Binary, Octal, Decimal, and Hexadecimal Explained (With Conversion Methods)

We write numbers in base 10 because we have ten fingers. Computers work in base 2, and programmers often use base 8 and base 16 as compact ways to write binary. Once you see the pattern, conversions become straightforward.

What a base means

In any base, each digit position is worth a power of the base. In decimal, 345 means 3 x 100 + 4 x 10 + 5 x 1. In binary, each position is a power of 2, so 1101 means 1 x 8 + 1 x 4 + 0 x 2 + 1 x 1, which is 13.

The four common bases

• Binary (base 2): digits 0 and 1. How hardware stores everything.
• Octal (base 8): digits 0 to 7. Used for Unix file permissions (chmod 755).
• Decimal (base 10): digits 0 to 9. Everyday numbers.
• Hexadecimal (base 16): digits 0 to 9 and A to F. Colors, memory addresses, and byte values.

Decimal  Binary     Octal  Hex
10       1010       12     A
255      11111111   377    FF
4095     111111111111  7777  FFF

Converting decimal to another base

1. 1Divide the number by the target base and record the remainder.
2. 2Divide the quotient again, repeating until the quotient is 0.
3. 3Read the remainders from last to first.

Example: 255 in hex. 255 divided by 16 is 15 remainder 15, and 15 divided by 16 is 0 remainder 15. Both remainders are F, so 255 is FF.

Why hex is so common

One hex digit represents exactly four bits, so one byte is always two hex digits. That is why colors are written like #4338ca, and why 0xFF is 255, the largest value of a single byte. Converting between binary and hex is just grouping bits in fours.

Prefixes in code

• 0b1010 for binary, 0o12 for octal (or a leading 0 in some older languages), and 0xFF for hex.
• In JavaScript, parseInt('ff', 16) parses hex and (255).toString(2) gives binary.
• In Python, int('ff', 16), bin(255), oct(255), and hex(255) do the conversions.