Enter a whole-number denominator n from 1 to 1,000,000. The calculator returns two denominators, a = n+1 and b = n(n+1). Read the result as 1/n = 1/a + 1/b. For distinct unit fractions, use n ≥ 2.
Use the common denominator n(n+1). The first fraction becomes n/[n(n+1)], and the second is 1/[n(n+1)]. Adding them gives (n+1)/[n(n+1)], which simplifies to 1/n.
For example, enter n = 5. The results are a = 6 and b = 30, so 1/5 = 1/6 + 1/30. In thirtieths, this is 6/30 = 5/30 + 1/30.
| n | First denominator | Second denominator | Identity |
|---|---|---|---|
| 1 | 2 | 2 | 1 = 1/2 + 1/2 |
| 2 | 3 | 6 | 1/2 = 1/3 + 1/6 |
| 3 | 4 | 12 | 1/3 = 1/4 + 1/12 |
| 5 | 6 | 30 | 1/5 = 1/6 + 1/30 |
| 12 | 13 | 156 | 1/12 = 1/13 + 1/156 |
Use positive integers only. Fractional inputs do not give the intended integer-denominator unit fractions; outputs are displayed with zero decimal places for whole-number inputs. Zero and negative denominators are outside this tool’s scope. An unavailable result is not zero.
The maximum input keeps both calculated denominators within the exact integer range of ordinary numeric arithmetic. At n = 1,000,000, the denominators are 1,000,001 and 1,000,001,000,000.
This calculator performs one specific split of 1/n. It does not decompose an arbitrary fraction or search for all possible decompositions.
Why is n = 1 a special case?
The identity gives 1 = 1/2 + 1/2. It is correct, but the two unit fractions repeat. Egyptian fraction representations conventionally use distinct positive unit fractions, so use n ≥ 2 for a distinct two-term split.
Can I split one of the resulting fractions again?
Yes. Apply the same identity to its denominator. For example, splitting 1/6 in 1/5 = 1/6 + 1/30 gives 1/5 = 1/7 + 1/42 + 1/30. When building longer representations, check that no denominator repeats.
