Number Base Converter
Convert numbers between different bases including binary, octal, decimal, hexadecimal, and any base from 2 to 36.
Common bases: Binary (2), Octal (8), Decimal (10), Hexadecimal (16)
Extended bases: Use letters A-Z for digits 10-35 (bases 11-36)
What is a Number Base?
A number base (or radix) determines how many unique digits are used to represent numbers. Decimal (base 10) uses digits 0-9, while binary (base 2) uses only 0 and 1. Different bases are fundamental to computing, cryptography, and puzzle solving.
This tool converts numbers between any base from 2 to 36, using letters A-Z to represent values 10-35 in higher bases.
Common Number Bases
Binary (Base 2)
The foundation of all digital computing:
- Uses only 0 and 1
- Each digit is called a "bit"
- Example: 42 = 101010 in binary
- Essential for understanding computer operations
Octal (Base 8)
Historically used in computing:
- Uses digits 0-7
- Each octal digit = 3 binary bits
- Example: 42 = 52 in octal
- Common in Unix file permissions
Decimal (Base 10)
The standard human number system:
- Uses digits 0-9
- Based on ten fingers (digits)
- Universal for everyday counting
- The default base for most contexts
Hexadecimal (Base 16)
Widely used in computing and programming:
- Uses 0-9 and A-F (A=10, B=11, ... F=15)
- Each hex digit = 4 binary bits
- Example: 42 = 2A in hexadecimal
- Used for colors, memory addresses, MAC addresses
Number Bases in Geocaching
Geocachers encounter various number bases in puzzles:
Common Uses
- Binary coordinates: GPS numbers encoded in binary
- Hex puzzles: Coordinates as hexadecimal values
- Tech-themed caches: Computer science puzzles
- QR codes: Often contain hex or binary data
Puzzle Patterns
- Binary strings that decode to coordinate digits
- Hex color codes hiding numbers (#2A = 42)
- Octal Unix permissions as coordinate values
- Mixed base encoding for multi-step solutions
How Base Conversion Works
To Decimal (Base 10)
Multiply each digit by its place value and sum:
- Binary 101010: (1×32) + (0×16) + (1×8) + (0×4) + (1×2) + (0×1) = 42
- Hex 2A: (2×16) + (10×1) = 32 + 10 = 42
From Decimal
Repeatedly divide by the target base and collect remainders:
- 42 to binary: 42÷2=21r0, 21÷2=10r1, 10÷2=5r0, 5÷2=2r1, 2÷2=1r0, 1÷2=0r1 → 101010
- 42 to hex: 42÷16=2r10 → 2A
Extended Bases (11-36)
For bases higher than 10, letters represent additional digits:
- A = 10
- B = 11
- C-F = 12-15 (used in hex)
- G-Z = 16-35 (for bases 17-36)
Base 36 uses all digits 0-9 and letters A-Z, making it the highest single-character base.
Recognizing Different Bases
Clues to identify which base a number uses:
- Only 0s and 1s: Likely binary
- Contains A-F: Likely hexadecimal
- Only 0-7: Could be octal
- 0x prefix: Hexadecimal notation
- 0b prefix: Binary notation
- 0o prefix: Octal notation
Binary Representation
Understanding binary place values:
- Position 0 (rightmost): 1 (2⁰)
- Position 1: 2 (2¹)
- Position 2: 4 (2²)
- Position 3: 8 (2³)
- Position 4: 16 (2⁴)
- Position 5: 32 (2⁵)
- Position 6: 64 (2⁶)
- Position 7: 128 (2⁷)
Tips for Base Conversion Puzzles
- Check the digits: The highest digit tells you the minimum base
- Look for prefixes: 0x, 0b, and 0o indicate the base
- Try common bases first: Binary, octal, decimal, hex cover most cases
- Verify with context: Decoded numbers should make sense as coordinates
- Group binary: Groups of 4 bits convert nicely to hex
Related Number Tools
- Hexadecimal Converter: Convert text to/from hex encoding.
- Binary Converter: Convert text to/from binary representation.
- Roman Numerals: Convert ancient Roman number system.
- Digital Root: Calculate the digit sum of numbers.