How to Do a Signed Binary Addition

To perform signed binary addition, you’ll first need to understand how to represent positive and negative numbers using binary. Signed binary numbers are usually represented using the two’s complement system.

Here’s a step-by-step guide:

1. Convert Decimal to Binary: For positive numbers, convert the number to binary as you normally would. For negative numbers, convert the positive equivalent to binary, then find the two’s complement (invert the bits and add 1).
2. Align the Numbers: Line up the numbers you wish to add, making sure to include the sign bit. Each binary number should have the same number of bits. If necessary, pad with leading zeros.
3. Add the Numbers: Perform binary addition from right to left, just as you would with decimal numbers. Carry over any values greater than 1 to the next left bit. Remember, in binary, 1 + 1 = 10 (which is 0 with a carry of 1).
4. Check for Overflow: If the result exceeds the maximum representable value for the given number of bits, an overflow occurs. For example, in an 8-bit system, if you end up with a carry into the 9th bit, you’ve got an overflow.
5. Interpret the Result: If there is no overflow, you can interpret the result in two’s complement format. If the final result’s sign bit is 1, the result is negative; convert it back to decimal by finding its two’s complement again to find the positive equivalent.

Example:

Add +6 (0110) and -3 (two’s complement: 1101):

• Convert +6 to binary: 0110
• Convert -3 to binary: 3 in binary is 0011; invert the bits: 1100; add 1: 1101
• Align the numbers:

  0110
+ 1101
______

• Perform the addition:

  0110
+ 1101
______
  0011 (This is 3 in decimal, no overflow)

The result is +3, which is correct!