Two’s Complement

Two’s complement is a mathematical operation commonly used in x86-64 assembly to represent signed integers. It allows us to convert between binary numbers that represent positive and negative values.

In two’s complement representation, the most significant bit (MSB) of a binary number is used as the sign bit. If the most significant bit is zero (0) then the value represents a positive number. If it is one (1), then it represents a negative number.

Two’s complement is used for its simplicity and efficiency in representing signed integers. It allows both positive and negative numbers to be represented using the same binary notation, simplifying arithmetic and logic operations.

Additionally, two’s complement enables the implementation of arithmetic operations like addition, subtraction, multiplication, and division using the same algorithms for both positive and negative numbers.

Calculating the Two’s Complement

To calculate the two’s complement of a number, you first convert from decimal to binary, invert all the bits (change 0s to 1s and 1s to 0s), and then add 1 to the result. Let’s break down each step:

Step One: Convert from decimal to binary. For example if we want to find the two’s complement of the positive integer +2, we first convert this to binary. Using four bits, the binary representation of 2 is 0010.

Step Two: Invert all the bits. Every bit containing a ‘1’ is ‘flipped’ to ‘0’, and vice versa. Using our example, we can flip each bit in 0010 to obtain 1101. Mathematically, this can be done by performing an XOR operation with a number of equal length that is all one’s. Again in our example, to find the two’s complement of the binary number 0010 (which represents +2 in decimal), we can xor this number with 1111. The result of the operation will be 1101.

Step Three: Add one. In our example, 1101 + 0001 = 1110. This is the two’s complement of the original number, 0010.

To recap: We have calculated that the two’s complement of the number ‘2’ is 1110. This is how the number negative two (-2) is represented in binary.

Two’s complement is useful because it allows for the representation of both positive and negative numbers using the same binary representation, simplifying arithmetic operations like addition and subtraction in digital systems.