
0
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 0000 0000 1101 0001 1011 0010 0000 0001 0010 0001 0010 0110 0011 1010 1000(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 0 × 257 + 0 × 256 + 0 × 255 + 0 × 254 + 0 × 253 + 0 × 252 + 1 × 251 + 1 × 250 + 0 × 249 + 1 × 248 + 0 × 247 + 0 × 246 + 0 × 245 + 1 × 244 + 1 × 243 + 0 × 242 + 1 × 241 + 1 × 240 + 0 × 239 + 0 × 238 + 1 × 237 + 0 × 236 + 0 × 235 + 0 × 234 + 0 × 233 + 0 × 232 + 0 × 231 + 0 × 230 + 0 × 229 + 1 × 228 + 0 × 227 + 0 × 226 + 1 × 225 + 0 × 224 + 0 × 223 + 0 × 222 + 0 × 221 + 1 × 220 + 0 × 219 + 0 × 218 + 1 × 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 + 0 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 0 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 0 + 281 474 976 710 656 + 0 + 0 + 0 + 17 592 186 044 416 + 8 796 093 022 208 + 0 + 2 199 023 255 552 + 1 099 511 627 776 + 0 + 0 + 137 438 953 472 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 268 435 456 + 0 + 0 + 33 554 432 + 0 + 0 + 0 + 0 + 1 048 576 + 0 + 0 + 131 072 + 0 + 0 + 16 384 + 8 192 + 0 + 0 + 0 + 512 + 256 + 128 + 0 + 32 + 0 + 8 + 0 + 0 + 0)(10) =
(2 251 799 813 685 248 + 1 125 899 906 842 624 + 281 474 976 710 656 + 17 592 186 044 416 + 8 796 093 022 208 + 2 199 023 255 552 + 1 099 511 627 776 + 137 438 953 472 + 268 435 456 + 33 554 432 + 1 048 576 + 131 072 + 16 384 + 8 192 + 512 + 256 + 128 + 32 + 8)(10) =
3 688 999 253 337 000(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0000 0000 1101 0001 1011 0010 0000 0001 0010 0001 0010 0110 0011 1010 1000(2) = 3 688 999 253 337 000(10)
The number 0000 0000 0000 1101 0001 1011 0010 0000 0001 0010 0001 0010 0110 0011 1010 1000(2), signed binary in two's (2's) complement representation, converted and written as an integer in decimal system (base ten):
0000 0000 0000 1101 0001 1011 0010 0000 0001 0010 0001 0010 0110 0011 1010 1000(2) = 3 688 999 253 337 000(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.