**k-leftin**:
Given k*Z*^{+},
a k-leftin is defined as the ordered set
(^{...} d_{i}, ^{...} d_{1},d_{0},
d_{-1},d_{-2}, ^{...} d_{j} ^{...})
such that d_{i}** Z**
and 0<=d

k is referred to as the *base*. Usually k-leftins are simply called *leftins*.
Leftins are generally written as numbers with a decimal point between
d_{0} and d_{-1}. The d_{i} are called *digits*.
If n such that i>n d_{i}=0, then the leftin
is equivalent to a positive real number. Note that the definition of a k-leftin can be
generalized to bases other than positive integers; the rules guiding leftins would still apply.

** L^{K}** is the set of
all k-leftins for a given base k.

** L** is

Since there are an infinite number of digits both to the left and the right of the decimal point,
leftins are often written with ellipses (...) before and after the number to indicate there are
other digits not specified. ...48204.67915... is an example. As with real numbers,
if the remaining digits to the left or right are 0's, they need not be written.
*Leftin* is short for "left infinite number."

When the digits repeat infinitely to the left or right, this can be indicated by putting the repeating digits inside parentheses. ...(387)41.69(1)... is ...38738738741.6911111... where the specified digits continue repeating infinitely.

**Right-finite leftins**:
** L^{k}_{f}** = { a

Elements of ** L^{k}_{f}** may have an infinite number of digits to the
left of the decimal point, but they only have a finite number of non-zero digits to the right.

In general, the superscript of ** L** indicates the number base and the subscript
characterizes the subset. If the base isn't indicated, it's assumed to be unimportant.
If the subset isn't specified,

**Leftints**:
** L_{0}** = { a

Elements of ** L_{0}** are called

**Rational leftins**:
** L_{r}** = {a

Thus rational leftins have a finite number of digits to the right of the decimal point
but may have an infinite number to the left. The digits to the left eventually repeat
with a period *p*. This mirrors rational numbers in decimal notation.
...(391)42.761 is an example of a rational leftin.

*L _{r}*

Referring to the number of digits to either side of the decimal point,
** L** can be called infinite.infinite,

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© Scott Sherman 1999