About the Binary Subtraction Calculator
This binary subtraction calculator is a great tool to help you understand how to subtract binary numbers. Here you can find descriptions of the two primary methods that deal with the subtraction of binary numbers, namely the Borrow Method and the Complement Method. There is also a short note about the different representations of signed and unsigned binary numbers.
Subtraction of Binary Numbers
The subtraction of binary numbers is essentially the same as for decimal, hexadecimal, or any other
numeral system. Binary numbers are values containing only two types of digits: 0 or
1. Every digit refers to consecutive powers of 2 and whether it should be multiplied
by 0 or 1.
As an example, 13 in decimal notation is equivalent to 1101 in binary
notation, because 13 = 8 + 4 + 1 = 1×2³ + 1×2² + 0×2¹ + 1×2⁰.
So, how to subtract binary numbers, e.g., 1101 − 110? We can always convert
these values to decimal, classically subtract them, and then transform them back into binary:
Methods of Subtraction of Binary Numbers
Borrow Method
Align the numbers just like regular decimal subtraction. Operate with only two digits, not ten. Follow these rules column by column from right to left:
- 1 − 0 = 1
- 1 − 1 = 0
- 0 − 0 = 0
- 0 − 1 = 1 (borrow 1 from left: 10₂ − 1 = 1)
Complement Method
Uses the two's complement to convert subtraction into addition:
- Align and pad both numbers to the same length
- Find the two's complement of the subtrahend (invert bits + add 1)
- Add the minuend and the two's complement
- If there is a carry bit, remove it — positive result
- If no carry, take the two's complement of the result — negative result
How to Use the Binary Subtraction Calculator?
Simply enter the two binary numbers you want to subtract — the minuend (A) and the subtrahend (B) — and select the subtraction method. The calculator computes A − B and shows a full step-by-step solution.
If B > A (you're subtracting a larger number from a smaller one), the result is negative. The calculator handles this automatically:
Example: 110 0101₂ − 1000 1100₂ = −10 0111₂
Decimal check: 101₁₀ − 140₁₀ = −39₁₀ = −10 0111₂ ✓
How Do We Represent Sign in Binary Numbers?
Unsigned Binary
All bits represent magnitude. An 8-bit unsigned number can store values from 0 to 255. There is no concept of negative numbers — every combination of bits is a non-negative integer.
Signed Binary (Two's Complement)
The most significant bit (MSB) is the sign bit. A leading 1
means the number is negative; a leading 0 means positive. In an 8-bit signed
system, values range from −128 to 127.
FAQs
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What is binary subtraction?
Binary subtraction is the operation of finding the difference between two binary (base-2) numbers. It follows the same principle as decimal subtraction but uses only the digits 0 and 1.
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What is the borrow method in binary subtraction?
The borrow method works column by column from right to left. When a digit needs to subtract a larger digit (0 − 1), it "borrows" from the next column to the left, just like in decimal arithmetic, but values are in base 2 instead of base 10.
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What is the complement method?
The complement method converts subtraction A − B into addition: A + (two's complement of B). The two's complement of a binary number is found by inverting all bits (one's complement) and adding 1. This method is widely used in digital circuits and computer processors.
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What is two's complement?
Two's complement is the most common method of representing signed integers in binary. To find the two's complement of a number: (1) invert all bits to get the one's complement, then (2) add 1. For example, for 0110 (6): invert → 1001, add 1 → 1010. So −6 in two's complement (4-bit) is 1010.
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How does the calculator handle negative results?
When the minuend (A) is smaller than the subtrahend (B), the result is negative. In the Borrow Method, the calculator internally swaps A and B, computes B − A, then applies a minus sign. In the Complement Method, the absence of a carry bit signals a negative result, and the two's complement of the sum gives the magnitude.
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Can I use this calculator for large binary numbers?
Yes — this calculator supports binary numbers up to 32 bits in length. Both the minuend and subtrahend can have up to 32 binary digits (bits).