➖ Binary Subtraction Calculator
Subtract two binary numbers with step-by-step solution
How to Use This Calculator
Enter Binary Numbers
Input the minuend (number to subtract from) and subtrahend (number to subtract). The calculator accepts only 0s and 1s. The first number must be greater than or equal to the second. Examples: 1010 - 1100, 1111 - 1010.
Click Calculate Subtraction
Press the "Calculate Subtraction" button to perform binary subtraction. The calculator will automatically align the numbers and show the step-by-step process with borrow operations when needed.
Review Results
The result shows the binary difference, decimal equivalent, and a detailed step-by-step breakdown showing how each bit position was calculated, including borrow operations.
Formula
Binary Subtraction: Subtract from right to left, borrow when needed
If bit1 - bit2 - borrow < 0, add 2 and borrow 1
How it works:
Binary subtraction works similarly to decimal subtraction, but uses base 2. You subtract bits from right to left, and when the result would be negative, you borrow 1 from the next position (which adds 2 to the current position).
Binary Subtraction Rules:
- 0 - 0 = 0 (no borrow)
- 0 - 1 = 1 (borrow 1, becomes 2 - 1 = 1)
- 1 - 0 = 1 (no borrow)
- 1 - 1 = 0 (no borrow)
- When borrowing: current position gets +2
Example 1: 1010 - 0110 (10 - 6 = 4)
1010
- 0110
------
Step 1: 0 - 0 = 0 (no borrow)
Step 2: 1 - 1 = 0 (no borrow)
Step 3: 0 - 1 = 1 (borrow 1, becomes 2 - 1 = 1)
Step 4: 1 - 0 - borrow 1 = 0 (borrow was used)
Result: 0100 (binary) = 4 (decimal)
Example 2: 1100 - 0101 (12 - 5 = 7)
1100
- 0101
------
Step 1: 0 - 1 = 1 (borrow 1)
Step 2: 0 - 0 - borrow 1 = 1 (borrow 1, becomes 2 - 1 = 1)
Step 3: 1 - 1 - borrow 1 = 1 (borrow 1, becomes 2 - 1 - 1 = 0, wait that's wrong...)
Actually: After borrowing, position 2 has 0, borrow again: 2 - 1 = 1
Step 4: 1 - 0 - borrow 1 = 0
Result: 0111 (binary) = 7 (decimal)
Frequently Asked Questions
How does binary subtraction differ from decimal subtraction?
Binary subtraction uses base 2, so you borrow 2 (not 10) when needed. The process is the same: subtract from right to left, borrow when the result would be negative. When you borrow, the current position gets +2, just like borrowing in decimal gives +10.
What happens when I need to borrow in binary subtraction?
When you need to borrow, you take 1 from the next position. This adds 2 to the current position (since we're in base 2). For example, if you have 0 - 1, you borrow to get 2 - 1 = 1, and the next position decreases by 1.
Can I subtract a larger binary number from a smaller one?
This calculator requires the first number to be greater than or equal to the second number. For negative results, you would typically use two's complement representation. If you need to handle negative numbers, consider using two's complement subtraction.
What is two's complement and how does it relate to subtraction?
Two's complement is a method for representing negative numbers in binary. Instead of subtracting directly, you can add the two's complement of the subtrahend. This is how computers actually perform subtraction: A - B = A + (-B), where -B is the two's complement of B.
Why is binary subtraction important in computer science?
All arithmetic operations in computers use binary. Processors use subtraction circuits to perform subtraction operations. Understanding binary subtraction helps in understanding how computers perform arithmetic, and it's fundamental to understanding two's complement representation for negative numbers.
How do computers handle binary subtraction?
Modern computers typically don't use direct binary subtraction. Instead, they use addition with two's complement: A - B = A + (~B + 1), where ~B is the bitwise NOT of B. This allows using the same addition circuit for both addition and subtraction, making hardware design simpler.