Identify a Numeral’s Base

1) Read the numeral as a base-specific expression

A numeral like (321_4) is not just a number written with digits; it means the digits are interpreted in base 4. The base tells you which place values are allowed. In a valid base (b), every digit must be from (0) up to (b-1).

2) Use the digits to narrow the base

Look at the largest digit in the numeral. If the largest digit is (7), then the base must be at least (8), because base (7) would only allow digits (0) through (6). So the smallest possible base is one more than the largest digit.

3) Check for a unique base if needed

If the exercise asks for the exact base, use any extra information given. For example, if the numeral is written with a subscript, that subscript is the base. If only the digits are given, then you may only be able to identify the minimum possible base.

4) Verify your answer

Check that every digit is valid in the base you chose. Also make sure the place-value interpretation makes sense: each place should represent powers of the base.

Quick rule

• Largest digit (d) means base must satisfy (b>d).
• If the numeral includes a base label, read that label directly.