Number Base Converter Tool

Convert numbers between different bases with detailed step-by-step solution

Enter the number to convert
Current number base
Target number base
Quick Examples:

Number Base Systems and Applications

BaseNameDigitsApplicationsExample
Base 2Binary0, 1Computer memory, digital logic, binary arithmetic1010₂ = 10₁₀
Base 8Octal0–7Unix permissions, legacy systems, compact representation17₈ = 15₁₀
Base 10Decimal0–9Everyday arithmetic, human calculations, standard math255₁₀ = 255₁₀
Base 16Hexadecimal0–9, A–FMemory addresses, color codes, programmingFF₁₆ = 255₁₀

⚠ Common Mistakes to Avoid

Invalid Digits for Base

Each base has specific valid digits. For example, binary only uses 0 and 1, octal uses 0–7, and hexadecimal uses 0–9 and A–F. Using invalid digits like '8' in binary or 'G' in hexadecimal will cause errors.

Case Sensitivity in Hex

Hexadecimal digits A–F can be written in uppercase or lowercase (A–F or a–f), but mixing cases inconsistently can lead to confusion. It's best to use a consistent format.

Reading Remainders Wrong

When converting from decimal to another base using repeated division, the remainders must be read from bottom to top (last to first) to get the correct result. Reading from top to bottom will give the wrong answer.

How to Convert Between Number Bases

Number base conversion involves two main steps: converting the input number to decimal, then converting the decimal to the target base. Understanding this process is essential for computer science and programming.

Step 1: Convert to decimal using positional notation
N = dₙ × bⁿ + dₙ₋₁ × bⁿ⁻¹ + … + d₀ × b⁰

Step 2: Convert from decimal using repeated division

Conversion Algorithm

  1. Validate input — Check if digits are valid for the source base
  2. Convert to decimal — Use positional notation formula
  3. Convert to target base — Use repeated division method
  4. Read remainders — Read from bottom to top
  5. Format result — Apply proper formatting for target base

Key Properties of Number Bases

Positional Notation
In base b, each digit position represents a power of b.

Unique Representation
Every positive integer has a unique representation in any base b ≥ 2.

Examples

Decimal to Binary

255 (base 10) → base 2

Process:

  1. Convert 255 to binary
  2. Repeated division by 2
  3. Read remainders bottom to top
11111111₂

Hexadecimal to Decimal

FF (base 16) → base 10

Process:

  1. Convert FF to decimal
  2. F = 15, F = 15
  3. 15 × 16¹ + 15 × 16⁰
255₁₀

Binary to Hexadecimal

1010 (base 2) → base 16

Process:

  1. Convert 1010 to decimal
  2. Convert decimal to hex
  3. Group binary digits
A₁₆