Binary Subtraction Calculator
This calculator subtracts one binary number from another and shows the full working, not just a bare result. It suits computer science students, programmers and anyone learning number systems who wants to understand how subtraction actually happens at the bit level, since this underpins how computers handle arithmetic internally. You enter two binary numbers: the minuend (A), the number being subtracted from, and the subtrahend (B), the number being taken away, using only the digits 0 and 1 and any number of bits. The calculator instantly returns the binary result of A minus B, the signed decimal equivalent, and the decimal value of each input, updating live as you type. Below the results, a step-by-step working panel walks through the borrow method column by column from right to left, showing where each borrow is taken and passed along, plus a full breakdown table of the calculation. A second panel then verifies the same answer using the two's complement method, computing the one's complement of B, adding 1, adding that to A, and confirming the two approaches agree. Watch the sign: if B is larger than A the result is negative, and the calculator flags this clearly, showing the true magnitude in two's complement notation, matching how signed integers are actually stored in computer hardware.
1. Enter Binary Numbers
2. Quick Reference
Binary subtraction rules (one column at a time):
| A − B | Result | Borrow |
|---|---|---|
| 0 − 0 | 0 | 0 |
| 1 − 0 | 1 | 0 |
| 1 − 1 | 0 | 0 |
| 0 − 1 | 1 | 1 (borrow) |
When you borrow, the column to the left loses 1; the current column gains 2 (binary base).
Step-by-Step Borrow Method
Full Breakdown
Two's Complement Verification
How Binary Subtraction Works
Binary subtraction follows the same column-by-column process as decimal subtraction, but with only two digits: 0 and 1. The four possible single-bit subtractions are straightforward except for 0 minus 1, which requires a borrow from the next column to the left.
When you borrow in binary you take 1 from the next higher bit. That higher bit loses 1, but the current column gains 2 (because each position is worth twice the one to its right). So 10 (binary 2) minus 1 (binary 1) = 1, with a borrow of 1 passed to the left.
Worked Example: 1101 − 0101
Default inputs: A = 1101 (decimal 13), B = 0101 (decimal 5). Expected result: 1000 (decimal 8).
| Column (bit position) | Bit 3 (8s) | Bit 2 (4s) | Bit 1 (2s) | Bit 0 (1s) |
|---|---|---|---|---|
| A | 1 | 1 | 0 | 1 |
| B | 0 | 1 | 0 | 1 |
| Borrow in | 0 | 0 | 0 | 0 |
| Result bit | 1 | 0 | 0 | 0 |
Column 0 (rightmost): 1 − 1 = 0, no borrow. Column 1: 0 − 0 = 0, no borrow. Column 2: 1 − 1 = 0, no borrow. Column 3: 1 − 0 = 1. Result: 1000 (binary) = 8 (decimal). This matches 13 − 5 = 8.
The Borrow Method in Detail
Process each column from right (least significant bit) to left (most significant bit):
- Subtract the B bit and any borrow-in from the A bit.
- If the result is 0 or 1, write it as the result bit with no borrow-out.
- If A bit minus B bit minus borrow-in is −1: write result bit 1 and set borrow-out to 1 (you borrowed 2, spent 1, gave back 1).
- If A bit minus B bit minus borrow-in is −2: write result bit 0 and set borrow-out to 1.
- Continue to the next column to the left.
If after processing all columns there is still a borrow-out of 1, the result is negative. The true magnitude is the two's complement of the binary result.
Two's Complement Method
Computers typically implement subtraction A − B as A + (two's complement of B). The two's complement of B is formed by inverting all its bits (one's complement) and adding 1. The two's complement of a number N equals −N in signed binary representation, so adding it to A gives A − B. Any carry beyond the bit width is discarded for same-width arithmetic. This calculator shows both methods and confirms they agree.
Negative Results
If A is smaller than B, the result is negative. The calculator displays the signed decimal result (e.g. −3) and shows the binary representation as the two's complement of the magnitude. In signed integer arithmetic (as used in programming languages and computer hardware), negative numbers are stored in two's complement form: the most significant bit is 1 for negative, 0 for positive.
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Method: Direct borrow-method binary subtraction working column by column from least significant bit to most significant bit. Two's complement verification by inverting all bits of B and adding 1, then adding to A. Both methods produce the same signed decimal result.
This calculator works with unsigned binary integers of any length. Results are exact for all inputs. The decimal equivalents treat the inputs as unsigned positive integers; the subtraction result may be negative if B is greater than A.