Base-N Arithmetic Calculator

Frequently Asked Questions

Which bases are supported?

Any base from 2 to 36. Bases beyond 10 use letters: A is 10, B is 11, and so on up to Z is 35. The two operands can be in different bases, and you choose the output base independently.

How does it compute the result?

It converts both inputs to integers, performs the operation, then shows the answer in your chosen output base.

Is division integer or fractional?

Integer division with a separate remainder, which is the normal convention when working in binary or hexadecimal.

Why work in another base at all?

Binary and hex map directly to how computers store data, so base arithmetic is everyday work in low-level programming and digital logic.

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How This Calculator Works

A positional number system of base b assigns each digit a place value that is a power of b: the string d_k…d₁d₀ means ∑ d_ibⁱ. This tool first parses each input from the chosen base into an ordinary integer (multiply-accumulate: value = value × b + digit, rejecting any digit ≥ b), performs the arithmetic, then renders the result back in the same base by repeated division, collecting remainders from least to most significant. Bases up to 36 use digits 0-9 then A-Z. It is intentionally distinct from a base converter: a converter just changes the representation of one number, whereas this performs genuine addition, subtraction, multiplication, division, or modulo within a base and shows the working.

Worked Example: FF + 1A in Base 16

Hexadecimal FF parses as 15×16 + 15 = 255; 1A parses as 1×16 + 10 = 26. Their sum is 281 in decimal. Convert 281 back to base 16: 281 ÷ 16 = 17 remainder 9, then 17 ÷ 16 = 1 remainder 1, then 1 ÷ 16 = 0 remainder 1; reading remainders upward gives 119₁₆. Check: 1×256 + 1×16 + 9 = 281. The same column-carry logic you learned in base 10 applies in every base; only the "carry when you reach b" threshold changes (10 in decimal, 16 in hex, 2 in binary).

Where Base Arithmetic Is Used

Computing: binary (base 2) is the machine's native arithmetic; hex (base 16) compactly encodes bytes, colours (#FF8800), and memory addresses.

Networking: subnet masks and IPv6 are reasoned about in binary and hex directly.

Octal (base 8): Unix file permissions (chmod 755) are octal arithmetic.

Cryptography & encoding: base-16/base-32/base-36 represent hashes, tokens, and short IDs.

Common Mistakes & Edge Cases

Using a digit too large for the base - "1A" is invalid in base 10 and "8" is invalid in base 8. The tool flags this rather than guessing.

Division here is integer (quotient truncated toward zero); the separate remainder is shown so you can reconstruct the exact value.

Letters are case-insensitive: hex FF and ff are equal.

Modulo is normalised to a non-negative result, so (−5) mod 3 = 1, not −2 - the convention used in number theory and most languages' mathematical mod.

Leading zeros do not change the value: 007 = 7 in any base.

Carrying and Borrowing in Any Base

Column arithmetic works identically in every base; only the carry threshold changes. Adding in base 16, 9 + 8 = 17 decimal = 11₁₆, so you write 1 and carry 1 - the carry triggers at 16, not 10. Subtraction borrows a full b from the next column (16 in hex, 2 in binary). Multiplication is repeated shifted addition just as in long multiplication, and integer division is repeated subtraction yielding a quotient and remainder. The cleanest mental model is to convert each operand to its plain integer value, do the familiar base-10 operation, then re-express the answer in the target base by repeated division - which is exactly the algorithm this tool runs and displays in the step line so you can check the intermediate decimal yourself.

Related Tools & Limits

Pair this with a base converter when you only need to restate a value, and with a bitwise calculator for AND/OR/XOR/shift operations specific to base 2. This tool handles integers only - fractional digits (e.g. base-2 "binary points" like 0.101₂) are out of scope and need a separate fixed-point treatment. Results stay exact while operands fit in JavaScript's safe-integer range (about 2⁹³); beyond that, large multiplications silently lose low-order digits and a big-integer implementation would be required for cryptographic-scale numbers.

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