How positional number bases work
A base determines the available digits and the value of each position. Decimal uses ten digits and powers of 10; binary uses 0 and 1 with powers of 2; hexadecimal uses 0–9 and A–F with powers of 16. In base b, the positions to the left of the point represent b⁰, b¹, b² and so on.
For example, 101101₂ = 1×2⁵ + 0×2⁴ + 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 45₁₀.
Fractional digits and recurring results
Positions to the right of the radix point use negative powers: the first is b⁻¹, the second b⁻². Therefore .011₂ = 0×1/2 + 1×1/4 + 1×1/8 = 3/8. The converter stores the entered value as an exact fraction before producing other bases.
Some fractions terminate in one base but repeat in another. Decimal 0.1 terminates in base 10 but repeats in binary. Repeating digits are enclosed in parentheses when a cycle is found; an ellipsis means the selected precision was reached first.
Where base conversion is used
Debugging IDs, addresses and protocol fields
A log may print an identifier as decimal while a debugger or protocol document uses hexadecimal. Converting both representations confirms whether they describe the same value. Binary is useful when individual flag bits matter; hexadecimal is easier to scan because each hex digit represents exactly four bits.
Working with web colours and encoded bytes
An RGB channel value such as decimal 172 becomes AC in hexadecimal. That makes rgb(172, 60, 83) equivalent to #AC3C53. This is numeric conversion only: decoding bytes into text still requires an ASCII or Unicode tool.
Checking digital logic and embedded registers
Datasheets often show register addresses and masks in hex, while timing diagrams expose the individual binary lines. Converting a register value to padded binary makes it easier to see which status or control bits are set.
Learning place value and repeated division
Students can compare the exact place-value expansion with the result. Repeated division by the target base supplies integer remainders from right to left; repeated multiplication of a fractional remainder supplies digits after the radix point.
Conversion examples
| Value | Equivalent representations and context |
|---|---|
| 42₁₀ | 101010₂ = 52₈ = 2A₁₆ |
| FF₁₆ | 255₁₀ = 11111111₂, the largest unsigned 8-bit value |
| 10.5₁₀ | 1010.1₂ = A.8₁₆ |
| 0.1₁₀ | 0.0(0011)₂; the repeating group explains common binary floating-point approximations |
| 1,000,000₁₀ | F4240₁₆ = 11110100001001000000₂ |
| AC₁₆ | 172₁₀ = 10101100₂; useful for one RGB channel or an 8-bit register |
Limits and notation
This tool converts numerical values, not text encodings. “41” in hexadecimal is the number 65; interpreting that byte as the letter A requires an ASCII or Unicode tool. Very long output is limited by the fractional-precision setting, while the integer part uses arbitrary-precision arithmetic.