Binary / Decimal / Hex Base Converter

Enter a value in any base field and click Convert to see the equivalent in all other bases.

Decimal (Base 10)

Binary (Base 2)

Hexadecimal (Base 16)

Octal (Base 8)

Results will appear here.

Every positional numeral system expresses an integer N as a sum of digit × base^position:

N = d_n·bⁿ + d_n-1·b^n-1 + … + d₁·b¹ + d₀·b⁰

Any base → Decimal: Multiply each digit by its base raised to its positional power and sum.
Decimal → Binary (base 2): Repeatedly divide by 2; remainders (read bottom-up) form the binary digits.
Decimal → Octal (base 8): Repeatedly divide by 8; remainders (read bottom-up) form the octal digits.
Decimal → Hex (base 16): Repeatedly divide by 16; remainders map to 0–9, A–F.
Binary → Hex shortcut: Group binary digits in sets of 4 from the right; each group maps directly to one hex digit.
Binary → Octal shortcut: Group binary digits in sets of 3 from the right; each group maps to one octal digit.

Example — Decimal 255:

255 ÷ 2 → remainders: 1,1,1,1,1,1,1,1 → Binary: 11111111
255 ÷ 16 → 15 remainder 15 → Hex: FF
255 ÷ 8 → 31 remainder 7 → 3 remainder 7 → Octal: 377

Conversion uses JavaScript's built-in parseInt(value, base) and Number.prototype.toString(base), which are exact for integers within the safe integer range (±2⁵³ − 1).
Negative numbers are handled with a leading minus sign; two's complement representation is not applied.
Hexadecimal digits A–F are displayed in uppercase.
Bit length is the minimum number of bits required to represent the absolute value (e.g., 255 → 8 bits).
Byte length is ⌈bit length / 8⌉.
Reference: Knuth, D. E. — The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, §4.1 Positional Number Systems.
IEEE 754 floating-point is not used; all arithmetic is integer-based.

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