What are positional number systems?

Answer 1

A positional number system is one where the position of a digit is significant to its value. For instance, the Hindu-Arabic system denotes the decimal symbol 42 to mean forty tens and two units, the sum of which gives us the number forty-two. Note that 42 is not a number, it is a symbol; forty-two is the actual number represented by that symbol. The digits in the symbol are valued according to their position, hence the Hindu-Arabic system is a positional number system.
The Hindu-Arabic system can be applied to any numeric base, whether binary (base 2), octal (base 8), decimal (base 10), hexadecimal (base 16) or sexagesimal (base 60). When working with integer values (whole numbers), the digits from right-to-left represent an increasing power of the base, starting from 0. When working with fractions of a unit, the digits from left-to-right represent a decreasing power of the base, starting from -1. Multiplying the digit by its positional value gives its actual value within the symbol.

By contrast, the Roman numeral system is non-positional, because the physical position of a digit within a symbol has no bearing on its value. An X always denotes the value ten regardless of where it appears within the symbol.

Non-positional systems typically do not have a symbol to represent the value zero. The reason is not that the notion of "nothing" didn't exist, it was simply that you do not need a symbol for zero when counting or making a tally; you only count the things you have, not what you do not have. Hence the natural numbers (counting numbers) begin at one, not zero.
However, in positional systems, you need a place-holder to denote that a position has no value. In some systems a space would be used, but in Hindu-Arabic, the digit 0 is used. Thus the decimal symbol 402 denotes that the value has no tens.