Two's Complement: Binary -> Integer: 0111 0001 1101 0110 0000 0000 0010 1010 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 0111 0001 1101 0110 0000 0000 0010 1010(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.
0111 0001 1101 0110 0000 0000 0010 1010 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
1 229
1 228
1 227
0 226
0 225
0 224
1 223
1 222
1 221
0 220
1 219
0 218
1 217
1 216
0 215
0 214
0 213
0 212
0 211
0 210
0 29
0 28
0 27
0 26
0 25
1 24
0 23
1 22
0 21
1 20
0
5. Multiply each bit by its corresponding power of 2 and add all the terms up.
0111 0001 1101 0110 0000 0000 0010 1010(2) =
(0 × 231 + 1 × 230 + 1 × 229 + 1 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 1 × 224 + 1 × 223 + 1 × 222 + 0 × 221 + 1 × 220 + 0 × 219 + 1 × 218 + 1 × 217 + 0 × 216 + 0 × 215 + 0 × 214 + 0 × 213 + 0 × 212 + 0 × 211 + 0 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20)(10) =
(0 + 1 073 741 824 + 536 870 912 + 268 435 456 + 0 + 0 + 0 + 16 777 216 + 8 388 608 + 4 194 304 + 0 + 1 048 576 + 0 + 262 144 + 131 072 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 32 + 0 + 8 + 0 + 2 + 0)(10) =
(1 073 741 824 + 536 870 912 + 268 435 456 + 16 777 216 + 8 388 608 + 4 194 304 + 1 048 576 + 262 144 + 131 072 + 32 + 8 + 2)(10) =
1 909 850 154(10)
5. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:
0111 0001 1101 0110 0000 0000 0010 1010(2) = 1 909 850 154(10)
The signed binary number in two's complement representation 0111 0001 1101 0110 0000 0000 0010 1010(2) converted and written as an integer in decimal system (base ten):
0111 0001 1101 0110 0000 0000 0010 1010(2) = 1 909 850 154(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.