1. Is this a positive or a negative number?
1000 0000 0000 0000 0000 0000 1000 1010 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.
1000 0000 0000 0000 0000 0000 1000 1010 - 1 = 1000 0000 0000 0000 0000 0000 1000 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 0000 0000 0000 0000 0000 1000 1001) = 0111 1111 1111 1111 1111 1111 0111 0110
4. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
231
0 230
1 229
1 228
1 227
1 226
1 225
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
0 26
1 25
1 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 1111 1111 1111 1111 1111 0111 0110(2) =
(0 × 231 + 1 × 230 + 1 × 229 + 1 × 228 + 1 × 227 + 1 × 226 + 1 × 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 + 0 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20)(10) =
(0 + 1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 33 554 432 + 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 + 0 + 64 + 32 + 16 + 0 + 4 + 2 + 0)(10) =
(1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 33 554 432 + 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 + 64 + 32 + 16 + 4 + 2)(10) =
2 147 483 510(10)
6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1000 0000 0000 0000 0000 0000 1000 1010(2) = -2 147 483 510(10)
The signed binary number in two's complement representation 1000 0000 0000 0000 0000 0000 1000 1010(2) converted and written as an integer in decimal system (base ten):
1000 0000 0000 0000 0000 0000 1000 1010(2) = -2 147 483 510(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.