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). Arithmetic Addition Location numerals allow for a simple and intuitive algorithm for addition: join the numerals end-to-end when necessary, rearrange this conjoined numeral's digits so they are in ascending order abbreviate this rearranged and conjoined numeral For example, to add 157 = acdeh and 230 = bcfgh, join the numerals end-to-end: acdeh + bcfgh → acdehbcfgh rearrange the digits of the previous result (because the digits of acdehbcfgh are not in ascending order): acdehbcfgh → abccdefghh and abbreviate the previous result: abccdefghh → abddefghh → abeefghh → abffghh → abgghh → abhhh → abhi The final result, abhi, equals 387 (abhi = 20 + 21 + 27 + 28 = 1 + 2 + 128 + 256 = 387); this is the same result achieved by adding 157 and 230 in decimal notation. Subtraction Subtraction is also intuitive, but may require expanding abbreviated forms to extended forms to perform borrows. Write the minuend (the largest number you want to diminish) and remove from it all the digits appearing in the subtrahend (the smallest number). In case the digit to be removed does not appear in the minuend, then borrow it by expanding the unit just larger. Repeat until all the digit of the subtrahend have been removed. A few examples show it is simpler than it sounds : Subtract 5 = ac from 77 = acdg : acdg - ac = acdg = dg