Binary ↔ Decimal

What Different Number Bases Actually Are

Every number you write down is expressed in a base — a positional system that defines how many symbols are available and how place value works. Humans default to base 10 (decimal) because we have 10 fingers. Computers use base 2 (binary) at the hardware level because transistors have two states: on or off. Programmers frequently work in base 16 (hexadecimal) because it maps neatly to binary bytes.

This converter lets you type a number in any of the four main systems and see its equivalent in all others simultaneously — no switching tabs, no manual arithmetic.

The Four Systems Explained

Decimal (Base 10) — The everyday system. Digits: 0–9. Positional value: each place is 10× the one to its right. The number 255 means (2 × 100) + (5 × 10) + (5 × 1).

Binary (Base 2) — The hardware language. Digits: 0 and 1 only. Each place is 2× the previous. The number 11111111 in binary = 255 in decimal. Every file, pixel, and byte your computer processes is ultimately stored this way.

Octal (Base 8) — Digits 0–7. Primarily seen in Unix/Linux file permissions. When a developer types chmod 755, those three digits are octal — each representing three bits of the binary permission string. 7 = 111 binary = read+write+execute.

Hexadecimal (Base 16) — Digits 0–9, then A–F. A = 10, B = 11, ... F = 15. Each hex digit represents exactly 4 binary bits, making it a compact and human-readable way to express binary data. Two hex digits = one byte. Used everywhere: memory addresses, colour codes, cryptographic hashes.

The Conversion Logic

Every conversion routes through decimal as an intermediate:

Any base → Decimal: Multiply each digit by its positional value and sum.

Binary 1011 = (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1) = 11
Hex B    = 11
Octal 13  = (1 × 8) + (3 × 1) = 11

Decimal → Any base: Repeatedly divide by the target base and collect remainders.

11 ÷ 2 = 5 R1
 5 ÷ 2 = 2 R1
 2 ÷ 2 = 1 R0
 1 ÷ 2 = 0 R1 → Read remainders bottom-to-top: 1011

Where Each System Actually Appears

Binary: You won't often type raw binary — it's what CPUs work with internally. But you'll encounter it when reading about bit flags, network masks (a /24 subnet mask is 11111111.11111111.11111111.00000000), or working with embedded systems.

Octal: Mainly Unix file permissions. When you see drwxr-xr-x, that's 755 in octal: owner has 7 (read+write+execute), gro...