The reciprocal of a number is what you multiply it by to get exactly 1. For a fraction you simply flip it over; for decimals, whole numbers and mixed numbers you first write them as fractions. This calculator accepts all of those forms — including repeating decimals such as 0.(3) — and returns the reciprocal as an exact fraction, a mixed number and a decimal with any repeating digits marked.
How to use the reciprocal calculator
- Type a number in any of these forms:
5,0.4,3/8,2 1/2,-7/3,0.(3)for a repeating decimal, or2.5e-3. - Read the reciprocal on the tape as a fraction and a decimal, plus a mixed number when it is greater than 1.
- The steps show how the number was turned into a fraction, flipped, and checked by multiplying back to 1.
Reciprocal formula
| Number | As a fraction | Reciprocal | As a decimal |
|---|---|---|---|
| 5 | 5/1 | 1/5 | 0.2 |
| 0.4 | 2/5 | 5/2 | 2.5 |
| 3/8 | 3/8 | 8/3 | 2.666… |
| 2 1/2 | 5/2 | 2/5 | 0.4 |
| 0.125 | 1/8 | 8 | 8 |
| −7/3 | −7/3 | −3/7 | −0.428571… |
Worked example
Find the reciprocal of 3/8.
Flip: swap the numerator and denominator to get 8/3.
Mixed number: 8 ÷ 3 = 2 remainder 2, so 8/3 = 2 2/3.
Decimal: 8/3 = 2.666…, with the 6 repeating forever (written 2.6 with a bar over the 6).
Check: 3/8 × 8/3 = 24/24 = 1 ✓
For a decimal such as 0.4, write it as 4/10 = 2/5 first, then flip: 5/2 = 2.5. For a repeating decimal, 0.(3) means 0.333… = 1/3, whose reciprocal is exactly 3 — something you cannot get by flipping a rounded 0.333, which gives 3.003.
Where reciprocals are used
Dividing fractions
Dividing by a fraction means multiplying by its reciprocal: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8. This is the “keep, change, flip” rule, and it works because 2/5 × 5/2 = 1.
Converting rates
A speed of 60 miles per hour is 1/60 hour per mile, or one minute per mile. Fuel use of 25 miles per gallon is 0.04 gallons per mile. Whenever a rate is “per” something, its reciprocal flips the question around.
Slopes, resistors and lenses
Perpendicular lines have slopes that are negative reciprocals of each other (2 and −1/2). Resistors in parallel and thin-lens optics both add reciprocals: 1/R = 1/R1 + 1/R2.
Quick facts
- A number and its reciprocal always have the same sign.
- Numbers between −1 and 1 (except 0) have reciprocals larger in size than 1, and vice versa.
- Only 1 and −1 are their own reciprocals.
- The reciprocal of a reciprocal is the original number.
Related tools
For full fraction arithmetic, use the fraction calculator. The decimal to fraction calculator and fraction to decimal calculator convert either way, the mixed number to improper fraction calculator prepares mixed numbers for flipping, and the complex fractions calculator simplifies fractions divided by fractions.
Frequently asked questions
What is a reciprocal?
The reciprocal of a number x is 1/x, the number you multiply x by to get 1. It is also called the multiplicative inverse. The reciprocal of 4 is 1/4, and the reciprocal of 2/3 is 3/2.
How do I find the reciprocal of a mixed number?
Convert it to an improper fraction first, then flip it. 2 1/2 = 5/2, so its reciprocal is 2/5 = 0.4. Flipping only the fraction part, to get 2 2/1, is a common mistake.
Does zero have a reciprocal?
No. No number multiplied by 0 gives 1, and 1 ÷ 0 is undefined. Zero is the only real number without a reciprocal.
What is the reciprocal of a negative number?
It is negative too, because a negative times a negative is positive. The reciprocal of −4 is −1/4, since −4 × −1/4 = 1.
Why do reciprocals matter?
Dividing by a number is the same as multiplying by its reciprocal, which is why fraction division uses keep–change–flip. Reciprocals also convert rates, such as miles per hour into hours per mile.