1. ## Quickie #10

Prove that any positive rational number can be expressed as a finite sum

. . of distinct terms of the harmonic series: . $1,\:\frac{1}{2},\:\frac{1}{3},\:\frac{1}{4},\:\hdo ts,\:\frac{1}{n}$

2. I created my own Riddle.

I was going to take the following approach....but then I stopped....why?

We will show that $\mathbb{Q}$ is a vector space over $\mathbb{Z}$ spanned by the infinite dimensional set $S=\{1,1/2,1/3,...\}$. But then I stopped there. Why?

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I am not going to answer your riddle because I already knew that, let someone else try.

3. No one took up the challenge?

This is an oldie from the study of "Eqyptian fractions".

There is no hide/show feature at this site, so here's the solution.

The rational number $\frac{a}{b}$ can be written: . $\underbrace{\frac{1}{b} + \frac{1}{b} + \frac{1}{b} + \cdots + \frac{1}{b}}_{a\text{ terms}}$

We eliminate the duplications by repeatedly using the identity:
. . $\frac{1}{n} \:=\:\frac{1}{n+1} + \frac{1}{n(n+1)}$ .until all denominators are distinct.

Example: $\frac{3}{7}$

$\frac{3}{7} \;=\;\frac{1}{7}\quad\; +\quad \frac{1}{7}\qquad\quad\; +\qquad\quad\; \frac{1}{7}$

. . $= \;\frac{1}{7} + \overbrace{\left(\frac{1}{8} + \frac{1}{56}\right)} +\quad \overbrace{\left(\frac{1}{8}\qquad +\qquad \frac{1}{56}\right)}$

. . $= \;\frac{1}{7} + \left(\frac{1}{8} + \frac{1}{56}\right) + \overbrace{\left(\frac{1}{9} + \frac{1}{72}\right)} + \overbrace{\left(\frac{1}{57} + \frac{1}{3192}\right)}$

Therefore: . $\frac{3}{7}\;=\;\frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{56} + \frac{1}{57} + \frac{1}{72} + \frac{1}{3192}$

4. Yes, it is called the Greedy algorithm.

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Anybody want to try my riddle above?