学术活动

# Converting a fraction into a decimal: a complex version

2018/10/16 09:34  点击：[]

:2018年10月19   上午9:00--10:00

:理科楼 LD202

: A complex number $z=c+di$ is called a Gaussian integer if both $c$ and $d$ are integers. The quotient $\frac{z_{1}}{z_{2}}$ of two Gaussian integers $z_{1}$ and $z_{2}(\neq 0)$ is called a Gaussian rational. It can be shown that any Gaussian rational can be rewritten as $p+qi$ for some $p,q\in\mathbb{Q}$. It is known that every $z\in\mathbb{Z}$ can be expressed as $z=\sum_{i=-k}^{\infty}a_{i}b^{-i}$ where $b=-n+i$, $n$ is a positive integer, and $a_{i}\in D=\{0,1,...,|b|-1\}$. This expression (may not be unique) is called a radix expansion or representation of $z$, the number $b$ a base or radix, the set $D$ a digit set, and the elements of $D$ digits.

I   In this talk, we focus on the simplest case when $n=1$. Then $b=-1+i$ and $D=\{0,1\}$. We may think of $(b,D)$ a complex binary system. As for real rationals, we study the condition for a radix expansion of a Gaussian rational to be terminating (respectively purely recurring and mixd recurring). We also present a method for finding such a radix representation in each case.