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 0001 1101 0101 0110 1000 1010 0000 1101 0100 1010 0100 0010 1010 0000 1010 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
1 255
1 254
1 253
0 252
1 251
0 250
1 249
0 248
1 247
0 246
1 245
1 244
0 243
1 242
0 241
0 240
0 239
1 238
0 237
1 236
0 235
0 234
0 233
0 232
0 231
1 230
1 229
0 228
1 227
0 226
1 225
0 224
0 223
1 222
0 221
1 220
0 219
0 218
1 217
0 216
0 215
0 214
0 213
1 212
0 211
1 210
0 29
1 28
0 27
0 26
0 25
0 24
0 23
1 22
0 21
1 20
0
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 0001 1101 0101 0110 1000 1010 0000 1101 0100 1010 0100 0010 1010 0000 1010(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 0 × 257 + 1 × 256 + 1 × 255 + 1 × 254 + 0 × 253 + 1 × 252 + 0 × 251 + 1 × 250 + 0 × 249 + 1 × 248 + 0 × 247 + 1 × 246 + 1 × 245 + 0 × 244 + 1 × 243 + 0 × 242 + 0 × 241 + 0 × 240 + 1 × 239 + 0 × 238 + 1 × 237 + 0 × 236 + 0 × 235 + 0 × 234 + 0 × 233 + 0 × 232 + 1 × 231 + 1 × 230 + 0 × 229 + 1 × 228 + 0 × 227 + 1 × 226 + 0 × 225 + 0 × 224 + 1 × 223 + 0 × 222 + 1 × 221 + 0 × 220 + 0 × 219 + 1 × 218 + 0 × 217 + 0 × 216 + 0 × 215 + 0 × 214 + 1 × 213 + 0 × 212 + 1 × 211 + 0 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 0 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 0 + 4 503 599 627 370 496 + 0 + 1 125 899 906 842 624 + 0 + 281 474 976 710 656 + 0 + 70 368 744 177 664 + 35 184 372 088 832 + 0 + 8 796 093 022 208 + 0 + 0 + 0 + 549 755 813 888 + 0 + 137 438 953 472 + 0 + 0 + 0 + 0 + 0 + 2 147 483 648 + 1 073 741 824 + 0 + 268 435 456 + 0 + 67 108 864 + 0 + 0 + 8 388 608 + 0 + 2 097 152 + 0 + 0 + 262 144 + 0 + 0 + 0 + 0 + 8 192 + 0 + 2 048 + 0 + 512 + 0 + 0 + 0 + 0 + 0 + 8 + 0 + 2 + 0)(10) =
(72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 4 503 599 627 370 496 + 1 125 899 906 842 624 + 281 474 976 710 656 + 70 368 744 177 664 + 35 184 372 088 832 + 8 796 093 022 208 + 549 755 813 888 + 137 438 953 472 + 2 147 483 648 + 1 073 741 824 + 268 435 456 + 67 108 864 + 8 388 608 + 2 097 152 + 262 144 + 8 192 + 2 048 + 512 + 8 + 2)(10) =
132 126 804 048 882 186(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0001 1101 0101 0110 1000 1010 0000 1101 0100 1010 0100 0010 1010 0000 1010(2) = 132 126 804 048 882 186(10)
The number 0000 0001 1101 0101 0110 1000 1010 0000 1101 0100 1010 0100 0010 1010 0000 1010(2), signed binary in two's (2's) complement representation, converted and written as an integer in decimal system (base ten):
0000 0001 1101 0101 0110 1000 1010 0000 1101 0100 1010 0100 0010 1010 0000 1010(2) = 132 126 804 048 882 186(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.