Binary addition is the process of adding two binary numbers (base-2 numbers) together. Just like decimal addition (base-10), binary addition follows specific rules for carrying over digits when sums exceed the value of the base (in this case, base-2).

In binary, there are only two digits: 0 and 1. When adding these digits, the rules are similar to those used in decimal addition, but with simpler carry rules because the base is 2 instead of 10.

The Basic Rules of Binary Addition

When adding binary digits (bits), you follow these basic rules:

1. 0 + 0 = 0
   This is the simplest case; no carry occurs.

2. 0 + 1 = 1
   If one digit is 1 and the other is 0, the sum is 1. No carry.

3. 1 + 0 = 1
   This is the same as the previous case; the sum is 1, and no carry occurs.

4. 1 + 1 = 0 (with carry 1)
   If both digits are 1, the sum is 0, and you carry over 1 to the next column. In binary, 1 + 1 is equivalent to 2 in decimal, which is written as 10 in binary (0 in the current column, and carry 1 to the next column).

5. Carry over:
   If the sum of two bits is 2 (1 + 1), the result is 0 for the current column, and you carry 1 to the next column on the left. This process continues through all the columns being added.

Here's an example of adding two binary numbers, 1101 and 1011:

   1101
 + 1011
-------
  11000

So, 1101 + 1011 in binary equals 11000 in binary.

It is the job of the Arithmetic Logic Unit to perform binary addition. It is achieved through the use of two types of circuit:

A half adder

Half adders take 2 binary inputs and produced a sum and a carry. This circuit is used for the right most bit calculation.

A Full adder

Full adders take 3 binary inputs(the addition bits and the carry from the previous column) and produces a sum and a carry.

If the final result of the addition produces a carry that does not fit within the size of the register (for instance a result of an addition is 9 bits long and the register size is 8 bits) then this results in an overflow, which can be handled in multiple ways.

Unsigned binary subtraction works column by column from the right, just like denary subtraction. When you need to take 1 away from 0, you borrow from the next column to the left.

• 0 − 0 = 0
• 1 − 0 = 1
• 1 − 1 = 0
• 0 − 1 = 1, borrowing 1 from the next column

  1101   (13)
− 0110    (6)
  ----
  0111    (7)

Check your answer by converting all three numbers to denary.

Computers usually subtract by adding a negative number: A − B = A + (−B). Writing B in two's complement makes it negative.

1. Write both numbers with the same number of bits.
2. Find −B: invert every bit of B, then add 1.
3. Add A and −B.
4. Ignore any carry out of the leftmost bit.

  00001101   (13)
+ 11111010   (−6)
 ----------
 100000111 → 00000111 (7)

The same method works when the answer is negative; the result is then in two's complement.