Addition, subtraction and multiplication of leftins can be defined in terms of modulus arithmetic of the digits analagous to addition, subtraction and multiplication of real numbers given their decimal expansion. Thus we will simply give a few examples to illustrate the principles involved.
To add ...5972286 + ...9421769, you can follow the normal right-to-left algorithm for adding numbers together.
1 111 ...5972286 +...9421769 __________ ...5394055
Notice that both these leftins have 7 significant digits and their sum is also accurate to 7 significant digits. No matter what the unspecified digits to the left are, this partial sum is completely accurate.
The fact that there can be an infinite number of "carries" for the addition is of no particular concern since every carry is used to determine the next digit. There is nothing "left over."
Subtraction produces similar results. You can perform the subtraction from right to left and borrow from the next digit to the left.
1 7 ...5972286 -...9421769 __________ ...6550517
Multiplication is a little more involved, but leads to the same general conclusion.
...5972286 x...9421769 __________ ...3750574 ...5833716 ...1806002 ...5972286 ...1944572 ...3889144 ...3750574 ... _________________ ...9093934
This leads to a general principle for determining the accuracy of adding, subtracting or multiplying two leftins:
AP1: If, for a,bLf , n such that i<=n, the digit di is specified for a and b, then the digits of a+b, a-b and a*b can be accurately specified for all di, i<=n using the normal rules for +-* for real numbers.
The results of addition, subtraction and multiplication are unique and well defined for leftins in Lf since these operations can be carried out using simple right-to-left algorithms. Standard long division does not behave this way and thus does not necessarily lead to unique well defined results. Examples will be discussed later on.
Question: Can multiplication be defined on L in general?
© Scott Sherman 1999