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 0000 0010 1000 0100 0100 1000 0011 0000 0011 0000 0100 1000 0011 0101 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
0 254
0 253
0 252
0 251
0 250
0 249
1 248
0 247
1 246
0 245
0 244
0 243
0 242
1 241
0 240
0 239
0 238
1 237
0 236
0 235
1 234
0 233
0 232
0 231
0 230
0 229
1 228
1 227
0 226
0 225
0 224
0 223
0 222
0 221
1 220
1 219
0 218
0 217
0 216
0 215
0 214
1 213
0 212
0 211
1 210
0 29
0 28
0 27
0 26
0 25
1 24
1 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 0001 0000 0010 1000 0100 0100 1000 0011 0000 0011 0000 0100 1000 0011 0101(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 0 × 257 + 1 × 256 + 0 × 255 + 0 × 254 + 0 × 253 + 0 × 252 + 0 × 251 + 0 × 250 + 1 × 249 + 0 × 248 + 1 × 247 + 0 × 246 + 0 × 245 + 0 × 244 + 0 × 243 + 1 × 242 + 0 × 241 + 0 × 240 + 0 × 239 + 1 × 238 + 0 × 237 + 0 × 236 + 1 × 235 + 0 × 234 + 0 × 233 + 0 × 232 + 0 × 231 + 0 × 230 + 1 × 229 + 1 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 0 × 224 + 0 × 223 + 0 × 222 + 1 × 221 + 1 × 220 + 0 × 219 + 0 × 218 + 0 × 217 + 0 × 216 + 0 × 215 + 1 × 214 + 0 × 213 + 0 × 212 + 1 × 211 + 0 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 1 × 22 + 0 × 21 + 1 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 0 + 72 057 594 037 927 936 + 0 + 0 + 0 + 0 + 0 + 0 + 562 949 953 421 312 + 0 + 140 737 488 355 328 + 0 + 0 + 0 + 0 + 4 398 046 511 104 + 0 + 0 + 0 + 274 877 906 944 + 0 + 0 + 34 359 738 368 + 0 + 0 + 0 + 0 + 0 + 536 870 912 + 268 435 456 + 0 + 0 + 0 + 0 + 0 + 0 + 2 097 152 + 1 048 576 + 0 + 0 + 0 + 0 + 0 + 16 384 + 0 + 0 + 2 048 + 0 + 0 + 0 + 0 + 0 + 32 + 16 + 0 + 4 + 0 + 1)(10) =
(72 057 594 037 927 936 + 562 949 953 421 312 + 140 737 488 355 328 + 4 398 046 511 104 + 274 877 906 944 + 34 359 738 368 + 536 870 912 + 268 435 456 + 2 097 152 + 1 048 576 + 16 384 + 2 048 + 32 + 16 + 4 + 1)(10) =
72 765 989 572 331 573(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0001 0000 0010 1000 0100 0100 1000 0011 0000 0011 0000 0100 1000 0011 0101(2) = 72 765 989 572 331 573(10)
The number 0000 0001 0000 0010 1000 0100 0100 1000 0011 0000 0011 0000 0100 1000 0011 0101(2), signed binary in two's (2's) complement representation, converted and written as an integer in decimal system (base ten):
0000 0001 0000 0010 1000 0100 0100 1000 0011 0000 0011 0000 0100 1000 0011 0101(2) = 72 765 989 572 331 573(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.