How this calculator works
A unit fraction has numerator 1. Greedy repeatedly subtracts the largest unit fraction no greater than the remaining fraction. Pairing combines two equal unit fractions; splitting replaces a repeated unit fraction by two smaller ones until denominators are distinct. Exact integer arithmetic preserves equality. Explicit size limits keep difficult decompositions responsive.
Formula
Greedy chooses k = ceil(d/n) and replaces n/d by (nk-d)/(dk). Splitting uses 1/k = 1/(k+1) + 1/[k(k+1)]. For odd k, pairing uses 2/k = 1/[(k+1)/2] + 1/[k(k+1)/2].
Worked example
Greedy gives 4/13 = 1/4 + 1/18 + 1/468. For 2/3, pairing gives 1/2 + 1/6; multiplying through by 6 checks 4 = 3 + 1.
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Frequently asked questions
Is the shortest decomposition guaranteed?
No. Greedy, pairing and splitting construct valid representations; none is promised to minimize the number of terms.
Why can a method stop with an error?
Some fractions produce rapidly growing denominators or many steps. Try greedy or a smaller fraction when a stated limit is reached.
Can I enter an improper fraction?
This tool focuses on positive proper fractions. Separate the whole part first, then decompose the remaining proper fraction.