1. Is this a positive or a negative number?
1011 1110 0101 0011 1001 1010 0101 0001 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.
1011 1110 0101 0011 1001 1010 0101 0001 - 1 = 1011 1110 0101 0011 1001 1010 0101 0000
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:
!(1011 1110 0101 0011 1001 1010 0101 0000) = 0100 0001 1010 1100 0110 0101 1010 1111
4. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
231
0 230
1 229
0 228
0 227
0 226
0 225
0 224
1 223
1 222
0 221
1 220
0 219
1 218
1 217
0 216
0 215
0 214
1 213
1 212
0 211
0 210
1 29
0 28
1 27
1 26
0 25
1 24
0 23
1 22
1 21
1 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0100 0001 1010 1100 0110 0101 1010 1111(2) =
(0 × 231 + 1 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 1 × 224 + 1 × 223 + 0 × 222 + 1 × 221 + 0 × 220 + 1 × 219 + 1 × 218 + 0 × 217 + 0 × 216 + 0 × 215 + 1 × 214 + 1 × 213 + 0 × 212 + 0 × 211 + 1 × 210 + 0 × 29 + 1 × 28 + 1 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20)(10) =
(0 + 1 073 741 824 + 0 + 0 + 0 + 0 + 0 + 16 777 216 + 8 388 608 + 0 + 2 097 152 + 0 + 524 288 + 262 144 + 0 + 0 + 0 + 16 384 + 8 192 + 0 + 0 + 1 024 + 0 + 256 + 128 + 0 + 32 + 0 + 8 + 4 + 2 + 1)(10) =
(1 073 741 824 + 16 777 216 + 8 388 608 + 2 097 152 + 524 288 + 262 144 + 16 384 + 8 192 + 1 024 + 256 + 128 + 32 + 8 + 4 + 2 + 1)(10) =
1 101 817 263(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1011 1110 0101 0011 1001 1010 0101 0001(2) = -1 101 817 263(10)
The signed binary number in two's complement representation 1011 1110 0101 0011 1001 1010 0101 0001(2) converted and written as an integer in decimal system (base ten):
1011 1110 0101 0011 1001 1010 0101 0001(2) = -1 101 817 263(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.