Arithmetic circuits

Half adder

• What is a half adder

  • it’s an adder that adds 2 binary values and outputs the sum and a carry

• it uses an xor and an and gate to be the sum and carry

Full adder

• What is a full adder

• A full adder can also be realized by cascading two half adders and using an OR gate

  • $S_i=(x_i)$

Ripple-Carry Adder

• it’s like an adder but carries the “carry” to the input of the next in a “ripple

Unsigned integers

• How to multiply in a circuit

  • we can just add the thing thats being multiplied by n, we can just add n times

    • this is not that efficient

Signed integer numbers

• Signed bits give sign and magnitude representation

  • all leading 0/1s denote pos/neg respectively
    • the most significant digit is the one behind the leading integer
    • the least significant digit is the one that is trailing

• Complement Representation

  • For an n-digit number, N in radix r

    • its r ‘s complement defined as $r^n-N$

  • For an n-digit number, N in radix r

    • its (r-1 )complement defined as $r^n-1-N$

  • Simply 1s compliment is just the inverted integers of original

  • 2’s complement

    • we just get 1s compliment and add 1

Half Subtractor

• What is a half subtractor

  • it takes 2 binary values and subtracts them, placing the “negative” number into a “barrow”, the main way since there is no negatives it uses difference, and to know if it is negative you use the borrow

Full Subtractor

• What is a full subtractor

Addition and subtraction in 1 circuit