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?
1001 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1100 0101 is the binary representation of a negative 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.
1001 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1100 0101 - 1 = 1001 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1100 0100
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:
!(1001 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1100 0100) = 0110 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1011
4. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
263
0 262
1 261
1 260
0 259
1 258
0 257
0 256
0 255
0 254
0 253
0 252
0 251
0 250
0 249
0 248
0 247
0 246
0 245
0 244
0 243
0 242
0 241
0 240
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
0 26
0 25
1 24
1 23
1 22
0 21
1 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0110 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1011(2) =
(0 × 263 + 1 × 262 + 1 × 261 + 0 × 260 + 1 × 259 + 0 × 258 + 0 × 257 + 0 × 256 + 0 × 255 + 0 × 254 + 0 × 253 + 0 × 252 + 0 × 251 + 0 × 250 + 0 × 249 + 0 × 248 + 0 × 247 + 0 × 246 + 0 × 245 + 0 × 244 + 0 × 243 + 0 × 242 + 0 × 241 + 0 × 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 + 0 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 1 × 20)(10) =
(0 + 4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 0 + 576 460 752 303 423 488 + 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 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 32 + 16 + 8 + 0 + 2 + 1)(10) =
(4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 576 460 752 303 423 488 + 32 + 16 + 8 + 2 + 1)(10) =
7 493 989 779 944 505 403(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1001 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1100 0101(2) = -7 493 989 779 944 505 403(10)
The number 1001 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1100 0101(2), signed binary in two's (2's) complement representation, converted and written as an integer in decimal system (base ten):
1001 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1100 0101(2) = -7 493 989 779 944 505 403(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.