Is that true : Every positive rational number $q$ can be written as $q = \sum_{i=0}^{k}1/n_i$ , where $n_i,k$ are positive intergers and $n_i\not=n_j$ if $i\not=j$.

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Express the fraction $\dfrac{a}{b}$ as a sum of $a$ copies of $\dfrac{1}{b}$. (For technical reasons, if $b=1$, use $2a$ copies of $\dfrac{1}{2}$.) Major flaw: the denominators are not all different.

Using repeatedly the identity $$\frac{1}{k}=\frac{1}{k+1}+\frac{1}{k(k+1)},\tag{1}$$ we can express any $\dfrac{1}{n}$ as a sum of distinct unit fractions with all denominators as large as we wish.

So leave the first $\dfrac{1}{b}$ alone. Express the second one as $\dfrac{1}{b+1}+\dfrac{1}{b(b+1}$. For the third $\dfrac{1}{b}$, use Identity $(1)$ repeatedly to express $\dfrac{1}{b}$ as a sum of distinct unit fractions with denominators all greater than $b(b+1)$. Continue.

The algorithm we have described is quite inefficient. There are much better algorithms available, going back to Fibonacci's greedy algorithm.

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An Egyptian fraction is the sum of distinct unit fractions, such as . That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The value of an expression of this type is a positive rational number a/b; for instance the Egyptian fraction above sums to 43/48. Every positive rational number can be represented by an Egyptian fraction. Sums of this type, and similar sums also including 2/3 and 3/4 as summands, were used as a serious notation for rational numbers by the ancient Egyptians, and continued to be used by other civilizations into medieval times. In modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. However, Egyptian fractions continue to be an object of study in modern number theory and recreational mathematics, as well as in modern historical studies of ancient mathematics.