Number 1111 1111 1111 1111 1111 1011 1111 1111(2) converted from signed binary two's complement representation to an integer in decimal system (in base 10): 1111 1111 1111 1111 1111 1011 1111 1111(2) = -1 025(10)
Spaces used to group digits: for binary, by 4; for decimal, by 3.
How to convert signed binary numbers in two's complement representation from binary system to decimal
To understand how to convert a signed binary number in two's complement representation from the binary system to decimal (base ten), the easiest way is to do it by an example - convert binary, 1101 1110, to base ten:
In a signed binary two's complement, first bit (leftmost) indicates the sign, 1 = negative, 0 = positive. The first bit is 1, so our number is negative.
Get the signed binary representation in one's complement, subtract 1 from the initial number: 1101 1110 - 1 = 1101 1101
Get the binary representation of the positive 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: !(1101 1101) = 0010 0010
Write bellow the positive binary number representation in base two, and above each bit that makes up the binary number write the corresponding power of 2 (numeral base) that its place value represents, starting with zero, from the right of the number (rightmost bit), walking to the left of the number, increasing each corresonding power of 2 by exactly one unit:
powers of 2:
Build the representation of the positive number in base 10, by taking each digit of the binary number, multiplying it by the corresponding power of 2 and then adding all the terms up: