The reciprocal of a fraction is found by swapping the numerator and the denominator, so the reciprocal of 5/8 is 8/5. It is also called the multiplicative inverse, and it is the number that gives 1 when multiplied by the fraction. With it, dividing becomes multiplying, as in dividing fractions, whose intuition is explained in why dividing by a fraction can increase the result.
The only number without a reciprocal is 0.
| You have | Do this | Reciprocal |
|---|---|---|
| A fraction, 3/4 | Swap numerator and denominator | 4/3 |
| A whole number, 5 | Write it as 5/1 and swap | 1/5 |
| A mixed number, 3 1/7 | Convert to an improper fraction and swap | 7/22 |
| A negative fraction, −4/9 | Swap the terms and keep the sign | −9/4 |
| Zero | There is nothing to do | Does not exist |
The table covers every case in this guide. Only the preparation changes; the swap of terms is always the same.
A fraction greater than 1 is flipped the same way. The reciprocal of 9/4 is 4/9, which is less than 1. In general, the reciprocal of a positive fraction less than 1 is greater than 1, and the other way around.
The first step is done by the mixed number to fraction tool. The order matters because a mixed number is a sum of a whole number and a fraction, not a single quotient.
Whole numbers and negatives: the sign stays
Every whole number is a fraction with denominator 1, so the reciprocal of n is 1/n. That is why dividing by 5 is the same as multiplying by 1/5. With negative numbers the sign stays where it is. The reciprocal of −4/9 is −9/4, because the product of two negatives is positive and (−4/9) × (−9/4) = 36/36 = 1. If you changed the sign, the product would come out as −1.
Worked example: the reciprocal of 3 1/7
You want to flip a mixed number, made of a whole number and a fraction.
- Convert to an improper fraction. 3 × 7 + 1 = 22, so 3 1/7 = 22/7.
- Swap numerator and denominator. The 22 moves to the bottom and the 7 moves to the top.
- Write the result. 7/22.
The reciprocal is 7/22.
Check: 22/7 × 7/22 = 154/154 = 1.
Multiplicative inverse and opposite
The two inverses answer different questions. The opposite, or additive inverse, is what you add to reach 0, and you get it by changing the sign. The multiplicative inverse is what you multiply by to reach 1, and you get it by swapping the terms.
| Number | Opposite (adds to 0) | Reciprocal (multiplies to 1) |
|---|---|---|
| −4/9 | 4/9 | −9/4 |
| 5 | −5 | 1/5 |
| 0 | 0 | does not exist |
Zero is its own opposite but has no reciprocal, because 0 × x is 0 for any x and never 1. For the same reason 5/0 is not a number, as explained in why the denominator cannot be zero.
The two numbers that are their own reciprocal
A number x equals its reciprocal when x = 1/x. That equation requires x ≠ 0. Then, multiplying by x, you get x² = 1, which has two solutions, 1 and −1. Any other number differs from its reciprocal. The reciprocal of 2 is 1/2, and the reciprocal of 1/2 is 2. The smaller a positive fraction is, the larger its reciprocal.
What it is for: dividing
Dividing by a fraction is the same as multiplying by its reciprocal. For 2/7 ÷ 3/4 you flip only the divisor. So 2/7 × 4/3 = 8/21. You can check it backwards, since 8/21 × 3/4 = 24/84 = 2/7, the dividend you started with. If you also flipped the first fraction, the result would be a different number, so the dividend stays as it is.
The same move appears when you solve for an unknown that has a fraction as its coefficient and when you compute powers with a negative exponent. Both cases have their own guide and calculator: fraction equations and fraction powers.
Practice
Find the reciprocal
- 7/10
- −6
- 3 1/4
See the answers
- 10/7, because 7/10 × 10/7 = 1.
- −1/6, because (−6) × (−1/6) = 1.
- 4/13, because 3 1/4 = 13/4 and 13/4 × 4/13 = 1.
Sources
- Wikipedia, “Multiplicative inverse”: defines the multiplicative inverse of a number x as the number that gives 1 when multiplied by x, and states that zero has no reciprocal (accessed October 17, 2026).