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?
0011 0000 0100 1010 0000 0000 0111 0000 0000 0000 0000 0000 0000 0000 0100 1110 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
1 260
1 259
0 258
0 257
0 256
0 255
0 254
1 253
0 252
0 251
1 250
0 249
1 248
0 247
0 246
0 245
0 244
0 243
0 242
0 241
0 240
0 239
0 238
1 237
1 236
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
1 25
0 24
0 23
1 22
1 21
1 20
0
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0011 0000 0100 1010 0000 0000 0111 0000 0000 0000 0000 0000 0000 0000 0100 1110(2) =
(0 × 263 + 0 × 262 + 1 × 261 + 1 × 260 + 0 × 259 + 0 × 258 + 0 × 257 + 0 × 256 + 0 × 255 + 1 × 254 + 0 × 253 + 0 × 252 + 1 × 251 + 0 × 250 + 1 × 249 + 0 × 248 + 0 × 247 + 0 × 246 + 0 × 245 + 0 × 244 + 0 × 243 + 0 × 242 + 0 × 241 + 0 × 240 + 0 × 239 + 1 × 238 + 1 × 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 + 1 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20)(10) =
(0 + 0 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 0 + 0 + 0 + 0 + 0 + 18 014 398 509 481 984 + 0 + 0 + 2 251 799 813 685 248 + 0 + 562 949 953 421 312 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 274 877 906 944 + 137 438 953 472 + 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 + 64 + 0 + 0 + 8 + 4 + 2 + 0)(10) =
(2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 18 014 398 509 481 984 + 2 251 799 813 685 248 + 562 949 953 421 312 + 274 877 906 944 + 137 438 953 472 + 68 719 476 736 + 64 + 8 + 4 + 2)(10) =
3 479 594 143 133 466 702(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0011 0000 0100 1010 0000 0000 0111 0000 0000 0000 0000 0000 0000 0000 0100 1110(2) = 3 479 594 143 133 466 702(10)
The number 0011 0000 0100 1010 0000 0000 0111 0000 0000 0000 0000 0000 0000 0000 0100 1110(2), signed binary in two's (2's) complement representation, converted and written as an integer in decimal system (base ten):
0011 0000 0100 1010 0000 0000 0111 0000 0000 0000 0000 0000 0000 0000 0100 1110(2) = 3 479 594 143 133 466 702(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.