Number Base Converter — Convert Between Any Bases 2-36
Convert numbers between any two bases from binary (base 2) to base 36. Supports hexadecimal, octal, binary, and all custom bases.
Base Converter
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The Formula
A number in any base b is represented as a sequence of digits where each digit position represents a power of the base. Converting between bases involves first interpreting the number in its source base (expanding each digit multiplied by base raised to its positional power) and then expressing that decimal value in the target base by repeatedly dividing by the target base and collecting remainders.
Variable Definitions
Base
The number of unique digits used in a positional numeral system. Binary (b=2) uses 0-1, decimal (b=10) uses 0-9, hexadecimal (b=16) uses 0-9 and A-F, and base-36 uses all digits 0-9 and letters A-Z representing values 10-35.
Digit at Position i
Each digit in the number represents a coefficient at a specific positional power of the base. Position i=0 is the rightmost (units) position, i=1 is the next (b^1 position), and so on. The value contributed by digit d at position i is d × b^i.
Number in Base b
The complete number expressed in the positional numeral system with base b. The total decimal value is the sum of all digit contributions: Σ (d_i × b^i) for all positions i from 0 to n-1, where n is the digit count.
Digit Count
The number of digits in the converted result. Also called the length of the representation. A larger base generally produces fewer digits to represent the same value. For example, 255 decimal = FF in hex (2 digits) = 11111111 in binary (8 digits).
How to Use This Calculator
- 1
Enter the number you want to convert in the "Input Number" field. Use uppercase letters A-Z for digits above 9 (e.g., "FF" for 255 in hex).
- 2
Set "From Base" to the base of your input number (minimum 2, maximum 36). For example, set 16 for hexadecimal input, 2 for binary, 10 for decimal.
- 3
Set "To Base" to the target base you want to convert to. The calculator will display the equivalent value in the target base.
- 4
Review the converted result, the intermediate decimal value (useful for understanding the conversion process), the validation status of your input, and the digit count of the result.
- 5
For base conversions above 10, remember that letters A through Z represent digit values 10 through 35 respectively. "A" = 10, "B" = 11, ..., "Z" = 35.
Quick Reference
| From | To |
|---|---|
| Binary (base 2) | 0, 1 |
| Octal (base 8) | 0–7 |
| Decimal (base 10) | 0–9 |
| Hexadecimal (base 16) | 0–9, A–F |
Common Applications
- Computer science and programming — converting between binary, octal, hexadecimal, and decimal for low-level programming, memory addressing, and debugging.
- Digital electronics and embedded systems — working with register values, memory maps, and hardware configuration where hexadecimal and binary are standard.
- Networking — converting IP addresses, subnet masks, and MAC addresses between decimal and hexadecimal representations.
- Cryptography and data encoding — understanding base-64 encoding (related to base conversion) and working with hash values often displayed in hexadecimal.
- Educational settings — learning how positional numeral systems work and building intuition about number representation across different bases.
Base conversion is a two-step process: first parse the source representation into a decimal value, then convert that decimal to the target base by repeated division.
Understanding the Concept
Number base conversion is a fundamental concept in mathematics and computer science. Every positional numeral system represents numbers using a fixed set of digits and positional weights that are powers of the base. In base 10 (decimal), which we use in everyday life, the digits 0-9 are multiplied by powers of 10: the number 342 means 3×10² + 4×10¹ + 2×10⁰ = 300 + 40 + 2. The same principle applies to any base. Binary (base 2) uses only two digits (0 and 1) and powers of 2, making it the natural language of digital circuits where each bit represents an on/off state. Hexadecimal (base 16) is widely used in computing because one hex digit corresponds exactly to four binary digits (a nibble), making it much more compact for representing binary data. Base 36 extends hexadecimal by using all 26 letters of the English alphabet (A-Z) for digits 10-35, creating the most compact alphanumeric representation within the 0-9 and A-Z character set. Converting a number from its source base to a target base follows a two-step process. First, interpret the source representation by expanding: multiply each digit by the source base raised to its positional power and sum the results to get the decimal value. Second, convert the decimal to the target base: repeatedly divide the decimal number by the target base, collecting the remainders from right to left, which become the digits of the result. This algorithm works for any base between 2 and 36, with bases above 10 requiring letters to represent the extra digit values. The maximum practical base for human-readable representation is base 36, since it uses all ten decimal digits plus the twenty-six letters of the Latin alphabet. Bases beyond 36 require additional symbols. The digit count of a number across different bases follows an inverse logarithmic relationship: smaller bases require more digits (binary is the most verbose), while larger bases produce more compact representations.
Frequently Asked Questions
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