Base Calculator - Convert Numbers Between Different Number Bases

Our free online Base Calculator allows you to convert numbers between different number bases instantly. Whether you need to convert from decimal to binary, hexadecimal to octal, or any other base conversion, our tool provides accurate results with real-time conversion and comprehensive validation.

What is Number Base Conversion?

Number base conversion is the process of changing a number from one number system (base) to another. The most common number systems include:

Binary (Base 2): Uses only digits 0 and 1, fundamental in computer science
Octal (Base 8): Uses digits 0 through 7, commonly used in computing
Decimal (Base 10): The standard number system using digits 0-9
Hexadecimal (Base 16): Uses digits 0-9 and letters A-F, widely used in programming

How to Use the Base Calculator

Enter your number: Type the number you want to convert in the input field
Select source base: Choose the current base of your number (e.g., 10 for decimal)
Select target base: Choose the base you want to convert to (e.g., 2 for binary)
View result: The converted number will appear instantly in the result field

Conversion Examples

Decimal to Binary:

255 (base 10) = 11111111 (base 2)

Hexadecimal to Decimal:

FF (base 16) = 255 (base 10)

Octal to Binary:

377 (base 8) = 11111111 (base 2)

Mathematical Formula

The conversion process involves two main steps:

Convert to Decimal: \[ \text{Decimal} = \sum_{i=0}^{n-1} d_i \times b^i \] Where \(d_i\) is the digit at position \(i\) and \(b\) is the base.
Convert from Decimal: Divide the decimal number by the target base repeatedly and collect remainders.

Supported Number Bases

Our calculator supports conversion between bases 2 through 36:

Bases 2-10: Use digits 0 through (base-1)
Bases 11-36: Use digits 0-9 and letters A-Z
Special bases: Binary (2), Octal (8), Decimal (10), Hexadecimal (16)

Common Use Cases

Computer Programming: Converting between binary, octal, and hexadecimal
Digital Electronics: Working with binary representations
Mathematics Education: Understanding different number systems
Data Analysis: Converting data between different representations
Cryptography: Working with different base encodings

Features of Our Base Calculator

Real-time conversion: Results update instantly as you type
Input validation: Ensures valid characters for each number base
Error handling: Clear error messages for invalid inputs
Copy functionality: Easy copying of results to clipboard
Base swapping: Quick swap between source and target bases
Educational content: Built-in guide and examples

Frequently Asked Questions

What is the maximum number I can convert?

The calculator can handle very large numbers, limited only by JavaScript's number precision. For extremely large numbers, you might see some precision loss in the least significant digits.

Can I convert negative numbers?

Yes, the calculator supports negative numbers. Simply enter a negative sign before your number, and it will be converted while preserving the sign.

What characters are valid for different bases?

For bases 2-10, only digits 0 through (base-1) are valid. For bases 11-36, you can use digits 0-9 and letters A-Z (case insensitive). The calculator automatically converts lowercase letters to uppercase.

How accurate are the conversions?

The conversions are mathematically accurate. The calculator uses JavaScript's built-in parseInt() and toString() methods, which provide precise base conversion for all supported number ranges.

Can I convert between any two bases directly?

Yes, you can convert between any two supported bases (2-36) directly. The calculator first converts to decimal as an intermediate step, then converts to the target base.

What if I enter an invalid number for a base?

The calculator will display a clear error message indicating that the number contains invalid characters for the selected base. For example, entering "8" when converting from base 8 will show an error since base 8 only uses digits 0-7.