Editing Location arithmetic (section)

==== 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]].