# Signed Binary Using Two's Complement

#### Understanding Signed Binary Using Two’s Complement

**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 in Two’s Complement

- 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.

#### Negative Numbers in Two’s Complement

- 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 in Two’s Complement

- 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.

#### Bitwise NOT Operation

- 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

- 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

- 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.

#### Sign Bit

- 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.

#### Importance of Two’s Complement

- 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.