Number Base Converter

How This Calculator Works

A number base (or “radix”) is how many distinct digits a numeral system uses before it carries over to the next place value — decimal uses 10 (0–9), binary uses 2 (0–1), octal uses 8 (0–7), and hexadecimal uses 16 (0–9 then A–F). Enter a whole number in any of these four bases, and this calculator shows the same value written in all four at once.

Binary is how computers represent every value internally (each digit is one “bit”); hexadecimal is a common shorthand for binary in programming (each hex digit stands in for exactly 4 binary digits, so it’s far more compact to read and type); octal shows up in older Unix file permissions and some low-level contexts. Decimal is just the everyday base-10 system.

The Formula

Every positional numeral system works the same way: each digit is multiplied by the base raised to its place-value power, then summed:

$$\vA{\text{Value}} = \sum_{i=0}^{n} d_i \times \vB{\text{Base}}^i$$

where $d_i$ is the digit at position $i$ (counting from 0 at the rightmost digit). Converting to a different base repeatedly divides the decimal value by the new base, recording each remainder — the reverse of the sum above.

Worked Example

Converting binary 11111111 to its other three bases:

1. Read right to left, each 1 contributes its place value: $2^7 + 2^6 + 2^5 + 2^4 + 2^3 + 2^2 + 2^1 + 2^0 = 255$.
2. Decimal: 255.
3. Octal: dividing 255 by 8 repeatedly (255 ÷ 8 = 31 r7, 31 ÷ 8 = 3 r7, 3 ÷ 8 = 0 r3) reads the remainders bottom-to-top as 377.
4. Hexadecimal: 255 ÷ 16 = 15 r15, 15 ÷ 16 = 0 r15 — both remainders are 15 (F in hex), giving FF.