Advanced Unit Fraction Calculator

Calculator

Mode

n (for 1/n)

a (numerator)

b (denominator)

Operation

a (denominator for 1/a)

b (denominator for 1/b)

Max terms (Egyptian)

Decimal x

Max denominator

Show steps Try unit-fraction decomposition where possible

Reset

Tip: Use Egyptian decomposition for a proper fraction where |a| < b.

Example data table

Example	Mode	Input	Output (reduced)	Notes
1	Evaluate 1/n	1/8	1/8	Decimal = 0.125
2	Check & simplify	2/6	1/3	Becomes a unit fraction after reduction.
3	Operate	1/4 + 1/6	5/12	Not a unit fraction; can be decomposed.
4	Egyptian decomposition	5/12	5/12	5/12 = 1/3 + 1/12
5	Decimal → closest 1/n	0.18	1/6	Closest unit fraction within max denominator.

Formulas used

Unit fraction: a fraction with numerator 1, written as 1/n.
Reduction: a/b becomes (a/g)/(b/g) where g = gcd(a, b).
Add: 1/a + 1/b = (a + b)/(ab).
Subtract: 1/a − 1/b = (b − a)/(ab).
Multiply: 1/a × 1/b = 1/(ab).
Divide: (1/a) ÷ (1/b) = b/a.
Greedy Egyptian method: for proper a/b, choose x = ceil(b/a), subtract 1/x, and repeat.
Decimal match: pick n minimizing |1/n − x| for n ≤ max.

How to use this calculator

Select a mode from the dropdown.
Enter the required numbers or decimal.
Enable steps or decomposition if needed.
Press Submit to show results above the form.
Use the download buttons to export CSV or PDF.

FAQs

1) What is a unit fraction?

A unit fraction has numerator 1 and a positive integer denominator, like 1/7. It represents one equal part of a whole divided into n parts.

2) Why do results get reduced?

Reducing removes common factors using gcd(a, b). It gives an equivalent fraction in lowest terms, which makes checking “unit fraction” status accurate and keeps numbers smaller.

3) Can every fraction become a single unit fraction?

No. Only fractions equal to 1/n are unit fractions. However, many proper fractions can be written as a sum of unit fractions using an Egyptian decomposition.

4) What is Egyptian fraction decomposition?

It expresses a proper fraction as a sum of distinct unit fractions. This tool uses a greedy method that repeatedly subtracts the largest possible unit fraction until the remainder is zero.

5) Why does subtraction sometimes yield a negative fraction?

If 1/a is smaller than 1/b, then 1/a − 1/b is negative. The calculator still simplifies the result and reports the sign clearly.

6) How is the closest unit fraction to a decimal found?

The calculator searches denominators up to your limit and chooses n that minimizes the absolute error |1/n − x|. Higher limits can improve accuracy but may take longer.

7) What happens if I enter an improper fraction for decomposition?

Egyptian decomposition here expects a proper fraction where |a| < b. For improper fractions, reduce first and convert to a mixed number, then decompose the proper remainder.

8) Do CSV and PDF downloads include my exact inputs?

Yes. After you run a calculation, the tool stores the latest inputs and outputs for export. If you refresh before calculating, there is nothing to download.