Leftins

    k-leftin: Given kZ⁺, a k-leftin is defined as the ordered set (^... d_i, ^... d₁,d₀, d_-1,d_-2, ^... d_j ^...) such that d_iZ and 0<=d_i<k i from - to .

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 d₀ 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 L^K with an unspecified number base.

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 = { aL^K | n such that i<n d_i=0}

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, L_f is assumed. For example, L² is the same as L²_f .

    Leftints: L₀ = { aL | i<0 d_i=0}.

Elements of L₀ are called leftints. They are of the form ^...d_i^...d₁d₀.(0). Basically they look like integers with an infinite number of digits. An integer can be thought of as a leftint with an infinite number of 0's to the left of the other digits.

    Rational leftins: L_r = {aL | n such that i<n d_i=0 and m>=n and pZ⁺ such that j>=m, d_j = d_{m + (j-m) mod p}}

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 L_f L
L₀ L_f L
L₀ L_r

Referring to the number of digits to either side of the decimal point, L can be called infinite.infinite, L_f is infinite.finite, and L₀ is infinite.0. Most of this documentation is concerned with L_f and its subsets.

© Scott Sherman 1999