Editing Positional notation (section)

=== Exponentiation ===
Positional numeral systems work using [[exponentiation]] of the base. A digit's value is the digit multiplied by the value of its place. Place values are the number of the base raised to the ''n''th power, where ''n'' is the number of other digits between a given digit and the [[radix point]]. If a given digit is on the left hand side of the radix point (i.e. its value is an [[integer]]) then ''n'' is positive or zero; if the digit is on the right hand side of the radix point (i.e., its value is fractional) then ''n'' is negative.

As an example of usage, the number 465 in its respective base ''b'' (which must be at least base 7 because the highest digit in it is 6) is equal to:
: <math>4\times b^2 + 6\times b^1 + 5\times b^0</math>

If the number 465 was in base-10, then it would equal:
: <math>465_{10} = 4\times 10^2 + 6\times 10^1 + 5\times 10^0 = 4\times 100 + 6\times 10 + 5\times 1 = 465_{10}</math>

If however, the number were in base 7, then it would equal:
: <math>465_{7} = 4\times 7^2 + 6\times 7^1 + 5\times 7^0 = 4\times 49 + 6\times 7 + 5\times 1 = 243_{10}</math>

10<sub>''b''</sub> = ''b'' for any base ''b'', since 10<sub>''b''</sub> = 1×''b''<sup>1</sup> + 0×''b''<sup>0</sup>. For example, 10<sub>2</sub> = 2; 10<sub>3</sub> = 3; 10<sub>16</sub> = 16<sub>10</sub>. Note that the last "16" is indicated to be in base 10. The base makes no difference for one-digit numerals.

This concept can be demonstrated using a diagram. One object represents one unit. When the number of objects is equal to or greater than the base ''b'', then a group of objects is created with ''b'' objects.  When the number of these groups exceeds ''b'', then a group of these groups of objects is created with ''b'' groups of ''b'' objects; and so on. Thus the same number in different bases will have different values:

 241 in base 5:
    2 groups of 5<sup>2</sup> (25)           4 groups of 5          1 group of 1
    ooooo    ooooo
    ooooo    ooooo                ooooo   ooooo
    ooooo    ooooo         +                         +         o
    ooooo    ooooo                ooooo   ooooo
    ooooo    ooooo

 241 in base 8:
    2 groups of 8<sup>2</sup> (64)          4 groups of 8          1 group of 1
  oooooooo  oooooooo
  oooooooo  oooooooo
  oooooooo  oooooooo         oooooooo   oooooooo
  oooooooo  oooooooo    +                            +        o
  oooooooo  oooooooo
  oooooooo  oooooooo         oooooooo   oooooooo
  oooooooo  oooooooo
  oooooooo  oooooooo

The notation can be further augmented by allowing a leading minus sign. This allows the representation of negative numbers. For a given base, every representation corresponds to exactly one [[real number]] and every real number has at least one representation. The representations of rational numbers are those representations that are finite, use the bar notation, or end with an infinitely repeating cycle of digits.