Editing Location arithmetic (section)

=== 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''' = 2<sup>0</sup> + 2<sup>1</sup> + 2<sup>7</sup> + 2<sup>8</sup> = 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 [[Carry (arithmetic)|borrows]].

Write the [[Subtraction|minuend]] (the largest number you want to diminish) and remove from it all the digits appearing in the [[Subtraction|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''' = '''<s>ac</s>dg''' = '''dg''' = 8+64 = 72. 
 
* Subtract 3 = '''ab''' from 77 = '''acdg''' :
: '''acdg''' - '''ab''' = '''abbdg''' - '''ab''' = '''<s>ab</s>bdg''' = '''bdg''' = 2+8+64 = 74.

* Subtract 7 = '''abc''' from 77 = '''acdg''' :
: '''acdg''' - '''abc''' = '''abbccg''' - '''abc''' = '''<s>ab</s>b<s>c</s>cg''' = '''bcg''' = 2+4+64 = 70.

==== Doubling, halving, odd and even ====

Napier proceeded to the rest of arithmetic, that is multiplication, division and square root, on an abacus, as it was common in his times. However, since the development of micro-processor computer, a lot of applicable algorithms have been developed or revived based on doubling and halving.

Doubling is done by adding a numeral to itself, which mean doubling each of its digit. This gives an extended form, which has to be abbreviated if needed. This operation can be done in one step by changing each digit of a numeral to the next larger digit. For example, the double of '''a''' is '''b''', the double of '''b''' is '''c''', the double of '''ab''' is '''bc''', the double of '''acfg''' is '''bdgh''', etc.

Similarly, multiplying by a power of two, is just translating its digits. To multiply by '''c''' = 4, for example, is transforming the digits '''a''' → '''c''', '''b''' → '''d''', '''c''' → '''e''',...  
 
Halving is the reverse of doubling: change each digit to the next smaller digit. For example, the half of '''bdgh''' is '''acfg'''.

One sees immediately that it is only feasible when the numeral to be halved does not contain an '''a''' (or, if the numeral is extended, an odd number of '''a'''s). In other words, an abbreviated numeral is odd if it contains an '''a''' and even if it does not.

With these basic operations (doubling and halving), all the binary algorithms can be adapted starting by, but not limited to, the [[Bisection method]] and [[Dichotomic search]].

==== Multiplication ====

Napier performed multiplication and division on an abacus, as was common in his times. However, [[Egyptian multiplication and division|Egyptian multiplication]] gives an elegant way to carry out multiplication without tables using only doubling, halving and adding.

Multiplying a single-digit number by another single-digit number is a simple process. Because all letters represent a power of 2, multiplying digits is the same as adding their exponents. This can also be thought of as finding the index of one digit in the alphabet ('''a''' = 0, '''b''' = 1, ...) and incrementing the other digit by that amount in terms of the alphabet ('''b''' + 2 => '''d''').

For example, multiply 4 = '''c''' by 16 = '''e'''

'''c''' * '''e''' = 2^2 * 2^4 = 2^6 = '''g'''

or...

''AlphabetIndex''('''c''') = 2, so... '''e''' => '''f''' => '''g'''

To find the product of two multiple digit numbers, make a two column table. In the left column write the digits of the first number, one below the other. For each digit in the left column, multiply that digit and the second number and record it in the right column. Finally, add all the numbers of the right column together.

As an example, multiply 238 = '''bcdfgh''' by 13 = '''acd'''

:{|
|- 
| '''a''' || '''bcdfgh'''
|- 
| '''c''' ||  '''defhij''' 
|-
| '''d''' ||   '''efgijk''' 
|}

The result is the sum in the right column '''{{not a typo|bcdfgh defhij efgijk}}''' = '''{{not a typo|bcddeefffgghhiijjk}}''' = '''bcekl''' = 2+4+16+1024+2048 = 3094.

It is interesting to notice that the left column can also be obtained by successive halves of the first number, from which the even numbers are removed. In our example, '''acd''', '''<s>bc</s>''' (even), '''ab''', '''a'''. Noticing that the right column contains successive doubles of the second number, shows why the [[Egyptian multiplication and division|peasant multiplication]] is exact.

==== Division, remainder ====

Division can be carried out by successive subtractions: the quotient is the number of time the divisor can be subtracted from the dividend, and the remainder is what is left after all the possible subtractions.

This process, which can be very long, may be made efficient if instead of the divisor we subtract multiple of the divisor, and computations are easier if we restrict to multiple by a power of 2.

In fact, this is what we do in the [[long division]] method.