Binary Subtraction Calculator

Subtract binary numbers using direct borrow method and two's complement with step-by-step solutions.

Choose Method & Enter Values

Result

Binary Difference (A - B)
01111
binary
A (decimal)26
B (decimal)11
Difference (decimal)15
Method usedBorrow

Step-by-Step Solution

11010 - 01011 = 01111

Understanding Binary Subtraction

Binary subtraction is the process of subtracting one binary number from another. There are two primary methods: the direct borrow method (similar to decimal subtraction) and the two's complement method (which converts subtraction into addition). Both methods are fundamental to computer arithmetic and digital circuit design.

Binary Subtraction Methods

Direct Borrow Method

Subtract column by column from right to left, borrowing from the next column when needed.

0-0=0, 1-0=1, 1-1=0, 10-1=1

Two's Complement

Negate B by flipping bits and adding 1, then add A + (-B). Discard overflow carry.

A - B = A + (NOT B + 1)

One's Complement

Flip all bits of B, add to A, then add the end-around carry to the result.

A - B = A + NOT(B) + carry

Borrow Rules

When 0 - 1 occurs, borrow 1 from the next higher bit (making it 10 - 1 = 1).

Borrow makes 0 become 10 (= 2)

The Borrow Method Explained

The direct borrow method works exactly like decimal subtraction. Starting from the rightmost bit, subtract each bit of B from the corresponding bit of A. If A's bit is smaller, borrow from the next column to the left:

  • 0 - 0 = 0 (no borrow)
  • 1 - 0 = 1 (no borrow)
  • 1 - 1 = 0 (no borrow)
  • 0 - 1: Borrow from the next bit. The borrowed bit makes this column 10 - 1 = 1, and the next column's bit is reduced by 1.

Two's Complement Method

The two's complement method converts subtraction into addition, which is how most computers perform subtraction internally. The steps are:

  1. Find the one's complement of B (flip all bits: 0 becomes 1, 1 becomes 0).
  2. Add 1 to get the two's complement of B (this is -B in two's complement notation).
  3. Add A and the two's complement of B.
  4. If there is a carry out of the most significant bit, discard it. The result is positive.
  5. If there is no carry out, the result is negative; take the two's complement of the result to find its magnitude.

Why Computers Use Two's Complement

Two's complement is preferred in hardware because it allows the same adder circuit to perform both addition and subtraction. This simplifies the ALU design and eliminates the problem of having two representations for zero (positive zero and negative zero) that exists in one's complement.

Negative Results

When A is smaller than B, the result of A - B is negative. In the borrow method, this is indicated by a final borrow out. In two's complement, a negative result remains in two's complement form and can be interpreted by taking the complement and adding one to find the magnitude.

Tips for Binary Subtraction

  • Always pad both numbers to the same length with leading zeros before subtracting.
  • Track borrow bits carefully -- they propagate just like carry in addition.
  • Verify your answer by converting to decimal.
  • The two's complement method avoids borrow propagation entirely.
  • Practice with small numbers first before attempting larger calculations.