Number Base Converter

How number bases work

A base (or radix) is how many digits a number system uses. Decimal uses ten (0–9), binary two (0 and 1), octal eight (0–7) and hexadecimal sixteen (0–9 then A–F). Each position is worth the base times the position to its right, so hex 2F is 2 × 16 + 15 = 47.

Hex to decimal: the formula with an example

Multiply each hex digit by 16 raised to its position, counting from 0 on the right, and add the results. A to F stand for 10 to 15. For 1A3F:

1 × 16³ = 1 × 4096 = 4096
A × 16² = 10 × 256 = 2560
3 × 16¹ =  3 × 16  =   48
F × 16⁰ = 15 × 1   =   15
                     ----
                     6719

Binary to decimal with an example

Same idea with powers of 2: write 1, 2, 4, 8, 16… under the digits from the right and add the ones that sit under a 1. For 101101: 32 + 8 + 4 + 1 = 45.

Decimal to hex or binary

Divide by the target base repeatedly and write down the remainders; read them from the last to the first. 6719 to hex: 6719 ÷ 16 = 419 remainder 15 (F), 419 ÷ 16 = 26 remainder 3, 26 ÷ 16 = 1 remainder 10 (A), 1 ÷ 16 = 0 remainder 1, so the answer is 1A3F. For binary, divide by 2 instead: 45 gives the remainders 1, 0, 1, 1, 0, 1, read backwards as 101101.

Conversion table: 0 to 15

One hex digit is exactly four binary digits, so this table is all you need to convert between hex and binary by hand: 0x2F is 0010 1111.

Hex values worth knowing

A hex colour is three bytes: #2F6FED is red 2F = 47, green 6F = 111 and blue ED = 237.

Where you meet each base

• Binary: bit flags, network masks, bitwise operations and hardware registers.
• Hexadecimal: colours (#2F6FED), memory addresses, byte dumps, hashes and UUIDs. One hex digit is exactly four bits, so two digits make a byte.
• Octal: mainly Unix file permissions (chmod 755).
• Base 36: compact IDs and short links, since it uses every digit and letter.

Negative numbers and two's complement

Computers store negative integers in two's complement: the value −1 in 8 bits is 11111111 (0xFF), the same bits as unsigned 255. The table above shows both readings for 8, 16, 32 and 64-bit widths whenever the number fits. This explains why a byte read from a file might show as −56 in Java (whose byte is signed) but 200 elsewhere.

Converting in code

• JavaScript: (255).toString(16) and parseInt('ff', 16); use BigInt for values beyond 2⁵³.
• Java: Integer.toBinaryString(n), Long.parseLong("ff", 16), or new BigInteger(s, 36).
• Python: bin(n), hex(n), oct(n) and int('ff', 16).

Frequently asked questions