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?
1011 0111 0011 1010 1011 0110 0010 1111 1111 1111 1111 1111 1111 1111 1101 1001 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.
1011 0111 0011 1010 1011 0110 0010 1111 1111 1111 1111 1111 1111 1111 1101 1001 - 1 = 1011 0111 0011 1010 1011 0110 0010 1111 1111 1111 1111 1111 1111 1111 1101 1000
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:
!(1011 0111 0011 1010 1011 0110 0010 1111 1111 1111 1111 1111 1111 1111 1101 1000) = 0100 1000 1100 0101 0100 1001 1101 0000 0000 0000 0000 0000 0000 0000 0010 0111
4. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
263
0 262
1 261
0 260
0 259
1 258
0 257
0 256
0 255
1 254
1 253
0 252
0 251
0 250
1 249
0 248
1 247
0 246
1 245
0 244
0 243
1 242
0 241
0 240
1 239
1 238
1 237
0 236
1 235
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
0 23
0 22
1 21
1 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0100 1000 1100 0101 0100 1001 1101 0000 0000 0000 0000 0000 0000 0000 0010 0111(2) =
(0 × 263 + 1 × 262 + 0 × 261 + 0 × 260 + 1 × 259 + 0 × 258 + 0 × 257 + 0 × 256 + 1 × 255 + 1 × 254 + 0 × 253 + 0 × 252 + 0 × 251 + 1 × 250 + 0 × 249 + 1 × 248 + 0 × 247 + 1 × 246 + 0 × 245 + 0 × 244 + 1 × 243 + 0 × 242 + 0 × 241 + 1 × 240 + 1 × 239 + 1 × 238 + 0 × 237 + 1 × 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 + 0 × 23 + 1 × 22 + 1 × 21 + 1 × 20)(10) =
(0 + 4 611 686 018 427 387 904 + 0 + 0 + 576 460 752 303 423 488 + 0 + 0 + 0 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 0 + 0 + 0 + 1 125 899 906 842 624 + 0 + 281 474 976 710 656 + 0 + 70 368 744 177 664 + 0 + 0 + 8 796 093 022 208 + 0 + 0 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 0 + 68 719 476 736 + 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 + 0 + 0 + 4 + 2 + 1)(10) =
(4 611 686 018 427 387 904 + 576 460 752 303 423 488 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 1 125 899 906 842 624 + 281 474 976 710 656 + 70 368 744 177 664 + 8 796 093 022 208 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 68 719 476 736 + 32 + 4 + 2 + 1)(10) =
5 243 678 498 844 835 879(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1011 0111 0011 1010 1011 0110 0010 1111 1111 1111 1111 1111 1111 1111 1101 1001(2) = -5 243 678 498 844 835 879(10)
The number 1011 0111 0011 1010 1011 0110 0010 1111 1111 1111 1111 1111 1111 1111 1101 1001(2), signed binary in two's (2's) complement representation, converted and written as an integer in decimal system (base ten):
1011 0111 0011 1010 1011 0110 0010 1111 1111 1111 1111 1111 1111 1111 1101 1001(2) = -5 243 678 498 844 835 879(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.