1. Is this a positive or a negative number?
0000 0000 0000 0000 0001 0011 0100 0010 1100 0101 1111 1111 1111 1111 1101 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 = 1; 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
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
1 243
0 242
0 241
1 240
1 239
0 238
1 237
0 236
0 235
0 234
0 233
1 232
0 231
1 230
1 229
0 228
0 227
0 226
1 225
0 224
1 223
1 222
1 221
1 220
1 219
1 218
1 217
1 216
1 215
1 214
1 213
1 212
1 211
1 210
1 29
1 28
1 27
1 26
1 25
0 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 0000 0000 0000 0001 0011 0100 0010 1100 0101 1111 1111 1111 1111 1101 0101(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 + 1 × 244 + 0 × 243 + 0 × 242 + 1 × 241 + 1 × 240 + 0 × 239 + 1 × 238 + 0 × 237 + 0 × 236 + 0 × 235 + 0 × 234 + 1 × 233 + 0 × 232 + 1 × 231 + 1 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 1 × 226 + 0 × 225 + 1 × 224 + 1 × 223 + 1 × 222 + 1 × 221 + 1 × 220 + 1 × 219 + 1 × 218 + 1 × 217 + 1 × 216 + 1 × 215 + 1 × 214 + 1 × 213 + 1 × 212 + 1 × 211 + 1 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 1 × 26 + 0 × 25 + 1 × 24 + 0 × 23 + 1 × 22 + 0 × 21 + 1 × 20)(10) =
(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 17 592 186 044 416 + 0 + 0 + 2 199 023 255 552 + 1 099 511 627 776 + 0 + 274 877 906 944 + 0 + 0 + 0 + 0 + 8 589 934 592 + 0 + 2 147 483 648 + 1 073 741 824 + 0 + 0 + 0 + 67 108 864 + 0 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 65 536 + 32 768 + 16 384 + 8 192 + 4 096 + 2 048 + 1 024 + 512 + 256 + 128 + 64 + 0 + 16 + 0 + 4 + 0 + 1)(10) =
(17 592 186 044 416 + 2 199 023 255 552 + 1 099 511 627 776 + 274 877 906 944 + 8 589 934 592 + 2 147 483 648 + 1 073 741 824 + 67 108 864 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 65 536 + 32 768 + 16 384 + 8 192 + 4 096 + 2 048 + 1 024 + 512 + 256 + 128 + 64 + 16 + 4 + 1)(10) =
21 177 510 658 005(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0000 0000 0000 0000 0001 0011 0100 0010 1100 0101 1111 1111 1111 1111 1101 0101(2) = 21 177 510 658 005(10)
The signed binary number in two's complement representation 0000 0000 0000 0000 0001 0011 0100 0010 1100 0101 1111 1111 1111 1111 1101 0101(2) converted and written as an integer in decimal system (base ten):
0000 0000 0000 0000 0001 0011 0100 0010 1100 0101 1111 1111 1111 1111 1101 0101(2) = 21 177 510 658 005(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.