Editing Positional notation (section)

=== Base of the numeral system ===
In [[numeral system|mathematical numeral systems]] the [[radix]] {{mvar|r}} is usually the number of unique [[Numerical digit|digits]], including zero, that a positional numeral system uses to represent numbers. In some cases, such as with a [[negative base]], the radix is the [[absolute value]] <math>r=|b|</math> of the base {{mvar|b}}. For example, for the decimal system the radix (and base) is ten, because it uses the ten digits from 0 through 9. When a number "hits" 9, the next number will not be another different symbol, but a "1" followed by a "0". In binary, the radix is two, since after it hits "1", instead of "2" or another written symbol, it jumps straight to "10", followed by "11" and "100".

The highest symbol of a positional numeral system usually has the value one less than the value of the radix of that numeral system. The standard positional numeral systems differ from one another only in the base they use.

The radix is an integer that is greater than 1, since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit. Negative bases are rarely used. In a system with more than <math>|b| </math> unique digits, numbers may have many different possible representations.

It is important that the radix is finite, from which follows that the number of digits is quite low. Otherwise, the length of a numeral would not necessarily be [[logarithm]]ic in its size.

(In certain [[non-standard positional numeral systems]], including [[bijective numeration]], the definition of the base or the allowed digits deviates from the above.)

In standard base-ten ([[decimal]]) positional notation, there are ten [[decimal digit]]s and the number
: <math>5305_{\mathrm{dec}} = (5 \times 10^3) + (3 \times 10^2) + (0 \times 10^1) + (5 \times 10^0)</math>.
In standard base-sixteen ([[hexadecimal]]), there are the sixteen hexadecimal digits (0–9 and A–F) and the number
: <math>14\mathrm{B}9_{\mathrm{hex}} = (1 \times 16^3) + (4 \times 16^2) + (\mathrm{B} \times 16^1) + (9 \times 16^0) \qquad (= 5305_{\mathrm{dec}}) ,</math>
where B represents the number eleven as a single symbol.

In general, in base-''b'', there are ''b'' digits <math>\{d_1,d_2,\dotsb,d_b\} =:D</math> and the number
:<math>(a_3 a_2 a_1 a_0)_b = (a_3 \times b^3) + (a_2 \times b^2) + (a_1 \times b^1) + (a_0 \times b^0) </math>
has <math>\forall k \colon a_k \in D .</math>
Note that <math>a_3 a_2 a_1 a_0</math> represents a sequence of digits, not [[multiplication]].