Calculate the reciprocal of a number, fraction, or mixed number and see the result in simplified fraction and decimal form with precision options.
- All Math and Numbers Calculators
- Exponent Calculator
- Rounding Calculator
- Direct Variation Calculator
Reciprocal Formula
Reciprocal(n) = 1 / n
Reciprocal(a/b) = b / a
Reciprocal(w a/b) = b / (w*b + a)
- n – any nonzero number or decimal
- a – numerator of the fraction
- b – denominator of the fraction (nonzero)
- w – whole-number part of a mixed number
The reciprocal of 0 is undefined. A number multiplied by its reciprocal always equals 1. For a mixed number, convert to an improper fraction first, then swap the numerator and denominator. Sign stays with the numerator after swapping.
Common Reciprocals at a Glance
Use these to sanity-check your answer.
| Number | Reciprocal (fraction) | Reciprocal (decimal) |
|---|---|---|
| 1 | 1/1 | 1 |
| 2 | 1/2 | 0.5 |
| 4 | 1/4 | 0.25 |
| 5 | 1/5 | 0.2 |
| 8 | 1/8 | 0.125 |
| 10 | 1/10 | 0.1 |
| 100 | 1/100 | 0.01 |
| -3 | -1/3 | -0.3333 |
How the size of the input maps to the size of the result:
| Original number | Reciprocal range |
|---|---|
| Greater than 1 | Between 0 and 1 |
| Equal to 1 or -1 | Same value (self-reciprocal) |
| Between 0 and 1 | Greater than 1 |
| Negative | Negative (same sign) |
| 0 | Undefined |
Worked Examples
Whole number: The reciprocal of 8 is 1/8 = 0.125. Check: 8 × 0.125 = 1.
Fraction: The reciprocal of 3/7 is 7/3 ≈ 2.3333. Check: (3/7) × (7/3) = 1.
Mixed number: For 2 1/4, convert to 9/4, then swap to get 4/9 ≈ 0.4444.
Decimal: The reciprocal of 0.2 is 1 / 0.2 = 5.
Negative: The reciprocal of -5 is -1/5 = -0.2. The sign carries through.
FAQ
Is the reciprocal the same as the inverse? The reciprocal is the multiplicative inverse. The additive inverse is the negative of a number, which is different.
Why is the reciprocal of 0 undefined? Division by 0 has no defined result, so 1/0 cannot produce a number.
Which numbers equal their own reciprocal? Only 1 and -1.
How do you divide by a fraction? Multiply by its reciprocal. Dividing by 2/3 is the same as multiplying by 3/2.
