1. Is this a positive or a negative number?
1000 1101 0010 1010 is the binary representation of a negative integer, on 16 bits (2 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.
1000 1101 0010 1010 - 1 = 1000 1101 0010 1001
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:
!(1000 1101 0010 1001) = 0111 0010 1101 0110
4. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
215
0 214
1 213
1 212
1 211
0 210
0 29
1 28
0 27
1 26
1 25
0 24
1 23
0 22
1 21
1 20
0
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0111 0010 1101 0110(2) =
(0 × 215 + 1 × 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 + 0 × 20)(10) =
(0 + 16 384 + 8 192 + 4 096 + 0 + 0 + 512 + 0 + 128 + 64 + 0 + 16 + 0 + 4 + 2 + 0)(10) =
(16 384 + 8 192 + 4 096 + 512 + 128 + 64 + 16 + 4 + 2)(10) =
29 398(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1000 1101 0010 1010(2) = -29 398(10)
The signed binary number in two's complement representation 1000 1101 0010 1010(2) converted and written as an integer in decimal system (base ten):
1000 1101 0010 1010(2) = -29 398(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.