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?
0000 1100 1010 0100 1001 0101 0001 0010 0100 1001 0010 0100 1001 0010 0111 0101 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
0 260
0 259
1 258
1 257
0 256
0 255
1 254
0 253
1 252
0 251
0 250
1 249
0 248
0 247
1 246
0 245
0 244
1 243
0 242
1 241
0 240
1 239
0 238
0 237
0 236
1 235
0 234
0 233
1 232
0 231
0 230
1 229
0 228
0 227
1 226
0 225
0 224
1 223
0 222
0 221
1 220
0 219
0 218
1 217
0 216
0 215
1 214
0 213
0 212
1 211
0 210
0 29
1 28
0 27
0 26
1 25
1 24
1 23
0 22
1 21
0 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0000 1100 1010 0100 1001 0101 0001 0010 0100 1001 0010 0100 1001 0010 0111 0101(2) =
(0 × 263 + 0 × 262 + 0 × 261 + 0 × 260 + 1 × 259 + 1 × 258 + 0 × 257 + 0 × 256 + 1 × 255 + 0 × 254 + 1 × 253 + 0 × 252 + 0 × 251 + 1 × 250 + 0 × 249 + 0 × 248 + 1 × 247 + 0 × 246 + 0 × 245 + 1 × 244 + 0 × 243 + 1 × 242 + 0 × 241 + 1 × 240 + 0 × 239 + 0 × 238 + 0 × 237 + 1 × 236 + 0 × 235 + 0 × 234 + 1 × 233 + 0 × 232 + 0 × 231 + 1 × 230 + 0 × 229 + 0 × 228 + 1 × 227 + 0 × 226 + 0 × 225 + 1 × 224 + 0 × 223 + 0 × 222 + 1 × 221 + 0 × 220 + 0 × 219 + 1 × 218 + 0 × 217 + 0 × 216 + 1 × 215 + 0 × 214 + 0 × 213 + 1 × 212 + 0 × 211 + 0 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 1 × 22 + 0 × 21 + 1 × 20)(10) =
(0 + 0 + 0 + 0 + 576 460 752 303 423 488 + 288 230 376 151 711 744 + 0 + 0 + 36 028 797 018 963 968 + 0 + 9 007 199 254 740 992 + 0 + 0 + 1 125 899 906 842 624 + 0 + 0 + 140 737 488 355 328 + 0 + 0 + 17 592 186 044 416 + 0 + 4 398 046 511 104 + 0 + 1 099 511 627 776 + 0 + 0 + 0 + 68 719 476 736 + 0 + 0 + 8 589 934 592 + 0 + 0 + 1 073 741 824 + 0 + 0 + 134 217 728 + 0 + 0 + 16 777 216 + 0 + 0 + 2 097 152 + 0 + 0 + 262 144 + 0 + 0 + 32 768 + 0 + 0 + 4 096 + 0 + 0 + 512 + 0 + 0 + 64 + 32 + 16 + 0 + 4 + 0 + 1)(10) =
(576 460 752 303 423 488 + 288 230 376 151 711 744 + 36 028 797 018 963 968 + 9 007 199 254 740 992 + 1 125 899 906 842 624 + 140 737 488 355 328 + 17 592 186 044 416 + 4 398 046 511 104 + 1 099 511 627 776 + 68 719 476 736 + 8 589 934 592 + 1 073 741 824 + 134 217 728 + 16 777 216 + 2 097 152 + 262 144 + 32 768 + 4 096 + 512 + 64 + 32 + 16 + 4 + 1)(10) =
911 016 930 404 766 325(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 1100 1010 0100 1001 0101 0001 0010 0100 1001 0010 0100 1001 0010 0111 0101(2) = 911 016 930 404 766 325(10)
The number 0000 1100 1010 0100 1001 0101 0001 0010 0100 1001 0010 0100 1001 0010 0111 0101(2), signed binary in two's (2's) complement representation, converted and written as an integer in decimal system (base ten):
0000 1100 1010 0100 1001 0101 0001 0010 0100 1001 0010 0100 1001 0010 0111 0101(2) = 911 016 930 404 766 325(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.