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?
0111 1011 1100 0000 0000 1001 1001 0000 0000 0011 0010 0010 0011 0010 0011 0001 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
1 261
1 260
1 259
1 258
0 257
1 256
1 255
1 254
1 253
0 252
0 251
0 250
0 249
0 248
0 247
0 246
0 245
0 244
0 243
1 242
0 241
0 240
1 239
1 238
0 237
0 236
1 235
0 234
0 233
0 232
0 231
0 230
0 229
0 228
0 227
0 226
0 225
1 224
1 223
0 222
0 221
1 220
0 219
0 218
0 217
1 216
0 215
0 214
0 213
1 212
1 211
0 210
0 29
1 28
0 27
0 26
0 25
1 24
1 23
0 22
0 21
0 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0111 1011 1100 0000 0000 1001 1001 0000 0000 0011 0010 0010 0011 0010 0011 0001(2) =
(0 × 263 + 1 × 262 + 1 × 261 + 1 × 260 + 1 × 259 + 0 × 258 + 1 × 257 + 1 × 256 + 1 × 255 + 1 × 254 + 0 × 253 + 0 × 252 + 0 × 251 + 0 × 250 + 0 × 249 + 0 × 248 + 0 × 247 + 0 × 246 + 0 × 245 + 0 × 244 + 1 × 243 + 0 × 242 + 0 × 241 + 1 × 240 + 1 × 239 + 0 × 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 + 1 × 225 + 1 × 224 + 0 × 223 + 0 × 222 + 1 × 221 + 0 × 220 + 0 × 219 + 0 × 218 + 1 × 217 + 0 × 216 + 0 × 215 + 0 × 214 + 1 × 213 + 1 × 212 + 0 × 211 + 0 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 1 × 20)(10) =
(0 + 4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 0 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 8 796 093 022 208 + 0 + 0 + 1 099 511 627 776 + 549 755 813 888 + 0 + 0 + 68 719 476 736 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 33 554 432 + 16 777 216 + 0 + 0 + 2 097 152 + 0 + 0 + 0 + 131 072 + 0 + 0 + 0 + 8 192 + 4 096 + 0 + 0 + 512 + 0 + 0 + 0 + 32 + 16 + 0 + 0 + 0 + 1)(10) =
(4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 8 796 093 022 208 + 1 099 511 627 776 + 549 755 813 888 + 68 719 476 736 + 33 554 432 + 16 777 216 + 2 097 152 + 131 072 + 8 192 + 4 096 + 512 + 32 + 16 + 1)(10) =
8 917 137 776 326 095 409(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0111 1011 1100 0000 0000 1001 1001 0000 0000 0011 0010 0010 0011 0010 0011 0001(2) = 8 917 137 776 326 095 409(10)
The number 0111 1011 1100 0000 0000 1001 1001 0000 0000 0011 0010 0010 0011 0010 0011 0001(2), signed binary in two's (2's) complement representation, converted and written as an integer in decimal system (base ten):
0111 1011 1100 0000 0000 1001 1001 0000 0000 0011 0010 0010 0011 0010 0011 0001(2) = 8 917 137 776 326 095 409(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.