Understanding Number Bases

A number base (or radix) is how many distinct digits a counting system uses before it carries over to the next column. Decimal — the system taught in school — is base 10: ten digits, 0 through 9, and the column after 9 rolls over to 10. Binary, octal, and hexadecimal follow the exact same positional logic with a different digit count.

Computers do not have a native concept of decimal. Every value in memory is a sequence of on/off switches — binary — and octal and hexadecimal exist purely as more compact, human-readable ways to write down binary values without losing the exact bit pattern.

The three bases that matter

Hex won over octal for most modern uses because a byte (8 bits) splits evenly into two 4-bit hex digits, but not evenly into 8/3 octal digits — hex maps to bit patterns more cleanly.

Conversion table: 0 to 15

Past 9, hexadecimal keeps counting with letters: A = 10, B = 11, up to F = 15. One hex digit always covers exactly the same range as 4 binary digits, which is the whole reason it is convenient — 15 is the largest value 4 bits can hold, and F is the largest single hex digit.

Converting hex to binary is just substitution

Because one hex digit is exactly 4 bits, converting hex to binary (or back) needs no arithmetic — just replace each digit with its 4-bit pattern from the table above:

0xC0  →  1100 0000
0xA8  →  1010 1000
0xFF  →  1111 1111

Decimal does not divide evenly into binary this way, which is why converting decimal to binary needs actual division (repeatedly dividing by 2 and reading the remainders bottom-up), while hex-to-binary is pure lookup.

Reading number literals in code

Most C-family languages (JavaScript, Python, Rust, Go) recognize the same prefixes for non-decimal literals:

0xFF        // hexadecimal — 255
0b11111111  // binary — 255
0o377       // octal — 255
1_000_000   // underscore as a digit separator (readability only)

C and C++ are the outliers: a bare leading zero means octal (0755, not 0o755), a frequent source of bugs when a decimal number is accidentally zero-padded and silently reinterpreted.

Where each base shows up in practice

• Unix permissions — chmod 755 means owner rwx (7 = 111), group r-x (5 = 101), others r-x (5 = 101). Octal exists here because 3 bits maps exactly to the 3 permission flags (read/write/execute).
• CSS/HTML colors — #RRGGBB, each channel one byte (00–FF) in hex.
• Network addresses — 192.168.1.1 is 0xC0A80101 as a 32-bit hex integer; subnet masks like 255.255.255.0 are 24 binary ones followed by 8 zeros.
• Character encodings — ASCII and Unicode code points are conventionally written in hex: the letter A is 0x41 (decimal 65).
• Hashes and UUIDs — a SHA-256 digest or a UUID is displayed as hex because the raw value is binary and hex is the standard compact text representation for it.