Signed: Binary -> Integer: 0100 0000 1001 0000 0000 0000 0001 1110 Signed Binary Number Converted and Written as a Decimal System Integer (in Base Ten)
The signed binary (in base two) 0100 0000 1001 0000 0000 0000 0001 1110(2) to an integer (with sign) in decimal system (in base ten) = ?
1. Is this a positive or a negative number?
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.
0100 0000 1001 0000 0000 0000 0001 1110 is the binary representation of a positive integer, on 32 bits (4 Bytes).
2. Construct the unsigned binary number.
Exclude the first bit (the leftmost), that is reserved for the sign:
0100 0000 1001 0000 0000 0000 0001 1110 = 100 0000 1001 0000 0000 0000 0001 1110
3. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
230
1 229
0 228
0 227
0 226
0 225
0 224
0 223
1 222
0 221
0 220
1 219
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
1 20
0
4. Multiply each bit by its corresponding power of 2 and add all the terms up.
100 0000 1001 0000 0000 0000 0001 1110(2) =
(1 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 0 × 224 + 1 × 223 + 0 × 222 + 0 × 221 + 1 × 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 + 1 × 21 + 0 × 20)(10) =
(1 073 741 824 + 0 + 0 + 0 + 0 + 0 + 0 + 8 388 608 + 0 + 0 + 1 048 576 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 16 + 8 + 4 + 2 + 0)(10) =
(1 073 741 824 + 8 388 608 + 1 048 576 + 16 + 8 + 4 + 2)(10) =
1 083 179 038(10)
5. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0100 0000 1001 0000 0000 0000 0001 1110(2) = 1 083 179 038(10)
The number 0100 0000 1001 0000 0000 0000 0001 1110(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
0100 0000 1001 0000 0000 0000 0001 1110(2) = 1 083 179 038(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.
Convert signed binary numbers to integers in decimal system (in base ten)
The first bit (the leftmost) is reserved for the sign (1 = negative, 0 = positive) and it doesn't count when calculating the absolute value.
Binary number's length must be: 2, 4, 8, 16, 32, 64 - or else extra bits on 0 are added in front (to the left).
How to convert a signed binary number to an integer in base ten:
1) Construct the unsigned binary number: exclude the first bit (the leftmost); this bit is reserved for the sign, 1 = negative, 0 = positive and does not count when calculating the absolute value (without sign).
2) Multiply each bit of the binary number by its corresponding power of 2 that its place value represents.
3) Add all the terms up to get the positive integer number in base ten.
4) Adjust the sign of the integer number by the first bit of the initial binary number.