Fractions are Hard!

4.3 A Curious Fraction Tree

Here is something fun to think about.  Consider the following “fraction tree:”

 

f

Do you see how it works?  Each fraction has two “children.” The left child is always a number smaller than 1 and the right child is always a number larger than 1 and the box in the upper right shows how to construct the two children from a given parent.

 

a) Continue drawing the fraction tree for another two rows.

 

b) Explain why the fraction \(\dfrac{13}{20}\) will eventually appear in the tree. (It might be easier to figure out what the parent of \(\dfrac{13}{20}\) must be by first noticing that is a left child. Next, what is its grandparent? Its great grandparent?)

 

c) Explain why the fraction \(\dfrac{13}{20}\) cannot appear twice in the tree.

 

d) Will the fraction \(\dfrac{456}{777}\) eventually appear in the tree? Could it appear twice?

 

In general: Explain why each and every fraction of the form \(\dfrac{a}{b}\) with \(a\) and \(b\) positive integers sharing only \(1\) as a common factor is sure to appear precisely one time in the tree.

 

COMMENT: To learn more about this tree and the astounding results young high-school students discovered about it, have a look at Chapter 21 of Mathematics Galore! The First Five Years of the St. Mark’s Institute of Mathematics published by the Mathematical Association of America, 2012.

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