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?
1111 1111 1111 1111 1111 1111 1111 1111 1000 0001 0101 1011 0011 0011 0000 1010 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.
1111 1111 1111 1111 1111 1111 1111 1111 1000 0001 0101 1011 0011 0011 0000 1010 - 1 = 1111 1111 1111 1111 1111 1111 1111 1111 1000 0001 0101 1011 0011 0011 0000 1001
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:
!(1111 1111 1111 1111 1111 1111 1111 1111 1000 0001 0101 1011 0011 0011 0000 1001) = 0000 0000 0000 0000 0000 0000 0000 0000 0111 1110 1010 0100 1100 1100 1111 0110
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
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
1 229
1 228
1 227
1 226
1 225
1 224
0 223
1 222
0 221
1 220
0 219
0 218
1 217
0 216
0 215
1 214
1 213
0 212
0 211
1 210
1 29
0 28
0 27
1 26
1 25
1 24
1 23
0 22
1 21
1 20
0
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 0000 0000 0000 0000 0000 0000 0000 0111 1110 1010 0100 1100 1100 1111 0110(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 0 × 260 + 0 × 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 + 1 × 230 + 1 × 229 + 1 × 228 + 1 × 227 + 1 × 226 + 1 × 225 + 0 × 224 + 1 × 223 + 0 × 222 + 1 × 221 + 0 × 220 + 0 × 219 + 1 × 218 + 0 × 217 + 0 × 216 + 1 × 215 + 1 × 214 + 0 × 213 + 0 × 212 + 1 × 211 + 1 × 210 + 0 × 29 + 0 × 28 + 1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20)(10) =
(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 + 1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 33 554 432 + 0 + 8 388 608 + 0 + 2 097 152 + 0 + 0 + 262 144 + 0 + 0 + 32 768 + 16 384 + 0 + 0 + 2 048 + 1 024 + 0 + 0 + 128 + 64 + 32 + 16 + 0 + 4 + 2 + 0)(10) =
(1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 33 554 432 + 8 388 608 + 2 097 152 + 262 144 + 32 768 + 16 384 + 2 048 + 1 024 + 128 + 64 + 32 + 16 + 4 + 2)(10) =
2 124 729 590(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1111 1111 1111 1111 1111 1111 1111 1111 1000 0001 0101 1011 0011 0011 0000 1010(2) = -2 124 729 590(10)
The number 1111 1111 1111 1111 1111 1111 1111 1111 1000 0001 0101 1011 0011 0011 0000 1010(2), signed binary in two's (2's) complement representation, converted and written as an integer in decimal system (base ten):
1111 1111 1111 1111 1111 1111 1111 1111 1000 0001 0101 1011 0011 0011 0000 1010(2) = -2 124 729 590(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.