Math & Scientific

Find a Number Base: Place Values and Unknown Radices

Expand positional numbers as powers of a base, solve a simple unknown-base equation, and check valid digits with the local integer converter.

By Uddhav Gopani, FounderPublished 25 September 2026Reviewed 29 September 2026

Find a Number Base: Place Values and Unknown Radices

Identify which meaning of base is intended

In a positional numeral system, the base or radix tells you how much each place is worth compared with the place to its right. This guide concerns number bases such as binary, decimal and hexadecimal. It does not concern the base of a triangle, a logarithm or an exponential expression. A written string such as 11 is ambiguous until its base is stated: it represents three in binary, nine in octal and eleven in decimal. Always retain the radix with the digits when comparing representations or solving a base problem.

Expand the positional formula

For digits a, b and c in a three-place numeral written in base r, its decimal value is a×r²+b×r+c. More generally, the rightmost integer digit has weight r⁰, the next r¹, then r² and so on. Each digit must have a value from zero through r−1. The base is a whole number at least two in this ordinary positional convention. The letters used for larger digits are symbols for values: in hexadecimal, A means ten and F means fifteen, rather than algebraic variables to be solved independently.

Check binary and hexadecimal examples

Binary 1011 expands as 1×2³+0×2²+1×2¹+1×2⁰. The total is 8+0+2+1, or decimal 11. Hexadecimal 2F expands as 2×16+15, which equals decimal 47. In the local Number Base Converter, enter 1011 with source base 2 and target base 10 for the first check, or 2F with source base 16 for the second. These examples change the representation while preserving the integer value. They do not mean that the strings 1011 and 2F have the same value as ordinary decimal digit strings.

Solve a simple unknown base

Suppose the numeral 12 in an unknown base r is stated to equal decimal 11. Expanding it gives 1×r+2=11, so r=9. Check both requirements: the base is at least two, and the largest digit, 2, is less than 9. Then entering 12 from base 9 to base 10 reproduces 11. The converter requires a known source base; it does not solve this equation for you. The algebra establishes the candidate radix, and conversion checks that candidate against the original numeral and decimal value.

Reject roots that cannot be a radix

For an equation involving the numeral 101 in base r and decimal 17, expansion gives r²+1=17. Thus r²=16, with algebraic roots 4 and −4. Only positive integer base 4 satisfies the stated ordinary radix convention. As a different example, saying 19 in base r equals decimal 15 gives r+9=15 and r=6, but digit 9 is not permitted in base 6. That candidate must be rejected even though it solves the simplified linear equation. Digit constraints belong to the original problem and must be checked after solving.

Convert back by repeated division

To write decimal 47 in base 16, divide by 16: the quotient is 2 and the remainder is 15. The quotient supplies the leading digit 2, and remainder 15 is written F, giving 2F. For decimal 61 in base 8, division gives quotient 7 and remainder 5, so the result is octal 75. Reading remainders in the wrong order is a common error in longer conversions. Verify by expanding the final numeral: 7×8+5 is 61. A round trip through the converter should preserve the original integer value.

Use the local converter's actual scope

The form accepts signed integer digit strings and whole source and target bases from 2 through 36. It validates every digit and uses integer arithmetic for the conversion; it does not accept fractional radix points or automatically infer an unknown base. Enter the digits with the base supplied in its separate field rather than adding an unrequested prefix. In JavaScript, MDN notes that parseInt can stop at an invalid character and lose precision for large values, so that function alone is not a complete strict validator. The mathematical expansion and the form's supported input contract should remain distinct from a programming shortcut.