Same number.
Different numeral.
A number is an abstract value, like a quantity. A numeral is a written representation of that value. Change the writing system. The quantity stays the same.
One quantity. Many names.
Each numeral on the right represents the same number. The base label tells you how to read it.
Build a numeral.
A positional base-b system uses b digit symbols. Each position is worth b times the position to its right.
A digit is a single symbol. A numeral can contain one digit or several.
Small alphabet. Huge possibilities.
Each extra digit multiplies the number of possible patterns by the base.
We count unsigned whole numbers, allowing leading zeros in the slots. With up to d digits in base b, there are bᵈ distinct values, from 0 to bᵈ − 1. Negative numbers and fractions need additional conventions.
What if the numeral must have exactly d digits, with no leading zeros?
For one digit, zero is also a valid one-digit numeral, so all b symbols count.
Does 10 always mean ten?
In binary, what quantity does the numeral 10 represent?