# Concerning: presentations of rational numbers into sums

Problem: Prove that all positive rational numbers can be expressed as the finite sum of different numbers $\displaystyle \frac {1} {n}$ ($n$ is a natural number).

Example: $\displaystyle \frac {19}{16}=1+ \frac {1}{8} + \frac {1}{16}.$

*We cant sum numbers as $\displaystyle \frac {3}{16}$ (denominator > 1) but we can sum $\displaystyle \frac {1}{8}+ \frac {1}{16}.$

Any solutions? Suggestions?

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– J. M. May 15 '11 at 14:32
You might want to say positive rational numbers since negative rationals obviously cannot be formed with numerator 1 and natural-number denominators. (The Putnam problem cited by Chandru1 specifies that the numbers be positive.) – Fixee May 15 '11 at 15:17
@Fixee, natural numbers are positive. – quanta May 15 '11 at 15:18
@quanta: Exactly. Which is why it's unlikely you can form a negative rational number via a sum of fractions with 1 over a natural number. – Fixee May 15 '11 at 15:56

• Take a look at this article as well: J.C.Owings, American Mathematical Monthly Vol. 75 (1968), Pages $777-778$.