Number bases for beginners

A base (radix) tells you how many distinct digits exist before you carry to the next place. Decimal uses ten digits (0–9); binary uses two (0–1); hexadecimal uses sixteen (0–9 and A–F). The value of a numeral is the sum of each digit times the base raised to its place power, starting at 0 on the right.

Why bases matter in computing

• Binary (base 2) — maps directly to bits and logic gates.
• Octal (base 8) — groups three bits per digit; legacy Unix permissions still use octal. Understanding of legacy 12-bit addressing.
• Hexadecimal (base 16) — groups four bits per digit; compact for memory dumps and colours (#FF00AA).
• Decimal (base 10) — what humans type; computers convert internally.

Converting via decimal

To convert between bases, expand to decimal then divide out the target base. Our Number Base Conversions tool keeps decimal, hexadecimal, octal, and binary in sync as you type.

Worked example

Binary 1011 to decimal: (1×2³) + (0×2²) + (1×2¹) + (1×2⁰) = 8 + 0 + 2 + 1 = 11. Hex B is also 11; so 1011₂ = B₁₆ = 11₁₀.

Hex nibble trick

Each hex digit is exactly four bits. Split long binary strings into groups of four from the right, convert each nibble to hex, and concatenate. Example: 1101 1011 → D B → DB.

FAQs

What is base 36?

Digits 0–9 plus A–Z; sometimes used for compact IDs. This site’s converter covers decimal, hex, octal, and binary only.

Leading zeros?

They change width but not value; binary and hex displays often pad to byte boundaries.

Signed numbers?

Two’s complement is a separate topic — see our Signed & Unsigned Numbers tool.

Sources

Standard positional numeral systems; Knuth, The Art of Computer Programming, vol. 2 (seminumerical algorithms).