"how numbers are stored and used in computers"
Two's complement is a method for representing both positive and negative whole numbers in a binary number system. Positive numbers are represented intuitively, where each bit represents a power of two, from right to left.
To store a negative number, one bit must be allocated as a sign bit, which is 0 for positive numbers and 1 for negative numbers. In two's complement, a negative number (i.e. where the sign bit is
To get the two's complement of a number, first flip all the bits, so 0101 becomes 1010. Then you add 1, so 1010 + 1 = 1011. That result, 1011, is the integer part of a negative number (i.e.
It is advantageous to represent integers this way, because the sum of a number and its two's complement will be the correct two's complement result.
The biggest reason for using two's complement is that it allows for efficient arithmetic without special rules for subtraction. It also avoids the problem of having two zeroes (one with a positive sign, and another with a negative sign).