Binary Subtraction Calculator

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 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):

1. Subtract the B bit and any borrow-in from the A bit.
2. If the result is 0 or 1, write it as the result bit with no borrow-out.
3. 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).
4. If A bit minus B bit minus borrow-in is −2: write result bit 0 and set borrow-out to 1.
5. 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.