1. Is this a positive or a negative number?
1111 1111 1111 1101 1100 1011 0110 1001 is the binary representation of a negative integer, on 32 bits (4 Bytes).
In a signed binary, the first bit (the leftmost) is reserved for the sign,
1 = negative, 0 = positive. This bit does not count when calculating the absolute value.
2. Construct the unsigned binary number.
Exclude the first bit (the leftmost), that is reserved for the sign:
1111 1111 1111 1101 1100 1011 0110 1001 = 111 1111 1111 1101 1100 1011 0110 1001
3. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
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
0 216
1 215
1 214
1 213
0 212
0 211
1 210
0 29
1 28
1 27
0 26
1 25
1 24
0 23
1 22
0 21
0 20
1
4. Multiply each bit by its corresponding power of 2 and add all the terms up.
111 1111 1111 1101 1100 1011 0110 1001(2) =
(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 + 0 × 217 + 1 × 216 + 1 × 215 + 1 × 214 + 0 × 213 + 0 × 212 + 1 × 211 + 0 × 210 + 1 × 29 + 1 × 28 + 0 × 27 + 1 × 26 + 1 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 1 × 20)(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 + 0 + 65 536 + 32 768 + 16 384 + 0 + 0 + 2 048 + 0 + 512 + 256 + 0 + 64 + 32 + 0 + 8 + 0 + 0 + 1)(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 + 65 536 + 32 768 + 16 384 + 2 048 + 512 + 256 + 64 + 32 + 8 + 1)(10) =
2 147 339 113(10)
5. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
1111 1111 1111 1101 1100 1011 0110 1001(2) = -2 147 339 113(10)
The number 1111 1111 1111 1101 1100 1011 0110 1001(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
1111 1111 1111 1101 1100 1011 0110 1001(2) = -2 147 339 113(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.