= 8+64 = 72. Subtract 3 = ab from 77 = acdg : acdg - ab = abbdg - ab = abbdg = bdg = 2+8+64 = 74. Subtract 7 = abc from 77 = acdg : acdg - abc = abbccg - abc = abbccg = 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, forexample, 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 as). 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), we can adapt all the binary algorithms starting by, but not limited to, the Bisection method and Dichotomic search. Multiplication Napier proceeded to multiplication and division on an abacus, as it was common in his times. However, the Egyptian multiplication gives an elegant way to carry 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