Reciprocal Calculator
The reciprocal of a number is 1 divided by that number: for a fraction, you simply flip the numerator and the denominator. Enter a fraction, a decimal, a mixed number, or a fraction of a fraction below, and the calculator returns the reciprocal as a simplified fraction, as a mixed number when it is improper, and as a decimal approximation.
Result
Quick Answer
For 3/4, the reciprocal is 4/3, which is 1 1/3 as a mixed number and 1.3333 as a decimal. The check is instant: 3/4 × 4/3 = 1, and every nonzero number times its reciprocal equals 1. The calculator above is pre-filled with 3/4, so press Calculate to reproduce these exact lines.
Key takeaways
- The reciprocal of a fraction a/b is b/a, simplified. The reciprocal of a decimal is 1 divided by that decimal, read exactly (0.2 gives exactly 5, not a 16-digit fraction).
- Zero has no reciprocal. The tool says so explicitly instead of returning an error value.
- 1 and -1 are the only numbers that are their own reciprocals. A unit fraction 1/n flips to the whole number n.
- The sign never changes: the reciprocal of a negative number is negative.
How to use the reciprocal calculator
Pick the input type that matches your number, type the number, and click Calculate. The results stay hidden until you do, and the tool is pre-filled with 3/4 so you can test it immediately.
- Choose the input type. Fraction (a/b) takes entries like 3/4. Decimal takes entries like 2.5. Mixed number takes entries like 2 3/4, with a space between the whole number and the fraction. The "Fraction of a fraction" option flips a single fraction exactly like Fraction mode does; genuinely nested input such as 1/2/3 is rejected with "Could not parse that input", so type a single fraction instead.
- Enter your number. A leading minus sign is allowed, as in -2/5. Decimals are read exactly, so 0.75 is treated as 3/4 and returns 4/3. If you need the exact fraction form of a decimal for other work, the decimal to fraction calculator converts any decimal to its reduced fraction.
- Click Calculate (or press Enter). Five lines appear: the main reciprocal, the exact statement of the flip, the mixed-number form, the decimal approximation rounded to 4 places, and a note for special cases such as 1, -1, or unit fractions.
- Read the special-case notes. Entering 1 gives "1 is its own reciprocal: 1 x 1 = 1." Entering a whole number like 7 gives "The reciprocal of the whole number 7 is the unit fraction 1/7."
- Copy or share. "Copy my results" copies all five lines as plain text; "Share results" opens your device's share sheet, or copies the text if sharing is unavailable.
Entering zero, or a fraction with a zero denominator, never produces a number: you get the plain message "Zero has no reciprocal." or "Denominator cannot be zero." A number and its reciprocal multiply to 1, and as the OpenStax Prealgebra section on reciprocals puts it, there is no number r with 0 × r = 1, which is why zero is the one input with no answer.
The reciprocal formula
The reciprocal of any nonzero number x is 1 divided by x. For a fraction, that means swapping the top and the bottom.
Reciprocal of a/b = b/a (a ≠ 0, b ≠ 0)
Mixed number: convert to an improper fraction first, then flip (2 3/4 = 11/4, reciprocal = 4/11)
Sign rule: the reciprocal keeps the sign (reciprocal of -2/5 is -5/2)
The sign rule is worth stating because beginners often flip the sign along with the fraction. OpenStax states it directly: reciprocals must have the same sign, so a negative number's reciprocal stays negative. The calculator enforces this by construction, and if signed fractions are your daily work, the negative fraction calculator handles the rest of the arithmetic. Only two numbers equal their own reciprocals, 1 and -1, a fact the Ohio State Ximera precalculus notes prove from the definition; the tool calls both out in its note line.
Worked example: flipping 3/4
The pre-filled run is 3/4 in Fraction mode. Here is every line the calculator returns, verified against the tool, plus a mixed-number run and a decimal run that shows the exact-decimal handling.
| Run | Reciprocal | Mixed-number form | Decimal |
|---|---|---|---|
| 3/4 (Fraction) | 4/3 | 1 1/3 | 1.3333 |
| 2 3/4 (Mixed) | 4/11 | 4/11 | 0.3636 |
| 0.2 (Decimal) | 5 | 5 | 5 |
| -2/5 (Fraction) | -5/2 | -2 1/2 | -2.5 |
The 3/4 run also prints the line "Exact: reciprocal of 3/4 = 1 1/3", which states the whole flip in one sentence, and the check "3/4 × 4/3 = 12/12 = 1" closes the loop: a number and its reciprocal have a product of 1, as every textbook defines it.
Common reciprocals table
These are the flips that come up constantly in homework, cooking, and DIY math. Every row below was produced by the calculator itself, so the mixed-number forms and 4-place decimals match the tool exactly.
| Number | Reciprocal | Decimal |
|---|---|---|
| 1/2 | 2 | 2 |
| 1/3 | 3 | 3 |
| 1/4 | 4 | 4 |
| 1/5 | 5 | 5 |
| 1/6 | 6 | 6 |
| 1/8 | 8 | 8 |
| 1/10 | 10 | 10 |
| 2/3 | 1 1/2 | 1.5 |
| 3/4 | 1 1/3 | 1.3333 |
| 2/5 | 2 1/2 | 2.5 |
| 3/5 | 1 2/3 | 1.6667 |
| 4/5 | 1 1/4 | 1.25 |
| 5/6 | 1 1/5 | 1.2 |
| 7/8 | 1 1/7 | 1.1429 |
| 2 | 1/2 | 0.5 |
| 3 | 1/3 | 0.3333 |
| 0.5 | 2 | 2 |
| 0.25 | 4 | 4 |
| 0.2 | 5 | 5 |
| 0.125 | 8 | 8 |
| 1.5 | 2/3 | 0.6667 |
| 0.75 | 4/3 | 1.3333 |
The pattern to notice: unit fractions flip to whole numbers, and any proper fraction between 0 and 1 flips to something bigger than 1. The Ximera notes state the general version: a fraction with a small value has a reciprocal with a large value, and vice versa. To express any of these results as a percent instead, the fraction to percent calculator converts 4/3 to 133.33% in one step.
Where reciprocals actually get used
Most people meet the reciprocal when they divide fractions: dividing by a fraction is the same as multiplying by its reciprocal. That one move turns 6 ÷ (2/3) into 6 × 3/2 = 9, and the OpenStax summary of fraction operations lists it as the definition of fraction division.
- Dividing fractions: (3/4) ÷ (2/5) = (3/4) × (5/2) = 15/8. Once the divisor is flipped, the rest is plain multiplication.
- Solving equations: for (3/4)x = 24, multiply both sides by the reciprocal 4/3 to isolate x, the standard strategy in the OpenStax lesson on equations with fraction coefficients.
- Unit rates and conversions: if a machine makes 2/3 of a widget per minute, its rate per widget is the reciprocal, 3/2 minutes. Rates and their reciprocals always come in pairs like this.
- Checking your work: multiply the number by its reciprocal. If the product is not exactly 1, the flip was wrong. The calculator's "Exact" line prints this pairing for every run.
Once the flip is done, the remaining arithmetic is ordinary: the fraction addition calculator handles adding and subtracting fractions, including mixed numbers.
What competing calculators miss
The top-ranking reciprocal tools each stop one step short of a complete answer, and the gap is consistent across all of them:
- Omni Calculator's reciprocal tool: clean fraction and decimal modes, simplified output, and an explicit zero warning. But there is no mixed-number input and no mixed-number output line: 7/3 comes back as a bare fraction with no 2 1/3 form.
- Calculator.now: decimal, fraction and scientific-notation outputs with optional step-by-step working. Again no mixed-number input, and nothing like a common-reciprocals lookup.
- Calculators.sg: accepts decimal, fraction and mixed-number input, with decimal places configurable from 0 to 15 and a verification step. The reciprocal itself, though, is shown as a decimal and a fraction only: no dedicated mixed-number result line.
- OnlyCalculators' multiplicative inverse tool: the strongest of the set, with step-by-step working, a verification multiply, and even a modular-inverse mode. Its reciprocal output is fraction and decimal only, and it has no quick-reference table.
- HackMath: frames the reciprocal inside fraction division (1: 3/4) and spells the result out in words, which is charming, but there is no decimal approximation and no mixed-number display.
- The gap this page fills: mixed-number input and a dedicated mixed-number result line (the 7/3 → 2 1/3 case every rival skips), decimals read exactly so 0.2 returns 5, a common-reciprocals table for the values people look up repeatedly, and special-case notes for 1, -1, and unit fractions. Every number in the table above was produced by the tool itself.
Frequently asked questions
The reciprocal of 7 is 1/7, about 0.1429. In general, the reciprocal of any whole number n is the unit fraction 1/n, and the calculator prints a note for this case: The reciprocal of the whole number 7 is the unit fraction 1/7.
No. The reciprocal of 0 would be 1/0, which is undefined. The calculator says "Zero has no reciprocal." instead of returning a value, because no number r satisfies 0 x r = 1.
The original number. Flipping a fraction twice restores it: the reciprocal of 4/3 is 3/4, and the calculator's exact line shows both directions.
Convert the mixed number to an improper fraction, then flip. For 2 3/4, the improper form is 11/4, so the reciprocal is 4/11, about 0.3636. The calculator takes the mixed number directly and shows the improper flip, the mixed-number form, and the decimal.
Both options flip a single fraction the same way, so 3/4 gives 4/3 in either mode. Truly nested input such as 1/2/3 is rejected with the message "Could not parse that input." If you see it, type a single fraction instead.
Yes. The reciprocal of any positive number smaller than 1 is larger than 1: the reciprocal of 1/4 is 4. In general, small fractions have large reciprocals and large fractions have small ones.
The calculator reads decimals exactly, treating 0.2 as 2/10, which reduces to 1/5, so the reciprocal is exactly 5. This matches the decimal approximation line, which also shows 5.
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References
- OpenStax. Prealgebra, Section 4.2: Multiply and Divide Fractions (reciprocal definition, sign rule, zero has no reciprocal). Accessed October 23, 2026.
- OpenStax. Prealgebra, Section 4.5: Add and Subtract Fractions with Different Denominators (fraction division as multiplication by the reciprocal). Accessed October 23, 2026.
- OpenStax. Prealgebra, Section 4.7: Solve Equations with Fractions (solving fraction-coefficient equations by multiplying by the reciprocal). Accessed October 23, 2026.
- UTSA Pressbooks. NCB 0542 Corequisite Mathematics: Fractions (product-of-1 rule, sign rule, reciprocal rules summary). Accessed October 23, 2026.
- The Ohio State University, Ximera. Reciprocals (only 1 and -1 are their own reciprocals; small fractions have large reciprocals). Accessed October 23, 2026.