Different numeral bases appear throughout computing, science, history, and everyday life. Here is a reference to what each significant base is used for.
The foundation of all digital computing. Maps directly to transistor on/off states. Used in bitwise operations, boolean logic, and machine code.
Used in early Unix file permission masks (e.g., chmod 755). Each octal digit represents exactly three binary bits, making it a compact binary shorthand.
The universal human counting system, likely derived from ten fingers. Used in everyday arithmetic, currency, measurement, and scientific notation.
Appears in dozens, inches per foot, months per year, and hours on a clock face. Twelve has more divisors than ten, making fractions simpler.
Standard shorthand for binary in computing. One hex digit represents four bits exactly. Used in memory addresses, color codes (#FF5733), and debugging.
Used by the ancient Maya for their complex calendar system. Also appears in the French counting system (quatre-vingts = "four-twenties" = 80).
Used in Crockford Base32 and RFC 4648 encoding for checksums, TOTP tokens (two-factor authentication), and compact hash representations.
The largest base expressible with the standard alphanumeric set (0-9, A-Z). Used for URL shorteners, unique ID generation, and compact numeric encoding.
Used by ancient Babylonians. Persists today in time (60 seconds, 60 minutes) and angles (360 degrees = 6 x 60). Sixty has twelve divisors, enabling clean fractions.
Not a positional number system, but an encoding scheme. Represents binary data as printable ASCII text. Used in email attachments, data URIs, and JWT tokens.
The Complete Guide to Base-N Number Systems
Whether you are debugging a hexadecimal memory address, studying historical numeral systems, or building a cryptographic tool that needs compact number representations, understanding arbitrary base conversion is a core computer science skill. This guide explains the math, the history, and the practical use cases.
How to Use This Calculator
Type any number into the "Number Value" field. Select the base that number is currently written in using the "Starting Base" slider (default is Base-10, which is standard decimal). Then choose your "Target Base" using the second slider. The result appears instantly in the green output box below. The Quick Reference Matrix below that always shows the same value converted into Binary, Octal, Decimal, and Hex simultaneously.
For very large numbers, this calculator uses JavaScript BigInt internally, so it avoids the floating-point precision loss that affects standard parseInt() for values beyond 2^53. You can safely convert large hashes and identifiers without corrupting digits.
Click "Show Algorithm Breakdown" to see the full mathematical expansion of the conversion - each digit multiplied by its positional weight (the base raised to the digit's index), then summed.
The Mathematics of Positional Notation
Every positional number system works the same way: the value of a number is the sum of each digit multiplied by the radix raised to that digit's position index (starting at 0 from the right). For a number with digits d(n), d(n-1), ..., d(1), d(0) in base B, the decimal value is:
Value = d(n)*B^n + d(n-1)*B^(n-1) + ... + d(1)*B^1 + d(0)*B^0
For example, the hexadecimal number 1A3 in Base-16 equals (1 x 256) + (10 x 16) + (3 x 1) = 256 + 160 + 3 = 419 in decimal. The letters A-Z represent digit values 10-35 in bases higher than 10.
Converting TO a Target Base
To convert a decimal integer to any target base B, repeatedly divide the number by B and record each remainder. The remainders, read from last to first (bottom to top), form the digits of the result in base B. This calculator performs that process using BigInt arithmetic so the precision is exact at any length.
Why Base-36 is the Maximum for Standard Text
The alphanumeric character set gives us exactly 36 symbols: the digits 0-9 (ten symbols) plus the letters A-Z (twenty-six symbols). Base-36 uses all of them. Any base higher than 36 would require inventing new symbols (or using multi-character tokens), which breaks the simple one-digit-one-character mapping that makes positional notation readable. Base-64 encoding, for example, is not a true positional number system - it uses a separate symbol table and is used for encoding binary data as text, not for arithmetic.