## Fractions are Hard!

### 4.3 A Curious Fraction Tree

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

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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