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.