Editing Location arithmetic (section)

==== 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.