Positional Radix Number Systems Explained
In a positional numeral system with radix (base) $B$, any positive integer is represented as a sequence of coefficients:
N = d_k · B^k + d_{k-1} · B^{k-1} + ... + d_1 · B^1 + d_0 · B^0
where each digit $d_i$ satisfies $0 \le d_i < B$. In bases larger than 10, alphabetical characters $A$ through $Z$ denote digit values 10 through 35.
| Base System | Radix | Allowed Digits | Primary Computing Usage |
|---|---|---|---|
| Binary | Base 2 | 0, 1 | Digital hardware circuits, machine code, bitmasks |
| Octal | Base 8 | 0, 1, 2, 3, 4, 5, 6, 7 | Unix file permissions (e.g. 755, 644) |
| Decimal | Base 10 | 0 to 9 | Human commerce, arithmetic, standard science |
| Hexadecimal | Base 16 | 0 to 9, A to F | Memory addresses, HTML color codes, byte buffers |
| Base 32 / 36 | Base 32 / 36 | 0 to 9, A to Z | URL shorteners, compact alphanumeric identifiers |
Frequently Asked Questions
How does binary to hexadecimal direct grouping work?
Because 16 = 2^4, every 4 binary bits (nibble) map directly to one hexadecimal digit without converting to decimal first. For example, binary 1101 1010 groups into 1101 (= 13 = D) and 1010 (= 10 = A), giving 0xDA immediately.
What is the maximum integer supported by the converter?
The converter uses JavaScript BigInt precision, allowing exact conversions for arbitrarily large integers up to thousands of digits without floating point rounding error.