Binary Addition

Rules of Binary Addition

There are four key rules to remember when adding binary numbers:

Step	Binary Addition	Explanation
1	0 + 0 = 0	Nothing to carry, so we write 0.
2	0 + 1 = 1	No carry, so we write 1.
3	1 + 1 = 10	We write 0 and carry 1 (just like how 9 + 1 in denary is 10).
4	1 + 1 + 1 = 11	We write 1 and carry 1.

Example: Adding Two Binary Numbers

Let's add 010001 and 100001.

Step-by-step:

1. Align the two binary numbers.
2. Apply the binary addition rules starting from the rightmost column (just like denary addition). | | | | | | | | |---|---|---|---|---|---|---| | | 0 | 1 | 0 | 0 | 0 | 1 | | + | 1 | 0 | 0 | 0 | 0 | 1 | | | | | | | | | | Carried values | | 1 | | | | | | Result | 1 | 1 | 0 | 0 | 0 | 1 |

Steps Explained

• In the first column, 1 + 1 = 10, so write 0 and carry 1.
• In the second column, 1 + 0 + carry 1 = 10, so write 0 and carry 1.
• Continue the process until all columns are added.
• The result is 1100010.

Checking the Answer with Denary

• 010001 = 17 in denary.
• 100001 = 33 in denary.
• 17 + 33 = 50 in denary.
• 1100010 in binary = 50 in denary, so the addition is correct!

Overflow Errors

When adding two binary numbers, there is a limit to the size of the result when using 8 bits. The largest binary number that can be represented with 8 bits is 11111111 (which equals 255 in denary). If the result of an addition exceeds this, it causes an overflow error.

What is Overflow?

• Overflow happens when a result is too large to fit into the available number of bits.
• If we add 1 to 11111111, we get 100000000. Since we're limited to 8 bits, the leftmost 1 is lost, and the result is incorrect.

Example of Overflow

Let's add 11001010 and 10110010.

	1	1	0	0	1	0	1	0
+	1	0	1	1	0	0	1	0
Carried values	1	1	1	1	1	1
Result	1	1	1	1	1	1	0	0

Explanation

• Adding these two 8-bit numbers results in 11111100 with a carried 1 on the left.
• Since there's no space for this carried 1 in 8 bits, it's lost, causing overflow.
• The correct result would need 9 bits: 111111100.