Advanced Number Base Conversion Tool

Convert between binary, decimal, hexadecimal, and octal number systems instantly

Technical Disclaimer: This tool provides mathematical conversions for educational and programming purposes. While we strive for accuracy, always verify critical conversions in sensitive applications like financial or cryptographic systems.

Tip: Use the "Explain Conversion" button to see detailed steps for learning purposes.

Number to Convert

From Base

To Base

Understanding Number Bases: A Practical Guide

Common Bases Explained

Binary (Base 2): Uses 0-1, fundamental for computers
Octal (Base 8): Uses 0-7, historically important in computing
Decimal (Base 10): Uses 0-9, our everyday number system
Hexadecimal (Base 16): Uses 0-9 and A-F, compact representation for binary

Practical Applications

Binary: Computer processors, memory addressing
Hexadecimal: Color codes (HTML/CSS), memory dumps
Octal: Unix file permissions, aviation standards
Custom bases: Cryptography, encoding schemes

"Did you know? The ancient Babylonians used a base-60 (sexagesimal) number system, which is why we have 60 seconds in a minute and 60 minutes in an hour."

Number Base Conversion Techniques

Decimal to Other Bases

To convert from decimal to another base, use the division-remainder method. Repeatedly divide the number by the target base and record the remainders. The converted number is the remainders read in reverse order.

Example: Convert 25 to binary (base 2):

25 ÷ 2 = 12 R1

12 ÷ 2 = 6 R0

6 ÷ 2 = 3 R0

3 ÷ 2 = 1 R1

1 ÷ 2 = 0 R1

Read remainders upwards: 11001

Other Bases to Decimal

Convert from any base to decimal by expanding each digit as a power of the base. Multiply each digit by the base raised to its position power (starting from 0 on the right) and sum all values.

Example: Convert 1A3 (hex) to decimal:

1×16² + A×16¹ + 3×16⁰

= 1×256 + 10×16 + 3×1

= 256 + 160 + 3 = 419

Between Non-Decimal Bases

For conversions between two non-decimal bases (e.g., binary to hex), it's often easiest to first convert to decimal as an intermediate step, then convert to the target base. Alternatively, use shortcut methods like binary-hex grouping.

Binary to Hex shortcut:

Group binary digits in 4s (from right)

11010110 → 1101 0110

Convert each group: D 6 → D6

Frequently Asked Questions

What is the maximum base this converter supports?

Our tool supports bases from 2 (binary) up to 36. Base 36 uses digits 0-9 and letters A-Z. While mathematically you can have higher bases, standard conversion conventions typically stop at 36 due to character set limitations.

How does the tool handle fractional numbers?

The current version converts whole numbers only. For fractional conversions (like 0.5 decimal to binary), we recommend multiplying to create a whole number, converting, then adjusting the decimal point in the result.

Why would I need to use a custom base?

Custom bases have niche but important applications. Base64 is used for encoding binary data, base32 for human-readable codes, base60 in time/angle measurements, and base26 sometimes for alphabetical numbering systems.

How accurate are the conversions?

The conversions are mathematically precise for whole numbers within JavaScript's number precision limits (up to 2⁵³ - 1). For extremely large numbers (over 16 digits), precision may be affected by floating-point representation.

Can I convert negative numbers?

Yes, with the Two's Complement option enabled. This is particularly useful for computer science applications where negative numbers are represented using this system in binary.