Binary Addition
- Adding binary numbers follows a similar process to adding denary numbers
- The binary adding rules are:
- 0+0=0
- 0+1=1
- 1+1=10 (The 1 is carried into the next column on the left)
- 1+1+1=11 (The 1 is carried into the next column on the left)
Adding binary steps:
Step 1:
Start by writing the two binary numbers you want to add underneath each other, with the least significant bit (LSB) on the right.
Step 2:
Begin by adding the LSBs together. If the sum is less than or equal to 1, write it down in the sum column. If the sum is 2 or greater, write the remainder of the sum (i.e., the sum minus 2) in the sum column and carry over the quotient (i.e., 1) to the next column
Step 3:
Repeat this process for the next column to the left, adding the two bits and any carryover from the previous column. Again, if the sum is less than or equal to 1, write it down in the sum column; if the sum is 2 or greater, write the remainder of the sum in the sum column and carry over the quotient to the next column.
Step 4:
Continue this process for each subsequent column until you have added all the bits.
Step 5:
If the sum of the last two bits produces a carryover, add an additional bit to the left of the sum to represent the carryover.
Step 6:
Check the sum to make sure it fits within 8 bits. If it doesn't, you will need to use more bits to represent the sum.
Adding binary walkthrough:
- In this example, we start by adding the two LSBs: 0 + 0 = 0, which we write down in the sum column
- We then move to the next column to the left and add the two bits and the carryover from the previous column: 1 + 1 + 0 = 10
- We write down the remainder of the sum (i.e., 0) in the sum column and carry over the quotient (i.e., 1) to the next column
- We repeat this process for the next two columns, and end up with the sum 101110000
Overflow
- An overflow error occurs when the result of a binary addition exceeds the maximum value that can be represented. In the case of 8-bits, the maximum value is 255
- Overflow occurs when the addition of two numbers results in a carry bit that cannot be accommodated
- To avoid overflow errors, it's important to check the result of binary addition to ensure that it doesn't exceed the maximum value that can be represented
- Overflow errors can also occur in other operations besides addition, such as multiplication or division
Exam Tip
- You can convert your binary numbers to denary, then perform the calculation and then convert them back to check you’ve got the right answer. Label this as checking to make sure that the examiner knows this is a check and not part of your working out
