Editing Location arithmetic (section)

=== Abbreviated and extended form ===

As in any numeral system using [[sign-value notation]] (but ''not'' those using [[positional notation]]), digits (letters) can be repeated such that multiple numerals can represent a single number. For example:

: '''abbc''' = '''acc''' = '''ad''' = 9

Additionally, the order of digits does not matter. For example:

: '''abbc''' = '''bbca''' = '''bcba''' = ... = 9

Because each digit in a location numeral represents twice the value of its next-lower digit, replacing any two occurrences of the same digit with one of the next-higher digit does not change the numeral's numeric value. Thus, repeatedly applying the rules of replacement '''aa''' → '''b''', '''bb''' → '''c''', '''cc''' → '''d''', etc. to a location numeral removes all repeated digits from that numeral.

Napier called this process '''abbreviation''' and the resulting location numeral the '''abbreviated form''' of that numeral; he called location numerals containing repeated digits '''extended forms'''. Each number can be represented by a unique abbreviated form, not considering the order of its digits (e.g., '''abc''', '''bca''', '''cba''', etc. all represent the number 7).