### Addition, Subtraction and Multiplication of Leftins

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.

#### Addition

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

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

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,b*L*_{f} ,
n such that i<=n, the digit d_{i}
is specified for a and b, then the digits of a+b, a-b and a*b can be accurately
specified for all d_{i}, i<=n using the normal rules for +-* for real
numbers.

### Division

The results of addition, subtraction and multiplication are unique and well defined
for leftins in *L*_{f} 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?

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© Scott Sherman 1999