Binary adders

contributed by < Eric Lin >

Half adder

Overview

A half adder is a fundamental digital circuit that computes the sum S and carry C outputs based on two input bits.

The carry signal represents an overflow into the next digit of a multi-digit addition.

The formula for the sum and carry is as follows:

$S=A\oplus B$

$C=A\cdot B$

True Table

A	B	C	S
0	0	0	0
0	1	0	1
1	0	0	1
1	1	1	0

digital logic circuits

Limitation

It is not designed to handle the complexities of multi-bit addition, lacking the ability to accommodate the expansion required for more significant binary numbers.

Code implementation

#include <stdio.h>

int half_adder(int x, int y) 
{
    return x ^ y + (( x & y ) << 1);
}

int main() 
{
    int x, y, ans;

    printf("Enter x (0 or 1): ");
    scanf("%d", &x);

    printf("Enter y (0 or 1): ");
    scanf("%d", &y);

    ans = half_adder(x, y);

    printf("ans: %d\n", ans);

    return 0;
}

Full adder

Overview

A full adder is designed to perform the addition of three binary digits: two inputs (

$A$ and

$B$) and a carry input from the previous stage

$Cin$. It produces two outputs: a sum

$S$ and a carry

$Cout$.

The sum

$S$ is the result of adding the three inputs and is given by the XOR (exclusive OR) operation:

$S=A\oplus B\oplus Cin$

The carry

$Cout$:

$Cout=A\cdot B+Cin(A\oplus B)$

True Table

A	B	Cin	S	Cout
0	0	0	0	0
0	0	1	1	0
0	1	0	1	0
0	1	1	0	1
1	0	0	1	0
1	0	1	0	1
1	1	0	0	1
1	1	1	1	1

digital logic circuits

Cascading Full Adders

Full adders can be connected in series to create a ripple-carry adder, allowing the addition of multi-bit binary numbers. The carry output

$Cout$ from one full adder becomes the carry input

$Cin$ for the next, enabling the propagation of carry through multiple stages.

Code implementation

#include <stdio.h>

typedef char bit;

void full_adder(bit a, bit b, bit cin, bit *sum, bit *cout) 
{
    *sum = a ^ b ^ cin;
    *cout = a & b | cin & (a ^ b);
}

int main() 
{
    int x, y;

    // Input integer numbers x and y
    printf("Enter integer x: ");
    scanf("%d", &x);

    printf("Enter integer y: ");
    scanf("%d", &y);

    int sum = 0, multiplier = 1; // Used to process each bit position
    bit carry = 0;
    
    while (x > 0 || y > 0 || carry > 0) {
        bit bit_x = x & 0x1;
        bit bit_y = y & 0x1;
        bit bit_sum = 0;

        // Perform addition using full adder
        full_adder(bit_x, bit_y, carry, &bit_sum, &carry);

        // Update the sum using the current bit position
        sum += (bit_sum * multiplier);

        // Move to the next bit position
        x >>= 1;
        y >>= 1;
        multiplier <<= 1;
    }

    // Output the result
    printf("Sum: %d\n", sum);

    return 0;
}

Reading: How to simulate a 4-bit binary adder in C