1. Is this a positive or a negative number?
1100 0000 1111 1111 1111 1111 1110 0011 is the binary representation of a negative integer, on 32 bits (4 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.
1100 0000 1111 1111 1111 1111 1110 0011 - 1 = 1100 0000 1111 1111 1111 1111 1110 0010
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:
!(1100 0000 1111 1111 1111 1111 1110 0010) = 0011 1111 0000 0000 0000 0000 0001 1101
4. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
231
0 230
0 229
1 228
1 227
1 226
1 225
1 224
1 223
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
0 25
0 24
1 23
1 22
1 21
0 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0011 1111 0000 0000 0000 0000 0001 1101(2) =
(0 × 231 + 0 × 230 + 1 × 229 + 1 × 228 + 1 × 227 + 1 × 226 + 1 × 225 + 1 × 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 + 0 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20)(10) =
(0 + 0 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 33 554 432 + 16 777 216 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 16 + 8 + 4 + 0 + 1)(10) =
(536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 33 554 432 + 16 777 216 + 16 + 8 + 4 + 1)(10) =
1 056 964 637(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1100 0000 1111 1111 1111 1111 1110 0011(2) = -1 056 964 637(10)
The signed binary number in two's complement representation 1100 0000 1111 1111 1111 1111 1110 0011(2) converted and written as an integer in decimal system (base ten):
1100 0000 1111 1111 1111 1111 1110 0011(2) = -1 056 964 637(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.