Two's Complement: Binary -> Integer: 0010 0110 1000 1011 0011 0000 0011 0011 Signed Binary Number in Two's Complement Representation, Converted and Written as a Decimal System Integer (in Base Ten)
Signed binary in two's complement representation 0010 0110 1000 1011 0011 0000 0011 0011(2) converted to an integer in decimal system (in base ten) = ?
1. Is this a positive or a negative number?
In a signed binary in two's complement representation, the first bit (the leftmost) indicates the sign, 1 = negative, 0 = positive.
0010 0110 1000 1011 0011 0000 0011 0011 is the binary representation of a positive integer, on 32 bits (4 Bytes).
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.
* Not the case - the number is positive *
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:
* Not the case - the number is positive *
4. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:
231
0 230
0 229
1 228
0 227
0 226
1 225
1 224
0 223
1 222
0 221
0 220
0 219
1 218
0 217
1 216
1 215
0 214
0 213
1 212
1 211
0 210
0 29
0 28
0 27
0 26
0 25
1 24
1 23
0 22
0 21
1 20
1
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0010 0110 1000 1011 0011 0000 0011 0011(2) =
(0 × 231 + 0 × 230 + 1 × 229 + 0 × 228 + 0 × 227 + 1 × 226 + 1 × 225 + 0 × 224 + 1 × 223 + 0 × 222 + 0 × 221 + 0 × 220 + 1 × 219 + 0 × 218 + 1 × 217 + 1 × 216 + 0 × 215 + 0 × 214 + 1 × 213 + 1 × 212 + 0 × 211 + 0 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 1 × 20)(10) =
(0 + 0 + 536 870 912 + 0 + 0 + 67 108 864 + 33 554 432 + 0 + 8 388 608 + 0 + 0 + 0 + 524 288 + 0 + 131 072 + 65 536 + 0 + 0 + 8 192 + 4 096 + 0 + 0 + 0 + 0 + 0 + 0 + 32 + 16 + 0 + 0 + 2 + 1)(10) =
(536 870 912 + 67 108 864 + 33 554 432 + 8 388 608 + 524 288 + 131 072 + 65 536 + 8 192 + 4 096 + 32 + 16 + 2 + 1)(10) =
646 656 051(10)
5. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0010 0110 1000 1011 0011 0000 0011 0011(2) = 646 656 051(10)
The signed binary number in two's complement representation 0010 0110 1000 1011 0011 0000 0011 0011(2) converted and written as an integer in decimal system (base ten):
0010 0110 1000 1011 0011 0000 0011 0011(2) = 646 656 051(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.
Convert signed binary numbers in two's complement representation to decimal system (base ten) integers
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 in two's complement representation to an integer in base ten:
1) In a signed binary two's complement, first bit (leftmost) indicates the sign, 1 = negative, 0 = positive.
2) Get the signed binary representation in one's complement, subtract 1 from the initial number.
3) Construct the unsigned binary number: flip all the bits in the signed binary one's complement representation (reversing the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s.
4) Multiply each bit of the binary number by its corresponding power of 2 that its place value represents.
5) Add all the terms up to get the positive integer number in base ten.
6) Adjust the sign of the integer number by the first bit of the initial binary number.