Binary Calculator
Instructions
How to Use the Binary Calculator
- Step 1: Enter the first binary number in the "Binary Number 1" field.
- Step 2: Enter the second binary number in the "Binary Number 2" field.
- Step 3: Select the operation: Addition or Subtraction.
- Step 4: Click the "Calculate" button.
- Step 5: View the step-by-step explanation and final result in binary and decimal.
Binary Calculator
Result
Step-by-Step
Click "Calculate" to see the steps.
Educational Tips
Binary Addition Rules
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 0 (carry 1)
- 1 + 1 + 1 = 1 (carry 1)
Binary Subtraction Rules
- 0 - 0 = 0
- 1 - 0 = 1
- 1 - 1 = 0
- 0 - 1 = 1 (borrow 1)
Common Mistakes
Forgetting to handle the carry in addition or the borrow in subtraction is a frequent error. Always check the column to the left when you perform a carry or borrow operation. For subtraction, remember that borrowing from a '0' requires you to continue borrowing from the left until you find a '1'.
Check Your Answers
You can check your answers by converting each binary number into denary (decimal) and checking the calculation.
Example:
1101 (binary) = 13 (denary)
0101 (binary) = 5 (denary)
Binary addition: 1101 + 0101 = 10010
Denary check: 13 + 5 = 18. The binary result 10010 is also 18.
How it works
Binary addition
When two numbers are added together in denary (decimal), the first number is combined with the second number to produce a sum. For example, 1 + 2 = 3.
When two numbers are added together in denary (decimal), the first number is combined with the second number to produce a sum. For example, 1 + 2 = 3.
In binary addition, different rules are applied. There are four basic rules for adding two binary numbers:
Example – adding 01 + 10
1 + 0 = 1
0 + 1 = 1
Result in binary: 11 (which represents denary 3)
- 0 + 0 = 0
- 1 + 0 = 1
- 1 + 1 = 10 (binary equivalent of denary 2)
- 1 + 1 + 1 = 11 (binary equivalent of denary 3)
Example – adding 01 + 10
1 + 0 = 1
0 + 1 = 1
Result in binary: 11 (which represents denary 3)
Adding binary numbers is similar to adding denary numbers however, instead of carrying groups of ten, you carry groups of 2.
Example – adding 01 + 01
1 + 1 = 0, carry 1
0 + 0 + 1 = 1
Result in binary: 10 (which represents denary 2)
Example – adding 01 + 01
1 + 1 = 0, carry 1
0 + 0 + 1 = 1
Result in binary: 10 (which represents denary 2)
Note: When we add two 1s we need to carry over to the next column.
Checking your answer
You can check your answers by converting each binary number into denary (decimal) and checking the calculation.
Example:
1101 (binary) = 13 (denary)
0101 (binary) = 5 (denary)
Binary addition: 1101 + 0101 = 10010
Denary check: 13 + 5 = 18. The binary result 10010 is also 18.
Checking your answer
You can check your answers by converting each binary number into denary (decimal) and checking the calculation.
Example:
1101 (binary) = 13 (denary)
0101 (binary) = 5 (denary)
Binary addition: 1101 + 0101 = 10010
Denary check: 13 + 5 = 18. The binary result 10010 is also 18.
Overflow
Overflow errors occur when a calculation produces a result that exceeds the bit limit of the system. Using four bits, the largest value possible is 1111 (denary 15). When adding two four-bit numbers, the result may require more than four bits. For instance, adding 1011 (denary 11) and 1100 (denary 12) will result in the following:
Overflow errors occur when a calculation produces a result that exceeds the bit limit of the system. Using four bits, the largest value possible is 1111 (denary 15). When adding two four-bit numbers, the result may require more than four bits. For instance, adding 1011 (denary 11) and 1100 (denary 12) will result in the following:
The actual sum is 10111 (denary 23), which needs five bits. However, if only four bits are available, the result would be 0111 (denary 7). Overflow can affect the accuracy of calculations.
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