What is a reciprocal?
The reciprocal of a number is 1 divided by that number. It's also called the multiplicative inverse, because multiplying a number by its reciprocal always gives 1 — that product is the whole point of the idea.
Reciprocal of n = 1 ÷ nn × (1 ÷ n) = 1 For a fraction there's a shortcut: swap the numerator and the denominator. The reciprocal of 2/3 is 3/2, because flipping a fraction is what dividing 1 by it amounts to. Whole numbers work the same way once you write 5 as 5/1, which flips to 1/5. If you want to perform other operations with fractions, use the Fraction Calculator.
Reciprocal, opposite and opposite reciprocal
These three get mixed up constantly, and they're genuinely different operations. One flips, one changes sign, and one does both:
- Reciprocal — flip a fraction, or divide 1 by the number.
- Opposite — change the sign of the number. Also called the additive inverse.
- Opposite reciprocal — do both: flip it, then change the sign.
| Operation | What it does | Result |
|---|---|---|
| Reciprocal of 3/4 | Flip it | 4/3 |
| Opposite of 3/4 | Change the sign | −3/4 |
| Opposite reciprocal of 3/4 | Flip it and change the sign | −4/3 |
A quick way to tell which one you've been asked for: the reciprocal multiplies with the original to give 1, the opposite adds with it to give 0, and the opposite reciprocal multiplies with it to give −1.
Why "opposite reciprocal" comes up in geometry
Two lines with nonzero slopes are perpendicular when their slopes are opposite reciprocals of each other. A line with slope 3/4 is perpendicular to one with slope −4/3, and the test is that the two slopes multiply to −1.
How to find a reciprocal
- Integer: put it under 1. The reciprocal of 5 is 1/5, and of 12 is 1/12.
- Fraction: swap top and bottom. The reciprocal of 2/3 is 3/2.
- Mixed number: convert to an improper fraction first. 1 1/2 becomes 3/2, which flips to 2/3.
- Decimal: write it as a fraction, then flip. 0.25 is 1/4, so its reciprocal is 4. For converting decimal values to fractions directly, use the Decimal to Fraction converter.
- Negative number: the reciprocal keeps the sign. The reciprocal of −4 is −1/4.
When a fraction can be reduced, simplify it before or after taking the reciprocal. For example, 6/8 reduces to 3/4, whose reciprocal is 4/3. You can use the Simplify Fractions calculator to reduce a fraction to lowest terms.
Examples
| Number | Reciprocal | Check |
|---|---|---|
| 5 | 1/5 | 5 × 1/5 = 1 |
| 2/3 | 3/2 | 2/3 × 3/2 = 1 |
| 3/4 | 4/3 | 3/4 × 4/3 = 1 |
| −4 | −1/4 | −4 × −1/4 = 1 |
| 1 1/2 | 2/3 | 3/2 × 2/3 = 1 |
| 0.25 | 4 | 1/4 × 4 = 1 |
| 1 | 1 | 1 × 1 = 1 |
| −1 | −1 | −1 × −1 = 1 |
| 0 | Does not exist | — |
Why zero has no reciprocal
The reciprocal of 0 would have to be 1 ÷ 0, and dividing by zero is undefined. Put the other way round: the reciprocal of a number is whatever you multiply it by to get 1, and nothing multiplied by 0 gives 1 — every product with 0 is 0. So zero is the one number with no reciprocal at all.
Where reciprocals are used
- Dividing fractions: dividing by a fraction is the same as multiplying by its reciprocal. That's the whole basis of "keep, change, flip".
- Perpendicular slopes: two lines are perpendicular when their slopes are opposite reciprocals.
- Rates: if a machine takes 4 hours per job, it completes 1/4 of a job per hour. Rates and their inverses are reciprocals.
- Solving equations: multiplying both sides by a reciprocal cancels a fractional coefficient.
Frequently asked questions
What is the difference between a reciprocal and an opposite?
A reciprocal flips the number — 3/4 becomes 4/3 — and multiplying the two gives 1. An opposite changes the sign — 3/4 becomes −3/4 — and adding the two gives 0. They're different operations with different results, which is why the phrase "opposite reciprocal" means doing both: 3/4 becomes −4/3.
What is the opposite reciprocal of 3/4?
−4/3. Flip 3/4 to get 4/3, then change the sign. A useful check: a number times its opposite reciprocal gives −1, and 3/4 × −4/3 = −1. This is also the rule for perpendicular slopes.
Does a negative number have a reciprocal?
Yes, and it keeps the same sign. The reciprocal of −4 is −1/4, and multiplying them gives 1 because a negative times a negative is positive. Changing the sign is the opposite, not the reciprocal.
What is the reciprocal of 1?
1, since 1 ÷ 1 = 1. It's one of only two numbers that are their own reciprocal — the other is −1. Every other number has a reciprocal different from itself.
Is a reciprocal the same as a multiplicative inverse?
Yes, two names for the same thing. "Multiplicative inverse" is the formal term and explains the definition: the number you multiply by to get the multiplicative identity, which is 1. "Additive inverse" is the matching formal name for the opposite, where the target is 0.
How do I find the reciprocal of a mixed number?
Convert it to an improper fraction first, then flip. For 1 1/2: multiply the whole number by the denominator and add the numerator to get 3/2, which flips to 2/3. Flipping a mixed number without converting it gives the wrong answer.