Signed Binary Using Two's Complement – A Level Computer Science AQA Revision

Two’s complement is a method used for representing signed integers in binary.
It is used to allow binary arithmetic operations to work with both positive and negative numbers.
A positive number in two’s complement form is the same as its unsigned form, while a negative number in two’s complement form is obtained by inverting all the bits of its positive value and then adding one.

Positive numbers are represented in the same way as in the unsigned binary system.
For example, the positive decimal number 5 is represented in an 8-bit two’s complement binary form as 00000101.

The two’s complement of a number is obtained by flipping all the bits (a process called bitwise NOT or inversion) and then adding 1 to the result.
For example, the two’s complement binary form of the negative number -5 is obtained by inverting the binary form of 5 (00000101) to get 11111010, and then adding 1 to get 11111011.

Zero is represented by a string of all zero bits. Both positive and negative zero would have the same representation (00000000).
Zero does not have a negative counterpart in a two’s complement system.

The Bitwise NOT operation is a unary operation that flips the bits of a binary number.
It is the first step in obtaining the two’s complement of a positive number in order to represent its negative counterpart.

Addition in two’s complement is carried out in the same way as unsigned binary addition.
If the result overflows beyond the allocated bit capacity (e.g. 8 bits), the overflow bit is simply ignored.

Subtraction in two’s complement is performed by adding the two’s complement (negative) of the number to be subtracted.
For example, to subtract 5 from 7, convert -5 into two’s complement (11111011) and add it to the binary representation of 7 (00000111). The result is 00000010, which is the binary representation of 2, the correct result.

The most significant bit in a two’s complement binary number is sometimes called the sign bit.
If the sign bit is 1, the number is negative. If it is 0, the number is positive.
This concept is essential when interpreting binary numbers in a two’s complement system.

Two’s complement simplifies the hardware needed for arithmetic operations in computers since it incorporates negative numbers into a system originally intended for only positive numbers.
It is widely used in microprocessors and other logic circuitry because of its ease of implementation using digital logic gates.
Understanding two’s complement is important in understanding the manipulation and interpretation of data in computers and digital systems.