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[parent] any rational number is a sum of unit fractions (Derivation)

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 $ 0\le \frac{a}{b} <1$ as such a sum:

  1. Let
    $\displaystyle n = \left \lceil \frac{b}{a} \right \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
    $\displaystyle \frac{a'}{b'} = \frac{a}{b}-\frac{1}{n}.$

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 $ \frac{1}{n-1} < \frac{2}{n}$ - so the same $ n$ cannot be selected twice by the algorithm, as then $ n-1$ 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
$\displaystyle b \le an < b+a.$
So
$\displaystyle \frac{a}{b}-\frac{1}{n} = \frac{an-b}{bn}, $
and $ 0 \le an-b < a$ - by the induction hypothesis, the algorithm terminates for $ \frac{a}{b}-\frac{1}{n}$.
$ \qedsymbol$

Problems

  1. The greedy algorithm always works, but it tends to produce unnecessarily large denominators. For instance,
    $\displaystyle \frac{47}{60}=\frac{1}{3}+\frac{1}{4}+\frac{1}{5},$
    but the greedy algorithm selects $ \frac{1}{2}$, leading to the representation
    $\displaystyle \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
    $\displaystyle \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.



"any rational number is a sum of unit fractions" is owned by Mathprof. [ owner history (1) ]
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Other names:  Egyptian algorithm
Keywords:  greedy algorithm

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Attachments:
conjecture on fractions with odd denominators (Conjecture) by drini
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Cross-references: representation, denominators, induction hypothesis, induction, term, natural number, represent, algorithm, rational numbers, unit fractions, sum, rational number
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This is version 4 of any rational number is a sum of unit fractions, born on 2002-06-23, modified 2004-02-16.
Object id is 3127, canonical name is AnyRationalNumberIsASumOfUnitFractions.
Accessed 6271 times total.

Classification:
AMS MSC: 11A67 (Number theory :: Elementary number theory :: Other representations)
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