Binary Calculator - Add, Subtract, and Convert Base-2 Numbers

A binary calculator adds, subtracts, multiplies, divides, and converts base-2 integers - the 0s and 1s that every digital circuit actually stores. People search for how to add binary numbers, how to convert binary to decimal, and what 1011 is in decimal because those tasks show up in intro CS, digital-logic labs, and bitwise debugging long before anyone opens a full scientific calculator.

Binary is not a cryptic coding trick. It is place value with powers of two instead of powers of ten. The ones place is 2⁰ = 1, the twos place is 2¹ = 2, then 4, 8, 16, and so on. That is why 1011₂ is 8+0+2+1 = 11₁₀, and why adding 1011 + 1101 by hand produces 11000 - the same 11 + 13 = 24 you would get in decimal, just written with only two digits.

This free online binary calculator runs entirely in your browser with JavaScript BigInt. Nothing is uploaded. Arithmetic and bitwise modes reject any character other than 0 and 1 (a 0b prefix and spaces are stripped first). Convert mode also accepts decimal and hexadecimal so you can see the same integer in all three bases. The hard limit is 256 bits, about 77 decimal digits. That is enough for machine words, IPv6 pieces, and most homework; it is not unlimited-precision arithmetic. For thousands of decimal digits use the big number calculator.

Bitwise AND, OR, and XOR are included because they are the operations students confuse with addition. AND never carries. Addition always might. The default example is chosen so you can see both ideas on the same pair: 1011 + 1101 = 11000, while 1011 AND 1101 = 1001.

Using this base-2 adder and converter is straightforward:

1. Choose a mode. Convert shows binary, decimal, and hex for one integer. Add, Subtract, Multiply, and Divide take two binary integers. AND, OR, and XOR are bitwise on the same kind of unsigned integers.
2. Enter the number(s). Binary operands may look like 1011, 0b1011, or 1 011. Convert mode lets you set the from-base to 2, 10, or 16 (hex accepts 0x and A–F).
3. Read the result panel. The primary display is binary. Decimal and hex sit underneath so you can check a homework step without a second tool. Divide shows quotient and remainder (truncated toward zero, like JavaScript BigInt).
4. Copy the binary string, or clear the fields to start over. The last inputs are saved locally in your browser for up to 30 days.

Note: inputs and results are capped at 256 bits. Division by zero is an error. Subtracting a larger number from a smaller one yields a negative result shown with a minus sign in front of the magnitude - this tool does not invent a two's-complement bit pattern for negative answers. Bitwise operators are defined on non-negative BigInt values only.

To convert binary to decimal by hand, expand each 1 as a power of two. To add, work from the least-significant bit and carry whenever a column sums to 2 or more - exactly like decimal addition, except the digit set stops at 1.

Example - add 1011 + 1101 (the default):

• Rightmost: 1 + 1 = 10₂ → write 0, carry 1
• Next: 1 + 0 + carry 1 = 10₂ → write 0, carry 1
• Next: 0 + 1 + carry 1 = 10₂ → write 0, carry 1
• Left: 1 + 1 + carry 1 = 11₂ → write 11
• Result 11000₂ = 16+8 = 24₁₀, and 11 + 13 = 24 checks in decimal

Example - what is 1011 in decimal?

• 1×2³ + 0×2² + 1×2¹ + 1×2⁰
• 8 + 0 + 2 + 1 = 11
• Same value in hex is B (because 10 is A and 11 is B)

Example - 1011 AND 1101 versus 1011 + 1101:

• AND, no carry: 1∧1, 0∧1, 1∧0, 1∧1 → 1001 (9₁₀)
• Add, with carry: 11000 (24₁₀)
• OR would be 1111 (15). XOR would be 0110 (6). None of those bitwise results equal the sum.

One hex digit is exactly four bits, which is why programmers prefer hex once a bit string is longer than a byte. 11000₂ is 18₁₆, not 18₁₀ - mixing those two 18s is a classic exam trap. If your assignment is hex-first, switch to the hex calculator so both operands start as A–F instead of 0/1.

The most common error is treating 1011 as the decimal one thousand eleven. In binary it is eleven. Another is forgetting the carry: people write 1011 + 1101 = 2112, which is not a binary number at all. A third is using bitwise AND when the homework asked for a sum - 1001 looks "binary-ish" and is still the wrong answer for 1011 + 1101.

Hex digits sneak in. B, C, D, E, and F are valid hex, not valid binary; this calculator rejects them in arithmetic mode instead of silently interpreting B as eleven. Leading zeros do not change the value (001011 is still 11) but they do count toward the 256-bit input length if you paste a huge padded string.

Signed interpretations are another trap. In eight-bit two's complement, 11111111 is −1. This tool treats that bit string as the unsigned integer 255. If your course is about two's complement overflow, you still need to apply the wrap-around rule yourself after you read the unsigned magnitude.

Binary division is the same integer division you already know, written in base 2. Given two non-negative integers A and B with B ≠ 0, there is a unique pair (Q, R) such that A = Q×B + R and 0 ≤ R < B. This calculator reports both the quotient Q and the remainder R in binary and in decimal. Division by zero is an error, not Infinity.

Worked example: 1101 ÷ 101. In decimal that is 13 ÷ 5, so Q = 2 and R = 3, which is Q = 10₂ and R = 11₂. Another: 1011 ÷ 10 is 11 ÷ 2, so Q = 101₂ (5) and R = 1. The remainder is never “a fraction in binary”; if you wanted 11/2 = 5.5 you would need a different tool. Integer remainder is what ALUs, hash maps, and clock arithmetic use.

Truncation follows JavaScript BigInt: toward zero. For the non-negative inputs this page accepts, that is the same as flooring. Multiply mode is the inverse check: Q×B + R must equal A when A and B fit in 256 bits. If a multiply of two large operands would exceed 256 bits, the product is rejected even if each factor was legal on its own.

Search results for “binary calculator two’s complement” often want an 8-bit wrap: 11111111 meaning −1. This page does not do that. Bitwise AND, OR, and XOR run on non-negative BigInt values. 11111111 is two hundred fifty-five, not minus one. If you subtract a larger number from a smaller one, the result is shown with a minus sign in front of the magnitude (for example −10₂ for decimal −2), not as a padded two's-complement word.

The 256-bit cap is about 77 decimal digits (2²⁵⁶ − 1). That is far larger than a CPU register and far smaller than a crypto modulus. It exists so a pasted 10,000-bit dump cannot freeze the tab. Convert mode will still show decimal and hex for any integer that fits, including a 0x prefix from a debugger. Arithmetic modes still insist on 0 and 1 only; they will not silently treat B as eleven.

Spaces and a 0b prefix are stripped because that is how people paste from C and Python. Leading zeros do not change the value, but they do count toward the 256-bit input length. If you need thousands of exact decimal digits rather than bits, the big number calculator is the right sibling. If you need hex operands from the start, use the hex calculator.