any rational number is a sum of unit fractions
Representation
Any rational number^{} $\frac{a}{b}\in \mathbb{Q}$ between 0 and 1 can be represented as a sum of different unit fractions^{}. This result was known to the Egyptians, whose way for representing rational numbers was as a sum of different unit fractions.
The following greedy algorithm can represent any $$ as such a sum:

1.
Let
$$n=\lceil \frac{b}{a}\rceil $$ be the smallest natural number^{} for which $\frac{1}{n}\le \frac{a}{b}$. If $a=0$, terminate.

2.
Output $\frac{1}{n}$ as the next term of the sum.

3.
Continue from step 1, setting
$$\frac{{a}^{\prime}}{{b}^{\prime}}=\frac{a}{b}\frac{1}{n}.$$
Proof of correctness
Proof.
The algorithm can never output the same unit fraction twice. Indeed, any $n$ selected in step 1 is at least 2, so $$ – so the same $n$ cannot be selected twice by the algorithm, as then $n1$ could have been selected instead of $n$.
It remains to prove that the algorithm terminates. We do this by induction^{} on $a$.
 For $a=0$:

The algorithm terminates immediately.
 For $a>0$:

The $n$ selected in step 1 satisfies
$$ So
$$\frac{a}{b}\frac{1}{n}=\frac{anb}{bn},$$ and $$ – by the induction hypothesis, the algorithm terminates for $\frac{a}{b}\frac{1}{n}$.
∎
Problems

1.
The greedy algorithm always works, but it tends to produce unnecessarily large denominators. For instance,
$$\frac{47}{60}=\frac{1}{3}+\frac{1}{4}+\frac{1}{5},$$ but the greedy algorithm selects $\frac{1}{2}$, leading to the representation
$$\frac{47}{60}=\frac{1}{2}+\frac{1}{4}+\frac{1}{30}.$$ 
2.
The representation is never unique. For instance, for any $n$ we have the representations
$$\frac{1}{n}=\frac{1}{n+1}+\frac{1}{n\cdot (n+1)}$$ So given any one representation of $\frac{a}{b}$ as a sum of different unit fractions we can take the largest denominator appearing $n$ and replace it with two (larger) denominators. Continuing the process indefinitely, we see infinitely many such representations, always.
Title  any rational number is a sum of unit fractions 

Canonical name  AnyRationalNumberIsASumOfUnitFractions 
Date of creation  20130322 12:48:31 
Last modified on  20130322 12:48:31 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  7 
Author  Mathprof (13753) 
Entry type  Derivation 
Classification  msc 11A67 
Synonym  Egyptian algorithm 