Considering the Rational Numbers of the Cantor Set

Article: "The Rationals of the Cantor Set" by Ioana Mihaila in The College Mathematics Journal
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Article Summary


Inspired by the article “The Cantor Set Contains ¼? Really?”, Ioana Mihaila set abot mapping all of the rational numbers which are elements of the Cantor Set. She constructs the Cantor Set on the interval [0,1], and uses the middle-third method of construction, such that f1=[0,1/3]U[2/3,1]. The set itself is defined as C = infintersection fn. Elements of the Cantor set are traditionally represented in base three.

There are three types of rational numbers in C: terminating, purely periodic, and mixed periodic. They arise from the endpoints of successive sets and geometrical relationships with the same.

$\textbf{Terminating Numbers}$

Terminating numbers are rational numbers with finitely many 2s in the decimal expansion. They make-up the left end of the intervals not removed from the set. For example:

\begin{align} 0.02_3=\frac{2}{9} \end{align}

$\textbf{Purely Periodic}$
Purely periodic numbers have an infinite decimal expansion of 0s and 2s. In fraction form, they have equal numerators to the terminating case, but their denominator is smaller by 1, so they are just to the right of the left endpoints. For example:

\begin{align} \frac{2}{8}=\frac{1}{4} \end{align}

$\textbf{Mixed Periodic}$
Mixed periodic numbers have an infinite decimal expansion containing a pattern of 0s and 2s that begins after a finite sequence of a different pattern of 0s and 2s. They can be written “as sums of a terminating number and a purely periodic number divided by a power of 3 (253). For example:

\begin{align} \frac{4}{13}=\frac{8}{26} \end{align}

Here, we want to know whether there exist any families of the irrationals in the Cantor set. First, we present Borel’s definition of a normal number. A real number is normal in base b if its expansion in this base contains each sequence of k digits with a frequency asymptotic to $$b^{-k}$$The aurthor conjectures that for every base b, the base b expansion of any irrational algebraic number is normal, which means that no algebraic irrational numbers are in the Cantor set. All irrational numbers in the set must be transcendental. The Cantor set contains transcendental numbers because it contains some Liouville numbers, but not all Liouville numbers are transcendental. At best, we can conclude that the irrational numbers are not readily categorized and understood in the Cantor set.

Connection to Real Analysis

In real analysis, the Cantor set provides many interesting properties that can serve as counterexamples to explore. So many important properties are eloquently displayed in the set such as zero measure, uncountability, compactness, and non-integer dimension. Here, we explore the properties of the rational numbers as a subset of an uncountable set of rational and irrational numbers. Through an understanding of this arises the proof of the Cantor set's property of being uncountable.

Broader Context

The cantor set provides some insight into the world of fractals and topology at a very accessible level. The undergraduate student can comprehend the set construction and even visualize the properties of the Cantor set, which will prove invaluable later as the student expands the concept and applies it in larger dimensions.

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