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any rational number is a sum of unit
fractions∗
Mathprof†
2013-03-21 14:25:19
Representation
Any rational number ab ∈ 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 ≤ ab < 1 as such a sum:
1. Let
b
n=
a
be the smallest natural number for which
2. Output
1
n
1
n
≤ ab . If a = 0, terminate.
as the next term of the sum.
3. Continue from step 1, setting
a0
a
1
= − .
0
b
b
n
Proof of correctness
Proof. The algorithm can never output the same unit fraction twice. Indeed,
1
< n2 – so the same n cannot be
any n selected in step 1 is at least 2, so n−1
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.
∗ hAnyRationalNumberIsASumOfUnitFractionsi created: h2013-03-21i by: hMathprofi
version: h33127i Privacy setting: h1i hDerivationi h11A67i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
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For a > 0: The n selected in step 1 satisfies
b ≤ an < b + a.
So
a
1
an − b
− =
,
b
n
bn
and 0 ≤ an−b < a – by the induction hypothesis, the algorithm terminates
for ab − n1 .
Problems
1. The greedy algorithm always works, but it tends to produce unnecessarily
large denominators. For instance,
47
1 1 1
= + + ,
60
3 4 5
but the greedy algorithm selects 12 , leading to the representation
47
1 1
1
= + + .
60
2 4 30
2. The representation is never unique. For instance, for any n we have the
representations
1
1
1
=
+
n
n + 1 n · (n + 1)
So given any one representation of ab 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.
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