Introduction

A Laurent Series is a generalization of a power series and is given by

$$\displaystyle \sum _ { n = - \infty } ^ { \infty } a _ { n } z ^ { n } = \cdots + a _ { -1 } z ^ { - 1 } + a _ { 0 } z ^ { 0 } + a _ { 1 } z ^ { 1 } + \cdots$$

It turns out that our positional number system can be written as a Laurent Series where $z$ is the base and $a_{n}$ are the digits.

Connection between Decimals and Laurent Series

If we let $z=10$, and $a$ be some real number in base 10, then $a$ can be written as the Laurent Series,

$$a = \sum _ { n = - \infty } ^ { \infty } a _ { n } 10 ^ { n }$$

where $a_{n}$ are the digits of $a$.

Example 1

If $a = 24$, then we can write:

$$24 = \sum _ { n = - \infty } ^ { \infty } a _ { n } 10 ^ { n }$$

where

$$ a_{i}=\left{\begin{array}{ll}{4,} & {i=0} \ {2,} & {i=1} \ {0,} & {\text {otherwise}}\end{array}\right. $$

That is,

$$ \begin{align*} 24 &= \cdots 00024.000 \cdots \
&=\sum _ { n = - \infty } ^ { \infty } a _ { n } 10 ^ { n }
\
&= \cdots + a _ { -1 } 10 ^ { - 1 } + a _ { 0 } 10 ^ { 0 } + a _ { 1 } 10 ^ { 1 } + \cdots \
&= \cdots + 0 \cdot 10 ^ { - 1 } + 4 \cdot 10 ^ {0} + 2 \cdot 10 ^ { 1 } + 0 \cdot 10^{2} + \cdots \
&= \cdots + 0 + 4 + 20 + 0 \cdots \
&= 24 \end{align*} $$