Exact arithmetic directly in binary
A binary calculator works with base-2 integers written using only 0 and 1. The arithmetic rules are the same as decimal arithmetic, but carrying and borrowing happen at powers of two. This tool accepts signed binary integers, performs addition, subtraction, multiplication, or integer division, and reports the result in binary, decimal, and hexadecimal so you can verify the value from several perspectives.
The implementation uses BigInt instead of converting operands to JavaScript Number. That avoids the precision loss that appears above 53 significant bits. Long bit strings used in cryptography exercises, masks, identifiers, algorithm analysis, or programming tests remain exact. The only practical limit is the browser’s available memory and the time needed to display exceptionally large strings.
How to use the binary calculator
Enter integers, not floating-point binary fractions. The selected operation determines whether a remainder is shown.
- Choose addition, subtraction, multiplication, or integer division.
- Enter the first and second operands using 0 and 1. A leading plus or minus sign and an optional 0b prefix are accepted.
- Use spaces or underscores to group long bit strings for readability; formatting separators are ignored during parsing.
- Press Calculate to see normalized binary, exact decimal, and hexadecimal output.
- For division, read both quotient and remainder. The quotient is truncated toward zero, matching BigInt integer division.
Binary arithmetic examples
These examples illustrate carries, negative results, multiplication, and quotient/remainder behavior.
| Expression | Result | Explanation |
|---|---|---|
| 1010 + 11 | 1101 | Ten plus three equals thirteen; the carry propagates into the 8s place. |
| 1010 - 11 | 111 | Ten minus three equals seven. Leading zeros are removed from the normalized result. |
| 11 - 1010 | -111 | The calculator uses a leading minus sign for mathematical negative integers, not a fixed-width two’s-complement pattern. |
| 1010 × 11 | 11110 | Ten times three equals thirty, which is 11110 in binary. |
| 1101 ÷ 101 | 10 remainder 11 | Thirteen divided by five gives quotient two and remainder three. |
| -1101 ÷ 101 | -10 remainder -11 | BigInt division truncates toward zero, so the remainder has the same sign as the dividend. |
Accepted binary integer format
Each operand must be one whole binary integer. The accepted form is an optional plus or minus sign, an optional 0b or 0B prefix, and one or more binary digits. Spaces and underscores are treated as visual grouping characters. A decimal point, exponent, hexadecimal digit, or other symbol is rejected.
Output is normalized rather than fixed-width. Zero is written as 0, positive values have no plus sign, negative values use a leading minus sign, and unnecessary leading zeros are removed. If you need a fixed-width stored pattern, use the two’s-complement converter after calculating the mathematical result.
- Examples of valid input: 1010, 0b1010, -1010, and 1111_0000.
- Examples of invalid input: 102, 10.1, 2e10, and an empty operand.
- There is no implicit 8-bit, 32-bit, or 64-bit overflow.
- Division by zero is rejected before any result is produced.
How binary arithmetic is evaluated
After validation, each bit string is parsed as an exact BigInt. Addition and subtraction operate on the mathematical integer values, equivalent to column arithmetic with base-2 carrying or borrowing. Multiplication is exact and conceptually combines shifted partial products for every set bit in the multiplier.
Division returns an integer quotient and remainder satisfying dividend = divisor × quotient + remainder. BigInt truncates the quotient toward zero. This differs from floor division for negative values: -13 ÷ 5 gives quotient -2 and remainder -3, not quotient -3 and remainder 2.
The result is formatted independently in base 2, base 10, and base 16. This does not introduce rounding because all three strings come from the same exact integer. The calculator does not simulate a CPU register unless you later apply a specific width and mask.
Practical uses
Exact binary arithmetic is useful for education and for checking integer-heavy code without manually translating every operand.
Verify programming exercises
Check carries, borrows, multiplication steps, and long division while learning binary arithmetic or computer architecture.
Test large integer logic
Create reference results for BigInt code, arbitrary-precision libraries, parsers, and serialization tests beyond the safe Number range.
Analyze bit-oriented values
Add offsets, subtract addresses, multiply block counts, or divide binary capacities while preserving the original notation.
Cross-check representations
Compare the binary answer with decimal and hexadecimal output before inserting a constant into code or documentation.
Common mistakes and edge cases
The calculator is exact, but it deliberately avoids guessing about fractions, widths, or division conventions.
Including non-binary digits
Digits 2 through 9 are not valid in a base-2 operand. Convert a decimal value first instead of mixing notations in one field.
Expecting fixed-width overflow
Arbitrary-precision arithmetic keeps growing. To model an 8-bit or 32-bit register, use a width-limited bitwise operation or convert the result with an explicit mask.
Ignoring the remainder
Integer division does not produce a fraction. Use quotient and remainder together when the dividend is not evenly divisible.
Assuming floor division for negatives
The quotient truncates toward zero and the remainder follows the dividend’s sign, matching JavaScript BigInt semantics.
How this differs from other tools
Arithmetic changes the value, while many nearby converters only change representation.
Binary to decimal
A converter rewrites one value in another base. This calculator combines two operands through an arithmetic operation.
Bitwise calculator
Bitwise AND, OR, XOR, NOT, and shifts operate on individual fixed-width bits. They are not substitutes for ordinary addition or multiplication.
Two’s complement
This calculator writes negative results with a minus sign. Two’s complement maps them into an exact fixed-width storage pattern.
IEEE-754 converter
IEEE-754 handles rounded floating-point values, including fractions and special values. This calculator handles exact whole integers only.