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?
0001 1010 0101 0000 0000 1111 0000 0000 0000 0000 0000 0000 0000 0000 0101 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
1 259
1 258
0 257
1 256
0 255
0 254
1 253
0 252
1 251
0 250
0 249
0 248
0 247
0 246
0 245
0 244
0 243
1 242
1 241
1 240
1 239
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
1 25
0 24
1 23
1 22
0 21
1 20
0
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0001 1010 0101 0000 0000 1111 0000 0000 0000 0000 0000 0000 0000 0000 0101 1010(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 1 × 260 + 1 × 259 + 0 × 258 + 1 × 257 + 0 × 256 + 0 × 255 + 1 × 254 + 0 × 253 + 1 × 252 + 0 × 251 + 0 × 250 + 0 × 249 + 0 × 248 + 0 × 247 + 0 × 246 + 0 × 245 + 0 × 244 + 1 × 243 + 1 × 242 + 1 × 241 + 1 × 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 + 1 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20)(10) =
(0 + 0 + 0 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 0 + 144 115 188 075 855 872 + 0 + 0 + 18 014 398 509 481 984 + 0 + 4 503 599 627 370 496 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 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 + 64 + 0 + 16 + 8 + 0 + 2 + 0)(10) =
(1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 144 115 188 075 855 872 + 18 014 398 509 481 984 + 4 503 599 627 370 496 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 64 + 16 + 8 + 2)(10) =
1 896 031 935 797 395 546(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0001 1010 0101 0000 0000 1111 0000 0000 0000 0000 0000 0000 0000 0000 0101 1010(2) = 1 896 031 935 797 395 546(10)
The number 0001 1010 0101 0000 0000 1111 0000 0000 0000 0000 0000 0000 0000 0000 0101 1010(2), signed binary in two's (2's) complement representation, converted and written as an integer in decimal system (base ten):
0001 1010 0101 0000 0000 1111 0000 0000 0000 0000 0000 0000 0000 0000 0101 1010(2) = 1 896 031 935 797 395 546(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.